AN INTRODUCTION to GEOMETRIC MEASURE THEORY and an APPLICATION to MINIMAL SURFACES ( DRAFT DOCUMENT) Academic Year 2018/19 Francesco Serra Cassano

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AN INTRODUCTION TO GEOMETRIC MEASURE THEORY AND AN APPLICATION TO MINIMAL SURFACES ( DRAFT DOCUMENT) Academic Year 2018/19 Francesco Serra Cassano

Contents I. Recalls and complements of measure theory. I.1 Measures and outer measures, approximation of measures I.2 Convergence and approximation of measurable functions: Severini- Egoroff’s theorem and Lusin’s theorem. I.3 Absolutely continuous and singular measures. Radon-Nikodym and Lebesgue decomposition theorems. I.4 Signed vector measures: Lebesgue decomposition theorem and polar decomposition for vector measures. I.5 Spaces Lp(X, µ) and their main properties. Riesz representation theorem. I.6 Operations on measures. I.7 Weak*-convergence of measures. Regularization of Radon measures on Rn. II. Differentiation of Radon measures. II.1 Covering theorems and Vitali-type covering property for measures on Rn. II.2 Derivatives of Radon measures on Rn. Lebesgue-Besicovitch differentiation theorem for Radon measures on Rn. II.3 Extensions to metric spaces. III. An introduction to Hausdorff measures, area and coarea formulas. III.1 Carathéodory’s construction and definition of Hausdorff measures on a metric space and their elementary properties; Hausdorff dimension. III.2 Recalls of some fundamental results on Lipschitz functions between Euclidean spaces and relationships with Hausdorff measures. III.3 Hausdorff measures in the Euclidean spaces; H1 and the classical notion of length in Rn; isodiametric inequality and identity Hn = Ln on Rn; k-dimensional densities. III.4 Area and coarea formulas in Rn and some applications. III.5 Extensions to metric spaces. IV. Rectifiable sets and blow-ups of Radon measures. IV.1 Rectifiable sets of Rn and their decomposition in Lipschitz images. IV.2 Approximate tangent planes to rectifiable sets. IV.3 Blow-ups of Radon measures on Rn and rectifiability. IV.4 Extensions to metric spaces. V. An introduction to minimal surfaces and sets of finite perimeter. V.1 Plateau problem: nonparametric minimal surfaces in Rn, area functional and its minimizers. V.2 Direct methods of the calculus of variations and application to the existence of minimizers for the Plateau problem. V.3 Sets of finite perimeter, space of bounded variation functions and their main properties; sets of minimal boundary. 1 2

V.4 Structure of sets of ﬁnite perimeter and reduced boundary. V.5 Regularity of minimal boundaries. V.6 Extensions to metric spaces.

SOME BASIC NOTATION

If A, B are sets then the symmetric diﬀerence between A and B will be denoted by A∆B := (A \ B) ∪ (B \ A) . We shall tipically work in a metric space X with a metric d, although we will present some notions and results in more general settings. In some chapters however we mainly deal with the Euclidean n- space Rn. Here are the basic notations used in metric spaces throughout these notes. The closed and open balls with centre x ∈ X and radius r, 0 < r < ∞, are denoted by B(x, r) = {y ∈ X : d(x, y) 6 r} , U(x, r) = {y ∈ X : d(x, y) < r} . In Rn we also set

B(r) = B(0, r),U(r) = U(0, r),S(x, r) = ∂B(x, r) and S(r) = ∂B(0, r); If B = B(x, r) (respectively B = U(x, r)) and α > 0, we denote αB = B(x, α r) (respectively αB = U(x, α r)). When α = 5 we will call 5B an enlargement of B and we will denote it by Bˆ.

The diameter of a nonempty subset A ⊂ X is d(A) = diam(A) = sup {d(x, y): x, y ∈ A} . We agree d(∅) = 0. If x ∈ X and A and B are non-empty subsets of X, the distance from x to A and the distance between A and B are, respectively, d(x, A) = inf {d(x, y): y ∈ A} , d(A, B) = inf {d(x, y): x ∈ A, y ∈ B} . For > 0 the open -neighbourhood of A is

I(A) = {x ∈ X : d(x, A) < } . If A ⊂ Rn, then |A| = Ln(A) where Ln denote the n-dimensional Lebesgue outer measure.

1. Recalls and complements of measure theory ([AFP, GZ, Mag, R1, SC]). Motivation: The main goal is to recall and complement some notions and results of measure theory such as: outer measure, measure, signed measures and vector measure with their properties and relationships; measurable functions and their properties; Lp spaces and Riesz representation theorem; convergence of measures. 3

1.1. Measures and outer measures, approximation of measures. Measures and outer measures. Let us quickly recall some important notions and results of abstract measure theory (see [GZ]). Tipically there are two approachs in abstract measure theory: one by using measure, may be more ordinary in the literature, and one by outer measure due to Carath´eodory. Firstly let us introduce the so-called set-theoretic approach where we introduce the notion of ”measure” and ”measurable set”, only assuming that the environment X is a set. Deﬁnition 1.1. Let X denote a set and P(X) denote the class of all subsets of X. (i) A set function ϕ : P(X) → [0, ∞] is called an outer measure (o.m.) on X if (OM1) ϕ(∅) = 0 ,

(OM2) ϕ(A) 6 ϕ(B) if A ⊂ B (monotonicity) , (OM3) ∞ ∞ X ϕ(∪i=1Ai) 6 ϕ(Ai) for any sequence (Ai)i ⊂ X (countable subadditivity) . i=1 (ii) A set E ⊂ X is called ϕ-measurable (with respect to an o.m. ϕ on X), if ϕ(A) = ϕ(A ∩ E) + ϕ(A \ E) ∀ A ⊂ X.

The class of ϕ-measurable sets will be denoted by Mϕ. (iii)A σ-algebra M on X is a (nonempty) class of subsets M ⊂ P(X) satisfying the two following properties: (σA1) if E ∈ M, then X \ E ∈ M ;

∞ (σA2) for each sequence (Ei)i ⊂ M, then ∪i=1 Ei ∈ M . (iv) A measure µ on X is a set function µ : M → [0, ∞], where M is a σ-algebra on X, satisfying the following two properties: (M1) µ(∅) = 0 ;

∞ ∞ X (M2) µ(∪i=1Ei) = µ(Ei) (countable additivity) i=1

for each disjoint sequence (Ei)i ⊂ M. The structure composed by the triple (X, M, µ), or also the couple (X, M), is called measure space and the sets contained in M are called measurable sets. (v) Let (X, M, µ) be a measure space. The measure µ is said to be finite, if µ(X) < ∞; it is said to be σ-finite, if there exists a sequence (Xi)i ⊂ M such ∞ that X = ∪i=1Xi and µ(Xi) < ∞ for each i. (v) Let (X, M, µ) be a measure space. A point x ∈ M is said to be an atom if the singleton {x} ∈ M and µ({x}) > 0. The set of atoms of µ will be denoted by Sµ and µ is said to be atomic if Sµ 6= ∅. If µ is finite or σ-finite the set of atoms Sµ is at most countable. 4

(vii) A function f : X → R := [−∞, ∞] is called measurable with respect to an o.m. ϕ (respectively with respect to a measure µ : M → [0, ∞]) if f −1(U) is ϕ-measurable (respectively f −1(U) ∈ M) for each open set U in R. (viii) A simple function s : X → R is one that assumes only a ﬁnite number of values. More precisely, s is a simple function if and only if it can be represented as

k X s(x) = ai χAi (x) ∀ x ∈ X, i=1 k with ai ∈ R, Ai ⊂ X (i = 1, . . . , k), X = ∪i=1Ai and Ai ∩ Aj = ∅ if i 6= j. Based on the ideas of H. Lebesgue, it is well known that a theory of an abstract integration can be carried out on a general measure space (X, M, µ) and we refer to [GZ, Chap. 6] or [R1, Chap. 1] for its complete treatment. In particular, let us recall that, given a measurable function f : X → [−∞, ∞], it is possible to make a sense to the value integral of f with respect to µ, denoted Z f dµ ∈ [−∞, ∞] . X R When the integral is finite, that is X f dµ ∈ (−∞, ∞), f is said to be integrable or also summable. Example 1.2. Let (X, M) be a measure space, then we define the following set functions on M, which turns out to be measures, as it can easily be proved. (i) (counting measure) We define the set function # : M → [0, ∞] , #(∅) := 0, #(E) as the cardinality of E if it is finite, #(E) = ∞ otherwise. (iI) (Dirac measures) With each x ∈ X we associate the set function δx : M → [0, ∞] defined by δx(E) := 1 if x ∈ E, δx(E) := 0 otherwise. If (xh)h ⊂ X P∞ and (ch)h ⊂ [0, ∞) is a sequence such that the series h=1 ch is convergent, P∞ we can define the set function h=1 ch δxh : M → [0, ∞] ∞ ! X X ch δxh (E) := ch .

h=1 {h: xh∈E} Measures of this kind are called purely atomic. Now, we enrich the environment X, by adding a topology and we require compat- ibility between topology and measure. Deﬁnition 1.3. Let (X, τ) denote a topological space and denote B(X) the σ-algebra of Borel sets of X, i.e. the smallest σ-algebra of X which contains the open and closed sets of X. (i) An o.m. ϕ on X is called a Borel o.m. if the class of ϕ-measurable sets Mϕ ⊃ B(X). (ii) An o.m. ϕ on X is called a Borel regular o.m. if it is a Borel o.m. and for each A ⊂ X there exists B ∈ B(X) with B ⊃ A and ϕ(A) = ϕ(B). (iii) An o.m. ϕ on X is called a Radon o.m. if it is a Borel regular o.m. and ϕ(K) < ∞ for each compact set K ⊂ X. 5

(iv) An o.m. ϕ on a metric space (X, d) is called a Carathéodory o.m. (or also a metric o.m.) if ϕ(A ∪ B) = ϕ(A) + ϕ(B) ∀ A, B ⊂ X with d(A, B) > 0 , where d(A, B) := inf{d(a, b): a ∈ A, b ∈ B}. (v) A measure µ : M → [0, ∞] on X is called a Borel measure if M = B(X). (vi ) A measure µ : M → [0, ∞] on X is called a Radon measure if it is a Borel measure and µ(K) < ∞ for each compact set K ⊂ X. (vii ) An o.m ϕ on X(respectively a Borel measure µ : B(X) → [0, ∞]) is said to locally finite if for each x ∈ X there exist an open neiborghood Ux of x such that ϕ(Ux) < ∞ (respectively µ(Ux) < ∞). Remark 1.4. We stress that the notion of Radon o.m. (respectively Radon measure) in a general topological space (X, τ) may actually differ in the current literature from one given in Definition 1.3 (iii) (respectively Definition 1.3 (vi)). Indeed it could be requested that ϕ (respectively µ) must satisfy to be finite on compact sets and approximation properties (i) and (ii) of Theorem 1.14. The two notions agree on a separable, locally compact metric space (X, d) because of Theorem 1.14. The following basic properties of outer measures are well known (see [GZ]). Theorem 1.5. (i) Let ϕ be an o.m. on X. Then the class of ϕ-measurable sets Mϕ is a σ-algebra on X and ϕ : Mϕ → [0, ∞] is a measure. (ii) If ϕ(N) = 0, then N ∈ Mϕ. (iii) (Carathéodory’s criterion) Let ϕ be a Carathéodory o.m. on a metric space (X, d). Then ϕ is a Borel o.m. Remark 1.6. The property of Theorem 1.5 (ii) is characteristic to an outer measure but is not enjoyed by measures. A measure µ : M → [0, ∞] with the property that all subsets of sets of µ-measure zero are measurable, is said to be complete and (X, M, µ) is called complete measure space. Not all measures are complete, but this is not a crucial defect since every measure can easily be completed by enlarging its domain of definition to include all subsets of measure zero, that is by replacing M with its completion denoted M∗ (see [R1, Theorem 1.36] or [GZ, Theorem 4.45]).

Example 1.7. (i) Let Ln denote the Lebesgue o.m. on Rn. Then Ln is a Radon n measure on R . The class Mn ≡ MLn is called the class of Lebesgue measurable sets. (ii) Let s ∈ [0, ∞) and Hs denote the s-dimensional Hausdorff measure on Rn. Then Hs is a Borel regular o.m. on Rn, but it is not a Radon o.m. unless s > n. We will deeply study these measures in Chapter III. (iii) Let A be a non Borel set of Rn (why does A exists?); let ϕ : P(Rn) → [0, +∞] be the set function defined as 0 if E ⊂ A ϕ(E) := . +∞ if E \ A 6= ∅ It is easy to see, by the definition, that ϕ is a Carathéodory outer measure on Rn. However it is not Borel regular because it does not exist a Borel set B ⊃ A such that ϕ(B) = ϕ(A). 6

(iv) Let ϕ : P(Rn) → [0, +∞] be the set function defined as 0 if E = ∅ ϕ(E) := . 1 if E 6= ∅ n n It is easy to see, by the definition, that ϕ is an o.m. on R and Mϕ = {∅, R }. In particular, it is not a Borel o. m. By Theorem 1.5 (i) we see that to every o.m. ϕ on X is associated the measure space (X, Mϕ, ϕ) Question: Given a measure space (X, M, µ) is there an associated o.m. µ∗ : P(X) → [0, ∞] such that µ∗ = µ on M ? There is a simple procedure due to Carathéodory to generate from a measure µ : M → [0, ∞] an outer measure µ∗. Moreover µ∗ is also unique (see [GZ, Theorems 4.47 and 4.48]). Theorem 1.8 (Carathéodory-Hahn extension theorem). Let (X, M, µ) be a measure space, let µ∗(E) := inf {µ(A): A ⊃ E,A ∈ M} for each E ⊂ X. Then (i) µ∗ is an o.m.; (ii) µ∗(A) = µ(A) whenever A ∈ M; (iii) M ⊂ Mµ∗ ; (iv) Let N be a σ-algebra with M ⊂ N ⊂ Mµ∗ and suppose that ν is a measure on N such that ν = µ on M. Then ν = µ∗ on N , provided that µ is σ-finite. µ∗ is called the o.m. generated by µ.

Let us recall three important results on approximation of measures by open and closed sets. The ﬁrst result is also contained in [GZ, Theorem 4.17]). Theorem 1.9 (Approximation of outer measures by open and closed sets). Let ϕ be a Borel (respectively a Borel regular) o.m. on a metric space (X, d) and let B ⊂ X be a Borel set (respectively a ϕ-measurable set). (i) Suppose that ϕ(B) < ∞, then for each > 0 there exists a closed set F ⊂ B such that ϕ(B \ F ) < . (ii) Suppose ∞ B ⊂ ∪i=1Vi where each Vi ⊂ X is an open set with ϕ(Vi) < ∞. Then there is an open set U ⊃ B such that ϕ(U \ B) < . Remark 1.10. Notice that, if ϕ(B) = ∞, then the conclusion of Theorem 1.9 (i) may fail. For instance, consider X = R, ϕ = #, B = (0, +∞). Then, for each closed set F ⊂ (0, +∞), #(B \ F ) = ∞. The conclusion of Theorem 1.9 (ii) may also fail by means of the same example if the assumptions are dropped. The ﬁrst important consequence of Theorem 1.9 is the following 7

Corollary 1.11. Let ϕ be a Borel (respectively a Borel regular) o.m. on a metric space (X, d). Suppose there exists a sequence of open sets (Vi)i ⊂ X such that ∞ (?) X = ∪i=1Vi with ϕ(Vi) < ∞ ∀ i .

Then for each B ∈ B(X) (respectively B ∈ Mϕ) (i) ϕ(B) = inf{ϕ(U): U ⊃ B,U open}; (ii) ϕ(B) = sup{ϕ(C): C ⊂ B,C closed}.

Proof. See, for instance, [SC, Corollary 1.19]. The second important consequence of Corollary 1.11 and the Carath´eodory-Hahn extension theorem (Theorem 1.8) is the following approximation result for Borel measures. Corollary 1.12 (Approximation of Borel measures by open and closed sets). Con- sider a measure space (X, B(X), µ) where X is a metric space and µ is a Borel measure. Suppose that the assumption (?) of Corollary 1.11 holds replacing ϕ with µ. Then for each B ∈ B(X) (i) µ(B) = inf{µ(U): U ⊃ B,U open}; (ii) µ(B) = sup{µ(C): C ⊂ B,C closed}.

Proof. See, for instance, [SC, Corollary 1.20]. When (X, d) is a separable, locally compact metric space and ϕ (respectively µ) is a Radon outer measure (respectively Radon measure) on X, the assumption (?) is satisﬁed. Thus the conclusions of Corollary 1.11 (respectively Corollary 1.12) hold. Moreover the approximation from below by means of closed sets can be replaced by compact sets. Let us recall Deﬁnition 1.13. Let (X, τ) be a topological space. (i)( X, τ) is said to be separable if it has a countable dense subset. (ii)( X, τ) is said to be locally compact if for each x ∈ X there is an open set O 3 x such that the closure of O, denoted by O, is compact . Theorem 1.14 (Approximation of Radon measures on l.c.s. metric spaces). Let (X, d) be a separable, locally compact metric space and ϕ (respectively µ) be a Radon outer measure (respectively Radon measure) on X. Then (i) for each B ⊂ X, ϕ(B) = inf{ϕ(U): U ⊃ B,U open} (respectively, for each B ∈ B(X), µ(B) = inf{µ(U): U ⊃ B,U open}); (ii) for each B ∈ Mϕ, ϕ(B) = sup{ϕ(K): K ⊂ B,K compact} (respectively, for each B ∈ B(X), µ(B) = sup{µ(K): K ⊂ B,K compact}). Remark 1.15. An immediate consequence of Theorem 1.14 is that, if two Radon measures on a locally compact metric space (X, d) agree on the class open set, then they have to agree on P(X). Before the proof of Theorem 1.14 we need the following topological results, whose the former is well known (see, for instance, [Ro, Proposition 7.6]). 8

Lemma 1.16. Let (X, d) be a separable metric space and let D = {xi : i ∈ N} ⊂ X be dense. Then the family of open sets

U := {U(xi, q): i ∈ N, q ∈ Q ∩ (0, ∞)} is a basis for the topology induced on X by the distance, where U(x, r) := {y ∈ X : d(x, y) < r} if x ∈ X and r > 0. Lemma 1.17. Let (X, d) be a separable, locally compact metric space. Then there exists an increasing sequence of open sets (Vi)i such that ∞ (1.1) X = ∪i=1Vi, Vi is compact for each i . Proof of Lemma 1.17. Recall that, by deﬁnition, (X, d) is locally compact if and only if ∀ x ∈ X ∃ rx > 0 such that U(x, rx) is compact. Let D = {xi : i ∈ N} ⊂ X be dense. From Lemma 1.16, the family U is a basis for the topology and let us enumerate U, that is assume that U = {Ui : i ∈ N}.Thus there exists a set I(x) ⊂ N for which U(x, rx) = ∪i∈I(x)Ui . In particular, there exists a choice function α : X → N satisfying:

(1.2) x ∈ Uα(x) and Uα(x) ⊂ U(x, rx) ⊂ U(x, rx) . Let J := α(X) ⊂ N and

Vi := ∪j∈(J∩{1,...,i})Uj if i ∈ N . Then, by (1.2), (1.1) follows. Proof of Theorem 1.14. Let us ﬁrst notice that, without loss of generality, we can assume that B ∈ B(X). Indeed, if not, since ϕ is a Borel regular o.m., we can replace B by a Borel set B˜ ⊃ B and ϕ(B˜) = ϕ(B). By Lemma 1.17, (?) of Corollary 1.11 is satisﬁed. Thus claim (i) follows at once from Corollary 1.11 (i) (respectively from Corollary 1.12 (i)). Let us now prove (ii) for a given ϕ Radon o.m. and B ∈ Mϕ. Since each compact set is also closed, from Corollary 1.11 (ii), it follows that ϕ(B) = sup{ϕ(C): C ⊂ B,C closed} > sup{ϕ(K): K ⊂ B,K compact} . Thus we have only to prove that (1.3) ϕ(B) = sup{ϕ(C): C ⊂ B,C closed} 6 sup{ϕ(K): K ⊂ B,K compact}

Let C ⊂ B be a closed set, let (Vi)i be the sequence of open sets in (1.1) and let i Ki := C ∩ (∪j=1Vj) .

Then (Ki)i is an increasing sequence of compact sets such that ∞ C := ∪i=1Ki . Observe now that

(1.4) ϕ(Ki) 6 sup{ϕ(K): K ⊂ B,K compact} for each i , and, by the continuity of o.m. ϕ on increasing sequences of measurable sets,

(1.5) lim ϕ(Ki) = ϕ(C) . i→∞ By (1.4) and (1.5), (1.3) follows. 9

1.2. Convergence and approximation of measurable functions: Severini- Egoroff’s and Lusin’s theorems. Theorem 1.18 (Severini-Egoroff). Let (X, M, µ) be a measure space with µ finite. Suppose fh : X → R (h = 1, 2,... ) and f : X → R are measurable functions that are finite µ-a.e.on X. Also, suppose that (fh)h converges pointwise µ-a.e. to f. Then for each > 0 there exists a set A ∈ M such that µ(X \ A) < and fh → f uniformly on A, that is

sup |fh(x) − f(x)| → 0 as h → ∞ . x∈A .

Proof. See [GZ, Theorem 5.15]. Remark 1.19. The hypothesis that µ(X) < ∞ is essential in Severini-Egoroﬀ’s theorem. Consider the case of Lebesgue measure L1 on R and deﬁne a sequence of functions by

fh = χ[h,∞) , for each positive integer h. Then, limh→∞ fh(x) = 0 for each x ∈ R, but (fh)h does not converge uniformly to 0 on any set A whose complement R\A has ﬁnite Lebesgue measure. Indeed, it would follow that R \ A does not contain any half-line [h, ∞); that is, for each h, there would exist x ∈ [h, ∞) ∩ A withfh(x) = 1, thus showing that (fh)h does not converge uniformly to 0 on A. Theorem 1.20 (Approximation by simple functions). Let (X, M) be a measure space and let f : X → [0, +∞] be a measurable function. Then there exists a sequence of measurable simple functions sh : X → [0, +∞) (h = 1, 2,... ) satisfying the properties:

(i)0 6 s1 6 s2 6 ... 6 sh 6 ... 6 f; (ii) limh→∞ sh(x) = f(x) ∀ x ∈ X. R In particular, if X f dµ < ∞, then Z |f − sh| dµ → 0 . X Proof. See [GZ, Theorem 5.25]. Let us now introduce two spaces of continuous functions which play an important role in measure theory. Deﬁnition 1.21. Let (X, τ) be a topological space. (i) 0 Cc (X) := {f : X → R : f is continuous and spt(f) is compact in (X, τ)} where (1.6) spt(f) := closure {x ∈ X : f(x) 6= 0} . (ii) 0 Cb (X) := {f : X → R : f is continuous and bounded} 10

Remark 1.22. Observe that, if a topological space (X, τ) is not locally compact, the 0 0 space Cc (X) could turn out to be meaningless, that is Cc (X) = {0}.Indeed Exercise: Let (X, k · k) be an infinite-dimensional normed vector space. Then 0 Cc (X) = {0}. Lusin’s theorem 1.23 (1912, form on locally compact metric spaces). Let µ be a Radon outer measure on a locally compact, separable metric space X . Let f : X → R be a µ-measurable function such that there exists a Borel set A ⊂ X with µ(A) < ∞, f(x) = 0 ∀ x ∈ X \ A and |f(x)| < ∞ µ − a.e. x ∈ X. 0 Then, for each > 0, there exists g ∈ Cc (X) such that µ ({x ∈ X : f(x) 6= g(x)}) < . Moreover g can be chosen such that sup |g(x)| 6 sup |f(x)| . x∈X x∈X Proof. See [R1, Theorem 2.23]. A consequence of Lusin’s theorem is the following useful approximation of measurable functions by means of Borel functions. Corollary 1.24. Let µ be a Radon outer measure on a locally compact, separable metric space X and let f : X → R be a µ-measurable function. Then there exist a Borel function g : X → R such that f = g µ-a.e. on X. Proof. See, for instance, [Fe, 2.3.6]. 1.3. Absolutely continuous and singular measures. Radon-Nikodym and Lebesgue decomposition theorems. Firstly, let us introduce some definitions and preliminary results. Definition 1.25. Let (X, M) be a measure space and let µ , ν : M → [0, ∞] be two measures. (i) The measure ν is said to be absolutely continuous with respect to the measure µ, written ν << µ, if it holds that µ(E) = 0 ⇒ ν(E) = 0 . (ii) The measures ν and µ are said to be mutually singular, written µ ⊥ ν, if there exists a measurable set E such that ν(E) = µ(X \ E) = 0 . The following result justifies why the word ”continuity” is used in this context. Theorem 1.26. Let ν be a finite measure and µ a measure on a measure space (X, M). Then the following are equivalent: (i) ν << µ; (ii) limµ(A)→0 ν(A) = 0, that is, for every ε > 0 ∃ δ = δ(ε) > 0 such that ν(E) < ε whenever µ(E) < δ.

Proof. See [GZ, Theorem 6.33]. 11

Let us now introduce two fundamental results of measure theory: the Radon- Nikodym and Lebesgue’s decomposition theorems. Let (X, M, µ) be a measure space and w : X → [0, ∞] be measurable. Then it is easy to see that µw is a measure, absolutely continuous w.r.t. µ. The very remarkable fact, content of the Radon-Nikodym theorem, is that essentially each measure ν, absolutely continuous w.r.t. µ is of this form. Theorem 1.27 (Radon-Nikodym). Let ν and µ be two measures on (X, M). Suppose that

(i) ν and µ are σ-ﬁnite, that is, there exists a sequence (Xi)i ⊂ M such that ∞ X = ∪i=1Xi and

ν(Xi) < ∞ and µ(Xi) < ∞ for each i . (ii) ν << µ. Then there exists a measurable function w : X → [0, ∞] such that ν = µw on M , that is, Z (RN) ν(E) = µw(E) := w dµ ∀E ∈ M . E Moreover the function w in (RN) is µ-a.e. unique. Definition 1.28. The function w in (RN) is called the Radon-Nikodym derivative of dν ν with respect to µ and denoted by w = . dµ Remark 1.29. Because ν is σ-finite, then w is also σ-integrable with respect to µ, that is, Z (σI) 0 6 w dµ < ∞ ∀ i , Xi where (Xi)i is the sequence in statement (i). Proof. See See [GZ, Theorem 6.38] and also [SC, Theorem 1.30]. Remark 1.30. We can actually weaken the assumptions of the Radon-Nikodym theorem. Indeed it is sufficient to require that only µ is σ-finite in order that (RN) holds (see, for instance,[Ro, Theorem 23, Chap. 11]). When µ is not σ-finite, the Radon- Nikodym theorem fails (see Exercise I.8). For Radon measures on locally compact, separable metric spaces, the Radon- Nikodym theorem has the following simpler and stronger version. Theorem 1.31 (Radon-Nikodym’s theorem for Radon measures). If X is supposed to be a locally compact, separable metric space and ν and µ are Radon measures on X dν with ν << µ, then (RN) holds and the Radon-Nikodym derivative w := dµ is locally 1 integrable on X, i.e. w ∈ Lloc(X, µ), where Z 1 Lloc(X, µ) := f : X → R : f is measurable, |f| dµ < ∞ for each compact K ⊂ X . K Proof. See [SC, Theorem 1.35]. 12

Historical notes: The ﬁrst version of Radon-Nikodym’s theorem is due to H. Lebesgue ([Le]) and to G. Vitali ([Vitali]) when X = R in 1904. Radon extended the result when X = Rn ([Ra]) in 1913. Eventually Nikodym ([Ni]) extended the result to the abstract setting in 1930. A consequence of the Radon-Nikodym theorem is the following Lebesgue decomposition theorem 1.32. Let ν and µ be σ-ﬁnite measures on a measure space (X, M). Then there is a decomposition of ν such that

ν = νac + νs , where νac and νs are still measures on (X, M) with νac << µ and νs ⊥ µ. The decomposition is unique.

Proof. See [GZ, Theorem 6.39] and also [SC, Theorem 1.36]. 1 Exercise I.10 Let consider µ = L , ν = δ0 as measures on the σ- algebra M1 of Lebesgue measurable sets in R,where δ0 denotes the Dirac measure at 0, that is, 1 if 0 ∈ E δ (E) := . Prove that the Lebesgue decomposition of ν with respect 0 0 if 0 ∈/ E. to µ, ν = νac + νs, is given by νac ≡ 0 and νs = ν.

1.4. Signed vector measures. Lebesgue decomposition theorem still holds for a more general class of measures. Namely for set functions ν : M → R which still verify basic properties of countable additivity. Definition 1.33 (Signed measures). Let (X, M) be a measure space. (i) An extended real valued set function ν : M → R is a signed measure if it satisfies the following three properties: (SM1) ν assumes at most one of the values +∞, −∞; (SM2) ν(∅) = 0; (SM3) For each sequence of disjoint sets (Ei)i ⊂ M, it holds that ∞ ∞ X ν(∪i=1Ei) = ν(Ei) i=1 where the series on the right either converges absolutely or diverges to −∞ or +∞. (ii) A signed measure ν : M → R is said to be absolutely continuous with respect to µ : M → [0, ∞] , written ν << µ, if ν(E) = 0 whenever µ(E) = 0. (iii) Two signed measure ν, µ : M → R are said to be mutually singular , written ν ⊥ µ, if there is E ∈ M such that ν(E) = µ(X \ E) = 0. (iv) A signed measure ν : M → R is said to be finite (respectively σ-finite) if |ν(X)| < ∞ (respectively there exists a sequence (Xi)i ⊂ M such that ∞ X = ∪i=1Xi and |ν(Xi)| < ∞ for each i). (v) A real valued signed measure ν : M → R, that is, if ν(E) ∈ R for each E ∈ M, is called a real measure. 13

Example 1.34 (Examples of signed measures). Let us introduce below two remarkable examples of signed measures on a given measure space (X, M). (i) Let µ : M → [0, ∞] be a measure and let f : X → R be a measurable function. Suppose at least one of f + := f ∨ 0 or f − := (−f) ∨ 0 is integrable, and let ν : M → R denote the extended real-valued function on M defined by Z ν(E) := f dµ ∀E ∈ M . E Then is easy to see that ν is a signed measure and ν << µ. If both f + and f − are integrable, or, equivalently, |f| is integrable, then ν is a real measure. (ii) Let µ1, µ2 : M → [0, ∞] be measures and assume that al least one of them is finite. Let ν : M → R denote the extended real-valued function on M defined by ν(E) := µ1(E) − µ2(E) ∀E ∈ M .

Then is easy to see that ν is a signed measure. If both µ1 and µ2 are finite, then ν is a real measure. Remark 1.35. Observe that a measure is a signed measure. In some contexts we will emphasize that a measure µ is not a signed measure by saying that it is a positive measure. Notice also that a signed (or also real) measure ν is not an increasing set function. Exercise: A signed measure ν is a real measure if and only if it is finite, that is |ν(X)| < ∞. Theorem 1.36 (Lebesgue decomposition theorem for signed measures). Let (X, M, µ) be a measure space with µ σ-finite, and ν : M → R be a σ-finite signed measure . Then there are two signed measures νac, νs : M → R such that

(LD) νac << µ, νs ⊥ µ, ν = νac + νs , and there exists a measurable function w : X → R such that either w+ or w− is integrable with respect to µ such that Z (RN) νac(E) = w dµ ∀ E ∈ M . E Moreover both decomposition (LD) and representation (RN) are unique. Proof. See [F, Theorem 3.8].

Remark 1.37. Notice that the sum of signed measures νac + νs is well defined in (LD) since νac and νs are mutually singular. Remark 1.38. Suppose ν is a real measure (observe that ν is also σ-finite), that is ν : M → R, and ν << µ with µ a given σ-finite positive measure on M. Applying Theorem 1.36, we have that Z ν(E) = νac(E) = w dµ ∀ E ∈ M , E and w : X → R is now an integrable function on X with respect to µ, that is R X |w| dµ < ∞. Indeed, since ν(E) ∈ R for each E ∈ M, it is easy to see that w must be integrable. 14

We recommend [GZ, Section 6.5] and [F, Chap. 6] for a complete treatment concerning signed measures. However we point out that signed measures in Example 1.34 are really the only examples: every signed measure can be represented in either of these two forms. An other important tool in GMT will turn out to be the notion of signed vector measure, which is an extension of the one of signed measure. Definition 1.39 (Vector signed measures). Let (X, M) be a measure space. m (i) A vector set function ν = (ν(1), . . . , ν(m)): M → R is a vector signed measure if its components ν(i) : M → R (i = 1, . . . , m) are signed measures (according to Definition 1.33). m (ii) A vector signed measure ν : M → R is a vector measure if it is Rm-valued vector measure, that is ν : M → Rm. (iii) If ν is a signed vector measure, we define its total variation |ν| : M → [0, ∞] as follows: ( ∞ ) X ∞ |ν|(E) = sup |ν(Eh)| :(Eh)h ⊂ M pairwise disjoint,E = ∪h=1Eh , h=1 where m |v|Rm if v ∈ R |v| := m m . ∞ if v ∈ R \ R (iv) If ν is a real measure, that is ν : M → R, we define its positive and negative parts respectively as follows: |ν| + ν |ν| − ν ν+ = and ν− = 2 2 Notation: In the following, we will say countable partition of a set E a pairwise ∞ disjoint sequence of sets (Eh)h such that ∪h=1Eh = E. m Remark 1.40. Observe that, according to Definition 1.33 (i), a R -valued signed vector measure ν = (ν(1), . . . , ν(m)) satisfies the following two properties: m (SVM1) ν(∅) = 0 := (0,..., 0) ∈ R ; (SVM2) For each sequence of disjoint sets (Eh)h ⊂ M, it holds that

∞ ∞ ∞ ! ∞ X X (1) X (m) ν(∪h=1Eh) = ν(Eh) := ν (Eh),..., ν (Eh) , h=1 h=1 h=1 P∞ (i) where the series on the right-hand side h=1 ν (Eh)(i = 1, . . . , m) either converges absolutely or diverges to −∞ or +∞. Notice also that, if ν is Rm-valued vector measure, then the absolute convergence in the series in (SVM2) is a requirement on the set function ν: in fact the sum of the series cannot depend on the order of its terms, as the union does not. Observe also that, when m = 1, the notion of signed vector measure (respectively vector measure) agrees with the one of signed measure (respectively real measure). Eventually notice m m that a R -valued set function ν = (ν1, . . . , νm): M → R is a vector measure if and only if νi : M → R (i = 1, . . . , m) is a real measure. 15

Remark 1.41. The introduction of the notion of total variation solves the problem of finding a positive measure µ which dominates a given signed vector measure ν on M in the sense that |ν(E)| 6 µ(E) for each E ∈ M, looking for keeping µ as small as we can. Every solution to this problem (if there is one at all) must satisfy ∞ ∞ X X µ(E) = µ(Eh) > |ν(Eh)| h=1 h=1 for each partition (Eh)h of any set E ∈ M, so that µ(E) is at least equal to quantity |ν|(E). This suggest the reason of total variation’s definition like in Definition 1.39 (iii). Let us show that the total variation of a signed vector measure (respectively vector measure) is a positive measure (respectively positive finite measure). Theorem 1.42. (i) Let ν be a signed vector measure on (X, M). Then its total variation |ν| is a positive measure. (ii) If ν is a vector measure, then |ν| is a positive finite measure, that is |ν|(X) < ∞. Proof. (i) We have to prove that (1.7) |ν|(∅) = 0 and (1.8) |ν| is countable additive . It is trivial, by definition, that (1.7) holds. Let us show (1.8). Let us firstly observe that |ν| : M → [0, ∞] is increasing , that is (1.9) |ν|(E) 6 |ν|(F ) if E ⊂ F.

Indeed let (Eh)h ⊂ M be a partition of E, then the family of sets {Eh : h} ∪ {F \ E} is a countable partition of F . Thus ∞ ∞ X X |ν(Eh)| 6 |ν(Eh)| + |ν(F \ E)| 6 |ν|(F ) . h=1 h=1 Then, taking the supremum on the partitions of E in the previous inequality, we get (1.9). Let us now show that (1.10) |ν| is countably subadditive and (1.11) |ν| is additive .

Observe that, from (1.9), (1.10) and (1.11), (1.8) follows. Indeed let (Eh)h ⊂ M be pairwise disjoint. Then by countable subadditivity (1.10), ∞ ∞ X (1.12) |ν|(∪h=1Eh) 6 |ν|(Eh) . h=1 On the other hand, by (1.9) and (1.11), 16

m ∞ m X (1.13) |ν|(∪h=1Eh) > |ν|(∪h=1Eh) = |ν|(Eh) ∀m ∈ N . h=1

Thus, by (1.12) and (1.14), (1.8) follows. Let us now prove (1.10). Let E,(Eh)h be in ∞ 0 M such that E ⊂ ∪h=1Eh. Let us deﬁne a pairwise disjoint sequence (Eh)h ⊂ M in 0 0 h−1 the following way E0 := E0 and Eh := Eh \ ∪i=1 Ei if h > 1. Let (Fi)i be a partition 0 of E, since (Eh ∩ Fi)h is a partition of Fi for ﬁxed i, by countable additivity (SVM2) and (1.9), we can infer

∞ ∞ ∞ ∞ ∞ X X X 0 X X 0 |ν(Fi)| = ν(Eh ∩ Fi) 6 |ν(Eh ∩ Fi)| i=1 i=1 h=1 i=1 h=1 ∞ ∞ ∞ ∞ X X 0 X 0 X = |ν(Eh ∩ Fi)| 6 |ν|(Eh ∩ E) 6 |ν|(Eh) . h=1 i=1 h=1 h=1 Taking the supremum on the partitions of E in the previous inequality, it follows that ∞ X |ν|(E) 6 |ν|(Eh) h=1 and (1.10) follows. Let us now prove (1.11). Let E, F ∈ M be disjoint. If at least one between |ν|(E) and|ν|(F ) is ∞, then, by (1.9), it is immediate that |ν|(E ∪ F ) = ∞ = |ν|(E) + |ν|(F ) . Thus, without loss of generality, we can assume that both |ν|(E) and|ν|(F ) are ﬁnite. By deﬁnition, for each > 0, there exist a partition (Eh)h ⊂ M of E and one (Fh)h ⊂ M of F such that

∞ ∞ X X |ν|(E) 6 |ν|(Eh) + , |ν|(F ) 6 |ν|(Fh) + . h=1 h=1

Observe now that the family of sets {Eh : h ∈ N} ∪ {Fh : h ∈ N} is a countable partition of E ∪ F . Then, by the previous inequality, it follows that, for each > 0, ∞ ∞ X X |ν|(E) + |ν|(F ) − 2 6 |ν|(Eh) + |ν|(Fh) 6 |ν|(E ∪ F ) . h=1 h=1 By countable subadditivity (1.10) and the previous inequality, (1.11) follows. (ii) It is suﬃcient to assume that ν is a real measure, that is m = 1 and ν : M → R. The Rm-valued case being an easy consequence of the following estimate m X |ν|(E) 6 |νi|(E) ∀E ∈ M , i=1 if ν = (ν1, . . . , νm). Suppose that for some E ∈ M has |ν|(E) = ∞. Let us then prove there exist two disjoint sets A, B ∈ M such that (1.14) E = A ∪ B, |ν(A)| > 1, |ν(B)| > 1, either |ν|(A) = ∞ or |ν|(B) = ∞. 17

By deﬁnition, there is a partition (Eh)h of E such that m X (1.15) |ν(Eh)| > 2(|ν(E)| + 1) . h=1

Let I := {1 6 h 6 m : ν(Eh) > 0} and J := {1 6 h 6 m : ν(Eh) < 0}. Since, by the additivity

m X X X |ν(Eh)| = ν(Eh) − ν(Eh) = ν(∪h∈I Eh) − ν(∪h∈J Eh) , h=1 h∈I h∈J by (1.15), we can infer that either |ν(∪h∈I Eh)| = ν(∪h∈I Eh) > (|ν(E)| + 1) or |ν(∪h∈J Eh)| = −ν(∪h∈J Eh) > (|ν(E)| + 1). Let A denote one between sets ∪h∈I Eh and ∪h∈J Eh such that |ν(A)| > (|ν(E)| + 1) and let B := E \ A. Then

|ν(B)| = |ν(E) − ν(A)| > |ν(A)| − |ν(E)| > 1 . By the additivity of |ν|, it is clear that either |ν|(A) = ∞ or |ν|(B) = ∞. Therefore (1.14) follows. Now if |ν|(X) = ∞, then we can apply (1.14) with E = X and split X into two sets A1 and B1 with |ν(A1)| > 1 and |ν|(B1) = ∞. Split B1 into two sets A2 and B2 with |ν(A2)| > 1 and |ν|(B2) = ∞. Continuing in this way, we get a countably inﬁnite disjoint family of sets (Ah)h with |ν(Ah)| > 1 for each h. The countable additivity of ν implies that

∞ ∞ X ν(∪h=1Ah) = ν(Ah) . h=1

But this series cannot converge since ν(Ah) does not tend to 0 as h → ∞. This contradiction shows that |ν|(X) < ∞. Remark 1.43. The above theorem shows that for any real measure ν, its positive and negative part are positive ﬁnite measures, hence the decomposition ν = ν+ − ν− holds; it is known as the Jordan decomposition of ν. We point out that a Jordan decomposition still hold for signed measures, by means of a suitable notion of positive and negative parts for a signed measure (see [GZ, Theorem 6.31] or [F, Theorem 3.4]).

Corollary 1.44. Let ν : M → R be a signed measure. Then ν is σ- ﬁnite if and only if so does its total variation |ν| : M → [0, ∞]. Proof. If |ν| is σ-ﬁnite, since

|ν(E)| 6 |ν|(E) ∀E ∈ M , according to Definition 1.33 (iv), ν is also σ-finite. Suppose that ν is σ- finite, that is there is a disjoint sequence (Xk)k ⊂ M such that |ν(Xk)| < ∞ for each k. For given k, let us define the set function νk : M → R defined by νk(E) := ν(E ∩ Xk). Notice that νk is a real measure. Thus, from Theorem 1.42 (ii), its total variation |νk| is a positive finite measure, that is |νk|(X) < ∞. Let us now prove that

(1.16) |νk|(X) = |ν|(Xk) 18 from which it will follow that ν is σ-ﬁnite and the proof is accomplished. By deﬁnition ( ∞ ) X |νk|(X) = sup |ν(Eh ∩ Xk)| :(Eh)h ⊂ M partition of X (1.17) h=1 ( ∞ ) X 6 sup |ν(Fh| :(Fh)h ⊂ M partition of Xk = |ν|(Xk) . h=1

Let (Fh)h ⊂ M be a partition of Xk and define the partition of X as E1 := X \ Xk, Eh := Fh−1 if h > 2.Then it trivial that ∞ ∞ ∞ X X X (1.18) |ν(Fh)| = |ν(Eh ∩ Xk)| = |νk(Eh)| 6 |νk|(X) . h=1 h=1 h=1 Therefore, by (1.17) and (1.18), (1.16) follows. Remark 1.45. It is immediate to check that Rm-valued vector measures can be added and multiplied by real numbers, hence they form a real vector space; moreover, an easy consequence of Theorem 1.42 is that the total variation is a norm on the space of measures, which turns out to be a Banach space . If X is a locally compact separable metric space, it will be identified with the dual of a space of continuous functions and this will give the completeness in another way (see Corollary 1.78 and Theorem 1.83). Example 1.46. According to the notation for positive measures (see (RN)), given a m measure space (X, M, µ) and a vector function w = (w1, . . . , wm): X → R , with each wi : X → R (i = 1, . . . , m) measurable functions such that either wi,+ or wi,− is m integrable. Let us define the vector set function µw : M → R defined as follows Z Z Z (1.19) µw(E) = w dµ := w1 dµ, . . . , wm dµ E ∈ M . E E E

Then it is easy to see that µw is a signed vector measure and its total variation is computed in the following proposition.

Proposition 1.47. Let (X, M, µ) be a measure space and let w = (w1, . . . , wm): m X → R , with each wi : X → R (i = 1, . . . , m) measurable functions such that either wi,+ or wi,− is integrable. Consider the vector signed measure µw in (1.19). Then Z (1.20) |µw|(E) = |w| dµ ∀E ∈ M . E Proof. It is easy to see that, by deﬁnition of total variation for a vector measure, Z |µw|(E) 6 |w| dµ ∀E ∈ M . E

Let us prove the reverse inequality. Let E ∈ M. If |µw|(E) = ∞ we are done. Then suppose |µw|(E) < ∞. Note that, from this assumption, we have that each wi R (i = 1, . . . , m) is integrable on E, that is E |wi| dµ < ∞. Without loss of generality, we can assume that each wi is a real-valued function on E and then we can consider m w = (w1, . . . , wm): E → R . 19

m−1 m Let D = {zh : h ∈ N} be a dense set in the unit sphere S := {y ∈ R : |y| = 1}. For any ∈ (0, 1) let us define σ : E → N σ(x) := min {h ∈ N : hw(x), zhi > (1 − )|w(x)|} x ∈ E, −1 and let Eh := σ (h). Then (Eh)h ⊂ M is a pairwise disjoint sequence and E = ∞ ∪h=0Eh. Therefore ∞ Z X Z (1 − ) |w| dµ = (1 − )|w| dµ 6 E h=0 Eh ∞ ∞ X Z X Z |hw(x), zhi| dµ 6 |w(x)| dµ 6 |µw|(E) ∀ ∈ (0, 1) . h=0 Eh h=0 Eh Thus, getting → 0 in the previous inequality, the proof is accomplished. Definition 1.48 (Integrals). Let (X, M) be a measure space. (i) Let ν : M → R be a real measure. If u : X → R is a |ν|-measurable function, we say that u is ν-integrable if u is |ν|-integrable and we set Z Z Z u dν := u dν+ − u dν− . X X X k If u = (u1, . . . , uk): X → R is a |ν|-measurable vector function,we say that u is ν-integrable if each its component ui (i = 1, . . . , k) is |ν|-integrable and we set Z Z Z u dν := u1 dν, . . . , uk dν . X X X m (ii) Let ν = (ν1, . . . , νm): M → R be a vector measure. If u : X → R is a |ν|-measurable function, we say that u is ν-integrable if u is |ν|-integrable and we set Z Z Z u dν := u dν1,..., u dνm . X X X (iii) Let E ∈ M, the integral of of a function u on E is defined by Z Z u dν := u χE dν , E X provided that the right-hand side makes sense. Remark 1.49. Notice that an immediate consequence of the above definition is the inequality Z Z

u dν 6 |u| d|ν| X X which holds for every extended real or vector valued summable function u and for every positive, real or vector measure ν. Deﬁnition 1.50 (Absolute continuity and singularity for signed vector measures). Let (X, M) be a measure space. 20

(i) Let µ be a positive measure and ν be a vector signed measure on the measure space (X, M). We say that ν is absolutely continuous with respect to µ, and write ν << µ, if |ν| << µ, as positive measures, that is, for every E ∈ M the following implication holds: µ(E) = 0 ⇒ |ν|(E) = 0 . m (ii) If µ or ν are signed R -valued signed measures on measure space (X, M), we say that they are mutually singular, and write µ ⊥ ν, if |µ| and |ν| are mutually singular, as positive measures, that is, there exists E ∈ M such that |µ|(E) = |ν|(X \ E) = 0. Remark 1.51. Observe that, given a positive measure µ on a measure space (X, M), then vector measure µw defined in (1.19) is trivially absolutely continuous with respect to µ by Proposition 1.47. Lebesgue decomposition theorem for vector signed measures 1.52. Let ν m and µ be respectively a R -valued σ-finite measure and a σ-finite positive measure on a measure space (X, M). Then there is a decomposition of ν such that

(1.21) ν = νac + νs , m where νac and νs are still R -valued signed measures on (X, M) with νac << µ and νs ⊥ µ. The decomposition is unique. Moreover there exists a unique vector function m w = (w1, . . . , wm): X → R with either wi,+ or wi,− (i = 1, . . . , m) integrable functions w.r.t. µ such that Z (1.22) νac(E) = µw(E) = w dµ ∀E ∈ M . E m Proof. Let ν = (ν1, . . . , νm): M → R . By deﬁnition, for each i = 1, . . . , m, νi : M → R is a σ-ﬁnite signed vector measure. By Theorem 1.36, there are two signed measures νi,ac, νi,s : M → R such that

(1.23) νi,ac << µ, νi,s ⊥ µ, νi = νi,ac + νi,s , and there exists a measurable function wi : X → R such that either wi,+ or wi,− is integrable with respect to µ such that Z (1.24) νi,ac(E) = wi dµ ∀ E ∈ M . E Moreover both decomposition (1.23) and representation (1.24) are unique. m Deﬁne the signed vector measures νac, νs : M → R

νac := (ν1,ac, . . . , νm,ac) and νs := (ν1,s, . . . , νm,s) m and the vector function w := (w1, . . . , wm): X → R . Then it is trivial to see that (1.21) and (1.22) now hold for the signed vector measure ν. Therefore the proof is accomplished. Each signed vector measure ν is trivially absolutely continuous with respect to its total variation |ν|. The following useful decomposition for vector measures immediately follows from the Lebesgue decomposition theorem for signed vector measures 1.52, Proposition 1.47 and Remark 1.38. 21

Corollary 1.53 (Polar decomposition for vector measures). Let ν be a Rm -valued measure on the measure space (X, M). Then there exists a unique measurable vector m function wν : X → R with |wν(x)| = 1 |ν| a.e. x ∈ X such that ν = |ν|wν , that is Z ν(E) = wν d|ν| ∀E ∈ M . E . Proof. Let us ﬁrst notice that, by Theorem 1.42 (ii), both ν and |ν| are ﬁnite and ν << |ν|. By Theorem 1.52 and Remark 1.38 there exists a vector function wν : m X → R such that Z (1.25) ν(E) = wν d |ν| ∀E ∈ M . E By (1.25) and Proposition 1.47 , we can infer that Z |ν|(E) = |wν| d |ν| ∀E ∈ M . E

Since |ν| is ﬁnite, we have that |wν(x)| = 1 |ν|-a.e. x ∈ X and the proof is accomplished. 1.5. Spaces Lp(X, µ) and their main properties. Riesz representation theorem. Completeness and dual space of Lp(X, µ) In this subsection we will only request that (X, M, µ) is a measure space. Let us introducel the space of p-integrable functions with respect to measure µ. Deﬁnition 1.54. Let p ∈ [1, ∞], p L (X, µ) := f : X → R : f is measurable and kfkLp < +∞ where  Z 1/p  |f(x)|p dµ(x) if 1 p < ∞ kfk p = kfk p := 6 . L L (X,µ) X  inf {M > 0 : |f(x)| 6 M µ − a.e. x ∈ X} if p = ∞ p The quantity kfkLp is called the L norm of f on measure space (X, M, µ). When n n X = Ω is an open subset of R , µ = L , M = Ω ∩ Mn, where Mn denotes the class of n-dimensional Lebesgue measurable sets of Rn and d the Euclidean distance, we will simply denote Lp(X, µ) as Lp(Ω). Remark 1.55. When dealing with measure- theoretic or functional-analytic prope- nies of functions and Lp spaces, it is often convenient to consider functions that agree a.e. as identical, thinking of the elements of Lp spaces as equivalence classes; in particular, this makes k · kLp a norm. We shall follow this path whenever our statements will depend only on the equivalence class without further mention, provided that this is clear from the context. Let us recall the following fundamental result concerning the completeness of Lp (see [GZ, Theorem 6.24]). 22

p Theorem 1.56 (Fisher-Riesz,1907). (L (X, µ), k · kLp ) is a B.s. if 1 6 p 6 ∞. Moreover L2(X, µ) turns out to be a Hilbert space with respect to the scalar product Z 2 (f, g)L2 := f g dµ f, g ∈ L (X, µ) . X As a consequence of the proof of Riesz- Fisher’s Theorem we have the following useful result. p p Theorem 1.57. Let (fh)h ⊂ L (X, µ) and f ∈ L (X, µ) with 1 6 p 6 ∞. Suppose that

(MC) lim kfh − fkLp(X,µ) = 0 . h→∞ p Then, there exist a subsequence (fhk )k and a function g ∈ L (X, µ) such that

(i) fhk (x) → f(x) µ − a.e. x ∈ X;

(ii) |fhk (x)| 6 g(x) µ − a.e. x ∈ X, ∀ k. Proof. See [GZ, Theorem 6.25].

Remark 1.58. The implication (MC)⇒ fh(x) → f(x) µ-a.e. x ∈ Ω may not hold. Historical notes: ([P, Sections 1.1.4,1.5.2,4.4.1]) Fisher and Riesz invented the Hilbert space L2([a, b]) in 1907, by proving its completeness. Both authors observed the significance of Lebesgue’s integral as the basic ingredient. Subsequently, in 1909, Riesz extended this definition to exponents 1 < p < ∞ and described how the interval [a, b] can be replaced by any measurable set of Rn. Lp spaces are sometimes called Lebesgue spaces, named after H. Lebesgue (Dunford & Schwartz 1958, III.3), although according to Bourbaki (1987) they were first introduced by Riesz. Definition 1.59. Let 1 6 p < ∞ and denote ( p if 1 < p < ∞ p0 := p − 1 ∞ if p = 1 p0 is called conjugate exponent of p. 0 Theorem 1.60 (Hölderinequality). Let p and p be conjugate exponents, 1 6 p < ∞ Let f ∈ Lp(X, µ) and g ∈ Lp0 (X, µ). Then f g ∈ L1(X, µ) and

kfgkL1(X,µ) 6 kfkLp(X,µ) kgkLp0 (X,µ) Proof. See [R1, Theorem 3.8] or [GZ, Theorem 6.20]. The Hölderinequality establishes a duality between Lp(X, µ) and the dual space of Lp(X, µ), denoted (Lp(X, µ))0 according to the notation of functional analysis: if 0 u ∈ Lp (X, µ), it is well defined the continuous linear functional T (u): Lp(X, µ) → R, that is T (u) ∈ (Lp(X, µ))0, by Z p T (u)(f) := hT (u), fi(Lp(X,µ))0×Lp(X,µ) := u f dµ ∀f ∈ L (X, µ) . X The question naturally arises whether all continuous linear functionals on Lp(X, µ) have this form, and whether the representation is unique. The answer is affirmative if 1 < p < ∞. It is also affirmative if p = 1, provided that an additional condition on measure µ. 23

Riesz representation theorem for the dual space of Lp 1.61. If 1 < p < ∞, then the mapping T : Lp0 (X, µ) → (Lp(X, µ))0, defined by Z p hT (u), fi(Lp(X,µ))0×Lp(X,µ) := u f dµ ∀f ∈ L (X, µ) , X is an isometric isomorphism, that is, T is a linear, one-to-one, onto mapping and p0 kT (u)k(Lp(X,µ))0 = kukLp0 (X,µ) ∀u ∈ L (X, µ) . If p = 1, the same conclusion holds under the additional assumption that µ is σ-finite. We will mean this feature by means of the identification (1.26) Lp0 (X, µ) ≡ (Lp(X, µ))0 . Proof. See [GZ, Theorem 6.43] or [F, Theorem 6.15], and [R1, Theorem 6.16]) if p = 1 provided µ is σ-finite. Remark 1.62. Identification (1.26) may fail in the other cases: see [F, section 6.2].

Historical notes:([P, Section 2.2.7]) The identity (Lp(a, b))0 = Lp0 (a, b) with 1 < p < ∞ was proved by F. Riesz in 1909. The limit case p = 1 is due to Steinhuas in 1919.

p Density of continuous functions in (L (X, µ), k · kLp ). Riesz representation theorem. The subject of this subsection concerns measure and integration theory on locally compact metric spaces. We have seen that the Lebesgue measure on Rn interacts nicely with the topology on Rn - measurable sets can be approximated by open or compact sets, and integrable functions can be approximated by continuous functions - and it is of interest to study measures having similar properties on more general spaces. Moreover, it turns out that certain linear functionals on spaces of continuous functions are given by integration against such measures. This fact constitutes an important link between measure theory and functional analysis, and it also provides a powerful tool for constructing measures. In this subsection we will only request that (X, B(X), µ) is a measure space with (X, d) locally compact, separable metric space (we will often use the abbrevia- tion l.c.s. in the following) and µ a Radon measure. Let us begin to deal with the approximation of continuous functions in Lp. Note 0 p that, under the above assumptions, Cc (X) ⊂ L (X, µ) for each p ∈ [1, ∞], provided that µ is a Radon measure on X. Theorem 1.63 (Approximation in Lp by continuous functions ). Let (X, B(X), µ) 0 be a measure space with (X, d) l.c.s. and µ Radon measure. Then Cc (X) is dense in p (L (X), k · kLp ), provided that 1 6 p < ∞. Proof of Theorem 1.63. The proof can be carried out as in the case of Lp(Ω), by means of the approximation by simple functions ( Theorem 1.20) and Lusin’s theorem (Theorem 1.23): see [R1, Theorem 3.14] and also [SC, Theorem 2.59] . ∞ p Remark 1.64. Assume that f ∈ L (X, µ) ∩ L (X, µ) (1 6 p < ∞). Then by 0 Theorems 1.63 and 1.57, it follows that there exists a sequence (fh)h ⊂ Cc (X) such 24 that p fh → f in L (X, µ) as h → ∞ and |fh(x)| 6 kfkL∞(X,µ) ∀ x ∈ X. 0 Indeed, by Theorems 1.63 and 1.57, there exist a sequence (gh)h ⊂ Cc (X) and a function g ∈ Lp(X, µ) such that p gh → f in L (X, µ) and |gh(x)| 6 g(x) µ-a.e. x ∈ X.

. Then, let deﬁne fh : X → R as

fh(x) = min{max{gh(x), −kfkL∞(X,µ)}, kfkL∞(X,µ)} We are now going to establish an important relationship between Radon measures and suitable linear bounded (or continuous) functionals on the space of compactly supported continuous functions, called Riesz representation theorem and due to F. Riesz. This relationship will turn out to be a fundamental bridge between measure theory and functional analysis. Let us ﬁrst recalls some preliminary topological results. Urysohn’s lemma 1.65 (1925). Let X be a locally compact metric space, let K ⊂ X and V ⊂ X be, respectively, a compact set and an open set such that K ⊂ V . Then 0 there exists a function ϕ ∈ Cc (X) such that 0 6 ϕ 6 1, ϕ ≡ 1 in K and spt(ϕ) ⊂ V. Proof. We skip the proof (see [GZ, Lemma 9.7]). Lemma 1.66 (Partition of unity). Let (X, d) be a locally compact metric space. Assume that V1,...,VN and K are respectively open sets and a compact set in X such that N K ⊂ ∪i=1Vi . 0 Then there exist fi ∈ Cc (X) (i = 1,...,N) such that

spt(fi) ⊂ Vi, 0 6 fi 6 1 ∀ i = 1,...,N ; N X fi(x) = 1 ∀ x ∈ K. i=1

A family of functions {f1, . . . , fN } satisfying previous properties is called a partition of unity subordinate to the open covering V1,...,VN of K.

Proof. See [GZ, Theorem 9.8] or [R1, Theorem 2.13]. Let µ be a Radon measure on measure space (X, B(X)). We can associate to µ a 0 linear functional Lµ : Cc (X) → R deﬁned by Z 0 (1.27) Lµ(u) := u dµ ∀u ∈ Cc (X) . X

Notice that Lµ is positive . Let us recall that 0 Deﬁnition 1.67. A linear functional L : Cc (X) → R is said to be positive (or also monotone) if L(u) > 0 whenever u > 0 (or, equivalently, L(u) > L(v) whenever u > v). 25

In this definition there is no mention of continuity, but it is worth noting that positivity itself implies a rather strong continuity property. More precisely, let us 0 m endow (Cc (X)) by the ∞- (or also uniform) norm defined by 0 (1.28) kuk∞ := sup |u(x)|Rm if u ∈ Cc (X) . x∈X For the sake of simplicity, we will denote Rm-norm simply by | · | from now on. 0 Proposition 1.68. If L : Cc (X) → R is a positive linear functional, for each compact K ⊂ X there is a positive constant CK (depending on K) such that 0 (1.29) sup |L(u)| : u ∈ Cc (X), kuk∞ 6 1, spt(u) ⊂ K 6 CK . Proof. See [F, Proposition 7.1]. We will assume inequality (1.29) as definition of continuity (or boundedness) for linear functionals on the compactly supported continuous functions, even in the vector 0 m case L :(Cc (X)) → R. 0 m Definition 1.69. A linear functional L :(Cc (X)) → R is said to be continuous (or bounded) if, for each compact set K ⊂ X, there is a positive constant CK (depending on K) such that (1.29) holds. Remark 1.70. For the sake of simplicity, assume that m = 1. Then it can be proved that the notion of continuity in Definition 1.69 is induced by a topology τ on 0 0 Cc (X) for which (Cc (X), τ) turns out to be a locally convex , complete topological vector space, which is not metrizable (see [T]). Indeed let (Ah)h be an increasing 0 sequence of relative compact open sets of (X, d) . Then, if we denote by C0(Ah) the space of continuous function vanishing at infinity on Ah (see Definition 1.80), 0 it can be shown that (C0(Ah), k · k∞) is a Banach space (see Proposition 1.81) and 0 ∞ 0 0 0 Cc (X) = ∪h=1C0(Ah) . If ih : C0(Ah) → Cc (X)(h = 1, 2,... ) denotes the inclusion 0 map, the topology τ turns out to be the strongest topology on Cc (X) for which maps 0 0 ih :(C0(Ah), k · k∞) → (Cc (X), τ) are continuous for each h. The existence of such 0 a topology τ can be provided meaning Cc (X) as the vector topological space, also 0 called LF-space, countable inductive strict limit of Banach spaces (C0(Ah), k · k∞)h (see, for instance, [T, Chap. XIII]). Moreover a notion of convergence can be induced by means of this topology. 0 m Exercise: A linear functional L :(Cc (X)) → R is continuous (according to 0 m Definition 1.69) if and only if limh→∞ L(uh) = L(u) for each (uh)h, u in (Cc (X)) satisfying (1.30) ∞ [ uh → u uniformly on X and there exists a compact K ⊂ X, sptu ∪ sptuh ⊂ K. h=1 0 m Definition 1.71. We write that uh → u in (Cc (X)) , if (1.30) holds. m−1 Example 1.72. If µ is a Radon measure on X and w = (w1, . . . , wm): X → S := {y ∈ Rm : |y| = 1} is a (Borel) measurable function, we may define a continuous 0 m linear functional wµ :(Cc (X)) → R setting Z Z m X 0 m (1.31) wµ(u) := (w, u)Rm dµ = wi ui dµ u = (u1, . . . , um) ∈ (Cc (X)) , X X i=1 26 which is trivially an extension of the functional defined in (1.27) when m = 1. We 0 m will see that each continuous linear functional L :(Cc (X)) → R can be represented by form (1.31) for suitable w and µ. Riesz representation theorem 1.73. Let (X, d) be a separable, locally compact 0 m metric space and let L :(Cc (X)) → R be a continuous linear functional. Then there exist a Radon measure µL : B(X) → [0, ∞] and a Borel measurable vector m−1 function wL : X → S such that Z 0 m (1.32) L(u) = (wL, u)Rm dµL ∀u ∈ (Cc (X)) , X that is, L = wLµ, and µL is characterized by the following identity: for each open set A ⊂ X m (1.33) µL(A) = sup {L(u): u ∈ (Cc(X)) , sptu ⊂ A, kuk∞ 6 1} . Moreover representation (1.32) is unique. Definition 1.74. Let (X, d) be a locally compact metric space and consider the measure space (X, B(X)). Let us also denote by Bcomp(X) the class of Borel sets which are relatively compact in X. m (i)A R -valued Radon vector measure ν on X is a set function ν : Bcomp(X) → m m−1 R for which there exist a Borel vector function wν : X → S and a positive Radon measure µ : B(X) → [0, ∞] such that Z (1.34) ν(E) = wν dµ ∀E ∈ Bcomp(X) . E (ii)A Rm-valued finite Radon vector measure is a set function ν : B(X) → Rm for m−1 which there exist a Borel vector function wν : X → S and a positive finite Radon measure µ : B(X) → [0, ∞] such that (1.34) holds for all E ∈ B(X).

We will write that ν = wν µ in both previous cases. m m m We denote by (Mloc(X)) (respectively (M(X)) ) the space of R -valued Radon (resp. Rm-valued finite Radon ) measures on X. Remark 1.75. Observe that, by Corollary 1.53, we could equivalently define a Rm- valued Radon vector measure (respectively a Rm-valued finite Radon vector measure) m as a set function ν : Bcomp(X) → R such that, for each compact K ⊂ X, ν : B(K) → Rm is a vector measure (respectively as a vector measure ν : B(X) → Rm) m Remark 1.76. It easy to see that (Mloc(X)) turns out to be vector space. Before the proof of the Riesz representation theorem we need some technical lemma.

0 m ∗ Lemma 1.77. Let L :(Cc (X)) → R be a continuous linear functional. Let µL : P(X) → [0, ∞] be the set function defined as ∗ (1.35) µL(E) := inf {µL(A): A open,A ⊇ E} if E ⊂ X ∗ where µL(A) is the quantity in (1.33). Then µL is a Radon outer measure on X. ∗ Proof. Denote ν = µL and µ = µL. Let us first observe that, if U is open, then (1.36) µ(U) = ν(U) , 27 that is the definition of ν is consistent. Indeed we immediately get that

µ(U) > ν(U) . If A is open and A ⊇ U, then by deﬁnition (see (1.33)), we get

µ(U) 6 µ(A) , which implies µ(U) 6 ν(U) . Thus (1.36) follows. Let us now divide the proof in three steps. 1st step. We prove that ν is an outer measure. Let us ﬁrst show that ν is countably subaddtive on open sets, that is, if (Ah)h is a ∞ sequence of open sets and A = ∪h=1Ah, then ∞ X (1.37) ν(A) 6 ν(Ah) . h=1 0 m Let u ∈ (Cc (A)) with kuk∞ 6 1 and let K := spt(ϕ). Since K ⊂ A is a compact N set, there exists an integer N such that K ⊂ ∪h=1Ah. Let us consider a partition of unity {ϕ1, . . . , ϕN } subordinate to {A1,...,AN } and K, that is

N 0 X ϕh ∈ Cc (Ah), 0 6 ϕh 6 1, ϕh(x) = 1 ∀ x ∈ K. h=1 PN 0 m Since u = h=1 u ϕh and uϕh ∈ (Cc (Ah)) with |uϕh| 6 1 N N ∞ X X X L(u) = L(uϕh) 6 ν(Ah) 6 ν(Ah) . h=1 h=1 h=1 0 m Then (1.37) follows passing to the supremum over all u ∈ (Cc (A)) with kuk∞ 6 1. Let us now prove that ν is countably subadditive, that is ∞ X ∞ (1.38) ν(E) 6 ν(Eh) if E ⊂ ∪h=1Eh . h=1 Let us ﬁrst observe that ν is non decreasing w.r. t. the inclusion, that is

ν(E) 6 ν(F ) if E ⊂ F.

Without loss of generality we can suppose ν(Eh) < ∞ for each h. For each > 0 and h there is an open set Ah ⊇ Eh such that ν(A ) < ν(E ) + . h h 2h Thus, by (1.37), ∞ ∞ ∞ X X ν(E) 6 ν(∪h=1Ah) 6 ν(Ah) 6 ν(Eh) + ∀ > 0 . h=1 h=1 Therefore (1.38) follows. 28

2nd step. Let us prove that ν is a Borel regular outer measure. In order to prove that ν is a Borel outer measure, by Carath´eodory’s criterion (see Theorem 1.5 (ii)), we have only to prove that (1.39) ν(E ∪ F ) > ν(E) + ν(F ) whenever d(E,F ) > 0 . 0 m ¯ Assume E, F are open, and let ϕ ∈ (Cc (E ∪ F )) with |ϕ| 6 1. Then, since E and F¯ are disjoint, 0 m 0 m ϕ = ϕ|E + ϕ|F , ϕ|E ∈ (Cc (E)) , ϕ|F ∈ (Cc (F )) , |ϕ|E| 6 1, |ϕ|F | 6 1 . Thus L(ϕ) = L(ϕ|E) + L(ϕ|F ) > ν(E) + ν(F ) , and (1.39) follows. In the general case, since 0 < d(E,F ) = d(E,¯ F¯), there exist open sets A1, A2 such that E ⊂ A1 and F ⊂ A2 with d(A1,A2) > 0. If A is open and E ∪F ⊂ A, then d(A1 ∩A, A2 ∩A) > 0 and E ⊂ A1 ∩A, F ⊂ A2 ∩A, so that (1.39)on open sets implies

ν(A) > ν ((A1 ∩ A) ∪ (A2 ∩ A)) > ν(A1 ∩ A) + ν(A2 ∩ A) > ν(E) + ν(F ) . As A is arbitrary, (1.39) follows. Hence ν is a Borel measure. Moreover ν is Borel regular, since, if E ⊂ X, ν(E) < ∞ and (Ah)h are open sets with E ⊂ Ah and ∞ limh→∞ ν(Ah) = ν(E), then B := ∩h=1Ah is a Borel set with E ⊂ B and ν(E) = ν(B). If ν(E) = ∞, we can choose as Borel envelope B = X. 3rd step. Let us prove that ν is ﬁnite on compact sets. Let us recall, that by Lemma 1.17, there exists an increasing sequence of open sets (Vi)i such that ∞ X = ∪i=1Vi, Vi compact for each i . In particular, since L is bounded, by (1.33) and (1.29), it follows that

ν(Vi) = µ(Vi) < ∞ ∀ i .

Given a compact set K ⊂ X, then there exists an integer i0 such that

i0 K ⊂ ∪i=1Vi . Therefore, since ν is subadditive, i X0 ν(K) 6 ν(Vi) < ∞ , i=1 and the proof is accomplished. ∗ Proof of the Riesz representation theorem 1.73. By Lemma 1.77, ν := µL is a Radon ˜ 0 outer measure on X. Let us now deﬁne functional L : Cc (X; [0, ∞)) → [0, ∞) as ˜ 0 m 0 L(ϕ) := sup L(u): u ∈ (Cc (X)) , |u| 6 ϕ, ϕ ∈ Cc (X; [0, ∞)) . We will divide the proof in three steps. In step one we will show that L˜ is additive, 0 positively homogeneous of degree one, and monotone on Cc (X; [0, ∞)). In step two, we will show the inequality Z ˜ 0 (1.40) L(ϕ) 6 ϕ dν ∀ ϕ ∈ Cc (X; [0, ∞)) . X Finally, in step three, by using the Riesz representation theorem for the dual of L1(X, µ) (see Theorem 1.61) we can conclude the proof. 29

0 1st step. We show that, whenever ϕi ∈ Cc (X; [0, ∞)) (i = 1, 2) and c > 0, we have ˜ ˜ ˜ (1.41) L(ϕ1 + ϕ2) = L(ϕ1) + L(ϕ2) , ˜ ˜ (1.42) L(c ϕ1) = c L(ϕ1), ˜ ˜ (1.43) L(ϕ1) 6 L(ϕ2), if ϕ1 6 ϕ2 . It si immediate that (1.43) follows by the deﬁnition of L˜. let us prove the remaining ˜ 0 m properties. By deﬁnition of L and the linearity of L, for each ui ∈ (Cc (X)) (i = 1, 2) with |ui| 6 ϕi, c > 0 , we can infer ˜ L(ϕ1 + ϕ2) > L(u1 + u2) = L(u1) + L(u2) , ˜ L(c ϕ1) > L(c u1) = c L(u1) , ˜ L(c ψ1) = c L(ψ1) 6 c L(ϕ1) , ˜ L(ui) 6 L(ϕi) i = 1, 2,. Then, by the previous inequalities, it follows respectively that ˜ ˜ ˜ (1.44) L(ϕ1 + ϕ2) > L(ϕ1) + L(ϕ2) , ˜ ˜ (1.45) L(c ϕ1) > c L(ϕ1) , ˜ ˜ (1.46) L(c ϕ1) 6 c L(ϕ1) . Therefore, by (1.45) and (1.46) we have proven (1.42), and we have only to prove 0 m the inverse inequality of (1.44) for showing (1.41) . Now let u ∈ (Cc (X)) be such that |u| 6 ϕ1 + ϕ2, and set  ϕi  u on {ϕ1 + ϕ2 > 0} ui := ϕ1 + ϕ2 i = 1, 2 . 0 otherwise

0 m Exercise: Prove that ui ∈ (Cc (X)) , |ui| 6 ϕi (i = 1, 2) and u = u1 + u2. Therefore ˜ ˜ L(u) = L(u1) + L(u2) 6 L(ϕ1) + L(ϕ2), and complete the proof by the arbitrariness of u. 0 2nd step. Let us now prove (1.40). Given ϕ ∈ Cc (X; [0, ∞)) and > 0, let t0, . . . , tN be real numbers such that (1.47) t0 < 0 < t1 < ··· < tN−1 < sup ϕ < tN , th+1 − th 6 with h = 1,...,N − 1 X and consider the partition {E1,...,EN } of spt(ϕ) by disjoint Borel sets, deﬁned as

Eh = {x ∈ spt(ϕ): th−1 < ϕ(x) 6 th} , 1 6 h 6 N.

Since ν is a Radon o.m., by Theorem 1.14, there exist open sets Ah with Eh ⊂ Ah and (1.48) ν(A ) ν(E ) + , 1 h N. h 6 h N 6 6 30

If necessary replacing Ah with {x ∈ Ah : ϕ(x) < th + }, we can also assume

(1.49) ϕ < th + on Ah .

Finally, let {f1, . . . , fN } be a partition of unity subordinated to the open covering 0 A1,...,AN of the compact set spt(ϕ), namely fh ∈ Cc (Ah) ,0 6 fh 6 1, and PN PN h=1 fh(x) = 1 on spt(ϕ). Since ϕ = h=1 fh ϕ , by step 1 and (1.49), we ﬁnd that

N N ˜ X ˜ X ˜ (1.50) L(ϕ) = L(fhϕ) 6 (th + ) L(fh) . h=1 h=1 0 m ˜ If ψ ∈ (Cc (X)) , and |ψ| 6 fh, then spt(ψ) ⊂ Ah and |ψ| 6 1. Hence, L(fh) 6 ν(Ah) and, by (1.47) and (1.48), we ﬁnd that

N X L˜(ϕ) (t + ) ν(E ) + 6 h h N h=1 N X (t + 2) ν(E ) + 6 h−1 h N h=1 N N N X X X = t ν(E ) + t + 2 ν(E ) + 22 h−1 h N h−1 h h=1 h=1 h=1 Z 2 6 ϕ dν + tN + 2 ν(spt(ϕ)) + 2 X Z 6 ϕ dν + sup ϕ + + 2 ν(spt(ϕ)) + 2 X X If we let → 0+ in the previous inequality, (1.40) follows. m−1 0 3rd step. Given e ∈ S , we deﬁne Le : Cc (X) → R by 0 Le(ϕ) := L(ϕ e) ϕ ∈ Cc (X) . 0 By (1.40), we ﬁnd that, for every ϕ ∈ Cc (X), 0 m Le(ϕ) 6 sup L(ψ): ψ ∈ (Cc (X)) , |ψ| 6 |ϕ| Z ˜ = L(|ϕ|) 6 |ϕ| dν . X By the approximation in Lp by continuous functions (see Theorem 1.63), we may 1 extend Le as a linear functional Le : L (X, ν) → R such that Z 1 |Le(u)| 6 |u| dν = kukL1(X,µ) ∀ u ∈ L (X, ν) . X Thus, by the Riesz representation theorem for the dual of L1(X, ν), there exists ∞ we ∈ L (X, ν) such that Z 1 L(u e) = u we dν ∀ u ∈ L (X, ν) . X 31

m If we set wL := (w1, . . . , wm): X → R , wi := wei , where {e1, . . . , em}denotes the m standard basis of R , then wL is bounded and ν-measurable, with m m X X Z Z m (1.51) L(ϕ) = Lei (ϕi) = wi ϕi dν = (wL, ϕ)R dν i=1 i=1 X X 0 m for every ϕ = (ϕ1, . . . , ϕm) ∈ (Cc (X)) . Let us now prove that

(1.52) |wL(x)| = 1 ν-a.e. x ∈ X. By (1.36) and (1.51), it follows that, for each bounded open set A ⊂ X, m ν(A) = sup {L(ϕ): ϕ ∈ (Cc(X)) , sptϕ ⊂ A, |ϕ| 6 1} Z m = sup (wL, ϕ)Rm dν : ϕ ∈ (Cc(X)) , sptϕ ⊂ A, |ϕ| 6 1 (1.53) X Z 6 |wL| dν . A From (1.53), it follows that

wL 1 m |wL| > 0-a.e. in X and u := χ{|wL|>0} ∈ (L (A, ν)) . |wL| 0 m By Remark 1.64, there exists a sequence (ϕh)h ⊂ (Cc (A)) such that 1 m ϕh → u in (L (A, ν)) and |ϕh| 6 1 . This implies that 1 (1.54) (wL, ϕh)Rm → |wL| in L (A, ν) Therefore, since Z ν(A) > (wL, ϕh)Rm dν , A by (1.54) and passing to the limit, as h → ∞, in the previous inequality, it follows that Z (1.55) ν(A) > |wL| dµ . A Thus, by (1.53) and (1.55), we can infer that Z ν(A) = |wL| dν for each bounded open set A ⊂ X A which implies (1.52). m−1 Let us prove that we can choose wL : X → S as a Borel measurable vector m function. Indeed, observe that, by Corollary 1.24, we can assume that wL : X → R is Borel measurable and satisfying (1.51). On the other hand, by (1.52), there exists a ν-null set N ⊂ X (which, a priori, could be not a Borel set) such that |wL(x)| = 1 for each x ∈ X \ N. Since ν is Borel regular, there exists a Borel ν-null set B ⊃ N. Thus, by changing the value of wL on B, for instance putting wL := e1 on B, we get m−1 that wL : X → S is still Borel measurable and satisﬁes (1.51). The uniqueness of ν and wL (ν-a-e.) follows in a standard way. 32

∗ Finally, without loss of generality, since ν(A) = µL(A) = µL(A) for each open set A ⊂ X, we still denote µL the outer measure defined in (1.35) and yield a Radon measure µL : B(X) → [0, ∞] satisfying the desired properties. Riesz representation theorem 1.73 provides a characterization of the measures m m spaces M(X) and (Mloc(X)) as dual spaces of suitable spaces of continuous functions. Observe first that functional (1.27) can make sense even for Radon signed measures. Indeed, if ν ∈ Mloc(X), that is, ν = wν µ with µ positive Radon measure on X and wν : X → {−1, 1} Borel function, then , according also to Definition 1.48 (i), it is well defined Z Z 0 (1.56) Lν(u) := u dν = wν u dµ ∀u ∈ Cc (X) . X X 0 The functional Lν : Cc (X) → R still turns out to be a linear continuous functional according to Definition 1.69. Thus a trivial consequence of Theorem 1.73 is the following

Corollary 1.78 (Characterization of Mloc(X)). Let (X, d) be a locally compact separable metric space and define (1.57) 0 0 0 (Cc (X)) := L : Cc (X) → R : L is linear and continuous w.r.t. Definition 1.69 . Let us define the map

0 0 (1.58) I : Mloc(X) → (Cc (X)) I(ν) := Lν . Then I is an isomorphism (between vector spaces).

If ν ∈ M(X) functional (1.56) is still well defined since M(X) ⊂ Mloc(X). In this case the functional is actually an element of the dual of normed vector space 0 0 0 (Cc (X), k · k∞) and we will write Lν ∈ (Cc (X), k · k∞) . 0 Proposition 1.79. Let ν ∈ M(X), and let Lν : Cc (X) → R be the functional in (1.56). Then 0 kL k 0 0 := sup |L (u)| : u ∈ C (X), kuk 1 = |ν|(X) . ν (Cc (X),k·k∞) ν c ∞ 6 Proof. It is a special case of (1.33). 0 Functional Lν can actually be extended to the completion of (Cc (X), k · k∞), that is the class of continuous functions on (X, d) vanishing at infinity. Definition 1.80 (Functions vanishing at infinity). Let (X, d) be a locally compact metric space. (i) A function u : X → R is said to vanish at infinity if for every > 0 there is a compact set K ⊂ X such that |u(x)| < ∀x ∈ X \ K.

0 (ii) The class of all continuous u : X → R which vanish at inﬁnity is called C0 (X), 33

0 0 It is clear that Cc (X) ⊂ C0 (X) and that the two classes coincide if (X, d) is compact. More precisely, if (X, d) is compact, then

0 0 0 Cc (X) = C0 (X) = C (X) . It is also well known that 0 0 Proposition 1.81. C0 (X) is the completion of normed vector space (Cc (X), k · k∞). Proof. See [R1, Theorem 3.17]. Proposition 1.82. Let ν ∈ M(X) and deﬁne (1.59) kνk := |ν|(X) . Then (M(X), k · k) is a real normed vector spaces.

Proof. [F, Proposition 7.16]. Theorem 1.83 (Characterization of M(X)). Let (X, d) be a locally compact separable metric space. Let I the map in (1.58). Then 0 0 (i) I(M(X)) = (C0 (X), k · k∞) ; 0 0 (iI) I :(M(X), k · k) → (C0 (X), k · k∞) is a topological isomorphism, that is an algebraic isomorphism, continuous with its inverse.

Proof. [R1, Theorem 6.19]. 0 0 Corollary 1.84. Let (X, d) be a compact metric space. Then (C (X), k · k∞) is isometrically isomporphic to M(X).

p Compactness in (L (Ω), k · kLp ). In this section we are going to deal with some compactness results in Lp spaces. We will only state these results, without proofs which can be found in [B, Section 4.5]. n n n Let f : R → R and v ∈ R be given, then we deﬁne by τvf : R → R the v-translated function of f deﬁned by

(τvf)(x) := f(x + v) . Theorem 1.85 (M. Riesz- Fréchét-Kolmogorov). Let F be a bounded subset in p n (L (R ), k · kLp ) with 1 6 p < ∞. Suppose that limv→0 kτvf − fkLp = 0 uniformly for f ∈ F, that is n ∀ > 0 ∃ δ = δ() > 0 such that kτvf − fkLp < ∀ v ∈ R with |v| < δ, (EN ) F ∀ f ∈ F . p Then F|Ω := {f|Ω : f ∈ F } is relatively compact in (L (Ω), k · kLp ), i.e. its closure p n is compact in (L (Ω), k · kLp ), for each open set Ω ⊂ R with finite Lebesgue measure. p From Theorem 1.85 it follows the following compactness criterion in (L (Ω), k·kLp ). If f :Ω → R, let us denote by f˜ : Rn → R the function defined as f(x) if x ∈ Ω f˜(x) := . 0 otherwise 34

Corollary 1.86. Let Ω ⊂ Rn be an open set with ﬁnite measure, let F ⊂ Lp(Ω) and let F˜ := {f˜ : f ∈ F}. Assume that p (i) F is bounded in (L (Ω), k · kLp ) with 1 6 p < ∞; ˜ ˜ ˜ (ii) limv→0 kτvf − fkLp = 0 uniformly for f ∈ F, that is, F satisﬁes ENF˜. p Then F is relatively compact in (L (Ω), k · kLp ). Proof. From Theorem 1.85, F˜ is relatively compact. Notice now that F˜ is relatively p n sequentially compact in (L (R ), k · kLp ) if and only if F is relatively sequentially p compact in (L (Ω), k · kLp ). Thus, the characterization of compact sets in metric spaces completes the proof. p n Eventually recall the following characterization of compactness in (L (R ), k · kLp ). p n Theorem 1.87. Let F ⊂ L (R ) with 1 6 p < ∞. Then F is relatively compact in p n (L (R ), k · kLp ) if and only if p n (i) F is bounded in (L (R ), k · kLp ); (ii) for each > 0 there exists r > 0 such that

p n kfkL (R \B(0,r)) < ∀ f ∈ F ;

(iii) limv→0 kτvf − fkLp = 0 uniformly for f ∈ F.

Remark 1.88. (i) The assumption (ENF ) is necessary in Theorem 1.85. Indeed, consider the family F := {fh : h ∈ N } where fh : R → R is deﬁned as fh(x) := ( 1 h if 0 6 x 6 h and let Ω := (0, 1). Then it is easy to see that kfhkL1(R) = 1 0 otherwise 1 for each h ∈ N and F|Ω is not relatively compact in (L (Ω), k·kL1 ), since there are no 1 subsequences of (fh)h converging in L (Ω) (see Exercise III.6). On the other hand, 1 for given v > 0, for each h > v Z 0 Z v kτvfh − fhkL1(R) > fh(x + v) dx = fh(x) = 1 . −∞ 0

Thus, (ENF ) does not hold for F. (ii) If Ω has not ﬁnite measure, then the conclusions of Theorem 1.85 need not hold. Indeed , consider the family F := {fh : h ∈ N } where fh : R → R is deﬁned as fh(x) := f(x + h) where f ∈ Lip(R) with spt(f) = [−a, a], a > 0, and f not identically vanishing. Then

(1.60) kfhkL1(R) = kfkL1(R) > 0 ∀ h .

Moreover F satisﬁes (ENF ), because

|τvf(x) − f(x)| = |f(x + v) − f(x)| 6 L |v| χ[−a−1,a+1](x) ∀ x ∈ R, v ∈ [−1, 1] , and

kτvfh − fhkL1(R) = kτvf − fkL1(R) ∀ h where L := Lip(f). Let Ω := R and observe now that F = F|Ω is not relatively 1 compact in (L (R), k·kL1 ). Otherwise a contradiction arises by (1.60), since fh(x) → 0 for each x ∈ R, 35

Historical notes:[HOH] A first compactness type-result was proved by Fréchet in 1908 in the setting of l2. In 1931, Kolmogorov proved the first result in this direction. The result characterizes the compactness in Lp(Rn) for 1 < p < ∞ , in the case where all functions are supported in a common bounded set. Condition (iii) of Theorem 1.87 is replaced by the uniform convergence in Lp norm of spherical means of each function in the class to the function itself. (Clearly, our condition (ii) is automatic in this case.) Just a year later, Tamarkin expanded this result to the case of unbounded supports by adding condition (ii) of Theorem 1.87. In 1933, Tulajkov expanded the Kolmogorov-Tamarkin result to the case p = 1. In the same year, and probably independently, Riesz proved the result for 1 < p < ∞, essentially in the form of our Theorem 1.87. Thus we feel somewhat justified in using the names Kolmogorov and Riesz in referring to the theorem, though we are perhaps being a bit unfair to Tamarkin and Tulajkov in doing so. In 1937, M.Fréchétreplaced conditions (i) and (ii) of Theorem 1.87 with a single condition (”equisummability”), and generalized the theorem to arbitrary positive p.

1.6. Operations on measures. In this section we discuss some useful and fundamental operations on measures and related notions: among them, we describe the product measures, and state the related Fubini and Tonelli theorems, together with some consequences. Deﬁnition 1.89 (Support). Let µ be a positive measure on a separable metric space X; we call the closed set of all points x ∈ X such that µ(U) > 0 for every neighbourhood U of x the support of µ denoted spt(µ). In other words,

sptµ = X \ ∪ {V : V open, µ(V ) = 0} (1.61) = X \{x ∈ X : ∃ r > 0 such that µ(B(x, r)) = 0} If ν is a signed or signed vector measure, we call the support of ν the support of |ν|. In the general case of a measure on a measure space (X, M), we say that ν is concentrated on S ⊂ X if S ∈ M and |ν|(X \ S) = 0. Notice that it is impossible in general to deﬁne a ”minimal” set where a measure is concentrated, hence the set S is not uniquely determined. However, for any pair ν1 and ν2 of mutually singular measures there exist pairwise disjoint M-measurable sets S1 and S2 such that νi is concentrated on Si (i = 1, 2). A property which is not shared by the support. Example: Consider, for instance, the measure space (R, B(R)) and measures µ1 := 1 P −h L and µ2 := h 2 δxh with (xh)h dense in R. Then spt(µ1) = spt(µ2) = R. Even if measure µ2 is concentrated on the elements of the sequence, but spt(µ1) contains every accumulation point of the sequence. Remark 1.90. Notice also that, if X is a separable metric space and ν is a Borel measure on X, then spt(ν) is the smallest closed set where ν is concentrated. Deﬁnition 1.91 (Restriction). Let ν be an outer measure X or a positive or vector signed measure on the measure space (X, M). Let E ⊂ X. 36

(i) If ν is an outer measure on X, the restricted outer measure of ν to E, written ν E, is the set function defined by (1.62) ν E(F ) := ν(E ∩ F ) ∀ F ⊂ X. (ii) If ν is a positive or vector signed measure on the measure space (X, M) and E ∈ M, the restricted measure of ν to E, still written ν E, is the set function defined in (1.94), but with F ∈ M. Theorem 1.92. (i) If ν is an outer measure on X, then so is ν E. Moreover every ν-measurable set is also ν E-measurable. (ii) If ν is a Borel regular outer measure on X and E is ν-measurable with ν(E) < ∞ , then ν E is a Radon outer measure. (iii) If ν is a positive or vector signed measure on a measure space (X, M) and E ∈ M, so is ν E. Proof. (i) It is clear that ν E is an outer measure. By using the definition of Carathéodory measurability, the second part of the statement follows, too. Note that E can be arbitrary here. (ii) Clearly (ν E)(K) 6 ν(E) < ∞ for each compact set K ⊂ X. Since, by previous claim (i), every ν-measurable is also (ν E)-measurable, (ν E) is a Borel outer measure. Thus we have only to show that ν E is a Borel regular outer measure. Since ν is Borel regular, there is a Borel set B such that E ⊂ B and ν(B) = ν(E). Then, since E is ν-measurable and ν(E) < ∞, ν(B \ E) = 0. Let us fix C ⊂ X. Then (ν E)(C) 6 (ν B)(C) = ν(C ∩ B) = ν(C ∩ B ∩ E) + ν((C ∩ B) \ E) 6 ν(C ∩ E) + ν(B \ E) = (ν E)(C) Therefore ν B = ν E, so we may as well assume that E is a Borel set. Given C ⊂ X, we must show that there exists a Borel set F such that C ⊂ F and (ν E)(F ) = (ν E)(C). Since ν is Borel regular, there is a Borel set D such that C ∩ E ⊂ D and ν(C ∩ E) = ν(D). Let F := D ∪ (X \ E). Since D and E are Borel sets, so is F . Moreover C ⊂ (E ∩ C) ∪ (X \ E) ⊂ F . Finally, since F ∩ E = D ∩ E, (ν E)(F ) = ν(F ∩ E) = ν(D ∩ E) 6 ν(D) = ν(C ∩ E) = (ν )E(C) . Thus (ν E)(F ) = (ν E)(C) and so ν E is Borel regular. (iii) The claim is trivial. Remark 1.93. By using the same proof of Theorem 1.92 (ii), we can infer that if ν is a Borel regular measure and E is a Borel set, then ν E is still Borel regular, even if ν(E) = ∞. Theorem 1.92 (ii) allows to generate Radon outer measures by restricting a given Borel regular outer measure to a measurable set of finite measure. An other interesting procedure for generating Radon outer measures, in a separable, locally compact metric space, by restriction of a given Borel regular outer measure is the following. Theorem 1.94. Let ν be a Borel regular outer measure on a separable, locally compact metric space (X, d). Let E be a ν-measurable set such that ν E is a locally finiteouter measure. Then ν E is a Radon outer measure. 37

Proof. Let us first recall that, from Lemma 1.17, there exists an increasing sequence of open sets (Vi)i such that ∞ (1.63) X = ∪i=1Vi, Vi is compact for each i . Let ϕ := ν E. Let us first prove that (1.64) ϕ(K) < ∞ for each compact set K ⊂ X. If K ⊂ X is a given compact set, by the local fineteness of ϕ, for each x ∈ K there exists an open ball U(x, rx) such that

(1.65) ϕ(U(x, rx)) < ∞ .

Since K is compact, there is a ﬁnite family of open balls U(x1, r1),...,U(xm, rm) such that m (1.66) K ⊂ ∪i=1U(xi, ri) . Thus, by (1.65),(1.66) and the subadditivity of ϕ,(1.64) follows. Since, by Theorem 1.92 (i), every ν-measurable set is also ϕ-measurable, ϕ is a Borel outer measure. Thus we have only to show that ϕ is a Borel regular outer measure, that is, given C ⊂ X, we must show that there exists a Borel set F such that C ⊂ F and ϕ(F ) = ϕ(C). Let ϕi := ϕ Vi = ν (E ∩ Vi) (if i ∈ N). Let us observe that

(1.67) ∃ ϕ(C) = lim ϕi(C) for all C ⊂ X. i→∞ Indeed, by the continuity of outer measures on increasing sequences of sets and (1.63),

lim ϕ(C) = lim (ν (E ∩ C))(Vi) = (ν (E ∩ C))(X) = ϕ(C) . i→∞ i→∞

On the other hand, since ν is Borel regular, E ∩ Vi is ν- measurable, and, by (1.64), ν(E ∩ Vi) 6 ν(Vi) < ∞, by Theorem 1.92 (ii) we can infer that ϕi = ν (E ∩ Vi) is a Borel regular outer measure. Thus, for each i and for a given C ⊂ X, there exists a Borel set Fi such that

(1.68) C ⊂ Fi and ϕi(Fi) = ϕi(C) . ∞ Let F := ∩i=1Fi. Then F is still a Borel set with F ⊃ C and

ϕi(C) 6 ϕi(F ) 6 ϕi(Fi) = ϕi(C) . Thus, it follws that there exists a Borel set F ⊃ C such tha

ϕi(C) = ϕi(F ) ∀ i . By taking the limit, as i → ∞, in the previous identity, we get that ϕ(C) = ϕ(F ) , and, then, the desired conclusion.

Given a measure space (X, M) and a measure on it, we see now how it can be carried on another set Y through a function f : X → Y . 38

Definition 1.95 (Push-forward of a measure, or image measure). Let (X, M) and (Y, N ) be measure spaces, and let f : X → Y be measurable, that is f −1(F ) ∈ M whenever F ∈ N . For any positive, real, or vector measure ν on (X, M) we define a measure f#ν in (Y, N ) by −1 f#ν(F ) := ν f (F ) From the previous definition the corresponding change of variable formula for integrals follows immediately: if u is a (reaI- or vector-valued) function on Y integrable with respect to f#ν, then u ◦ f in integrable with respect to ν, and we have the equality: Z Z (1.69) u d(f#ν) = u ◦ f dν Y X The very general definition given above can be easily seen to have good properties in l.c.s. spaces when f is assumed to be continuous and proper, i.e. such that f −1(K) is compact for any compact K ⊂ Y as the following remark shows. Remark 1.96. Let X, Y be l.c.s. metric spaces, f : X → Y continuous and proper : the continuity of f ensures that f −1(B) whenever B ∈ B(Y ), and since f is proper, if ν is a Radon measure on X, then f#ν is a Radon measure on Y . Hausdorff measures provide an important source of examples of Radon measures. We will deeply study them in Chapter III. We are going now to stress some relationships between the 1-dimensional Hausdorff measure and the classical notion of length measure for a curve in Rn. Example 1.97 (Push-forward of the classical length measure). A set Γ ⊂ Rn is a curve of Rn if there exists a continuous, injective function γ :[a, b] → Rn such that γ([a, b]) = Γ. The function γ is called a parametrization of Γ. Given a parametrization γ :[a, b] → Rn and a subinterval [c, d] ⊆ [a, b], we define the length of γ over [c, d] as ( N ) X (1.70) length(γ;[c, d]) := sup |γ(ti) − γ(ti−1| : t0 = c < t1 < ··· < tN = d i=1 where the supremum is taken over all finite partitions {t0 = c < t1 < ··· < tN = d} of [c, d]. It can be proved that, if Γ = γ1([a1, b1]) = γ2([a2, b2]) for two given n parametrizations γi :[ai, bi] → R (i = 1, 2), then length(γ1;[a1, b1]) = length(γ2;[a2, b2]). Thus we can define as length of Γ = γ([a, b]) the quantity length(Γ) := length(γ;[a, b]) . It is also well-known that, if Γ is a C1 regular curve, that is there exists a C1 parametrization γ :[a, b] → Rn of Γ with |γ0(t)|= 6 0 for each t ∈ [a, b], then

Z d (1.71) length(γ;[c, d]) = |γ0(t)| dt ∀[c, d] ⊂ [a, b] . c Let us recall that the 1-dimensional Hausdorff measure of a set E ⊂ Rn is defined as 1 1 1 H (E) := lim Hδ (E) = sup Hδ (E) , δ→0 δ∈(0,∞) 39 where ( ∞ ∞ ) 1 X [ Hδ (E) = inf diam(Ei): E ⊂ Ei, diam(Ei) 6 δ . i=1 i=1 Given a curve Γ, it holds that (1.72) H1(Γ) = length(Γ) , whether length(Γ) is finite or not (see Theorem 3.25). Assume now Γ = γ([a, b]) is C1 regular curve and define the measures 1 1 n ν(E) := H Γ(E) := H (Γ ∩ E) if E ∈ B(R ) , Z µ(E) := |γ0(t)| dt if E ∈ B([a, b]) . E Then, by (1.71) and (1.72), it follows that ν and µ are finite Radon measure respectively on (Rn, B(Rn)) and ([a, b], B([a, b]). Moreover, according to Definition 1.95,

(1.73) ν = γ#µ . In particular, by (1.73) and (1.69), it follows that

Z Z Z b Z b (1.74) ϕ dH1 = ϕ dν = ϕ ◦ γ dµ = ϕ(γ(t)) |γ0(t)| dt Γ Rn a a 0 n for each ϕ ∈ Cc (R ). We consider now two measure spaces and see the resulting structure on their carte- sian product. In particular we introduce Fubini and Tonelli theorems which general- izes the notion of iterated integration of Riemannian calculus. Deﬁnition 1.98 (Product σ-algebra). Let (X, M) and (Y, N ) be measure spaces. The product σ-algebra of M and N denoted by M × N is the σ-algebra generated in X × Y by G = {E × F : E ∈ M,F ∈ N } .

Remark 1.99. Let S ∈ M × N ; then for every x ∈ X the section Sx := {y ∈ Y : (x, y) ∈ S} belongs to N and for every y ∈ Y the section Sy := {x ∈ X :(x, y) ∈ S} belongs to M. In fact it is easily checked that the collections

SX := {S ∈ M × N : Sx ∈ N , ∀x ∈ X} , SY := {S ∈ M × N : Sy ∈ M, ∀y ∈ Y } are σ-algebras in X ×Y and contain G. Thus SX = SY = M ×N (see [R1, Theorem 8.2]). Remark 1.100. If (X, M) and (Y, N ) are complete measure spaces (see Remark 1.6), it need not be true that the product algebra M × N is complete. Indeed, suppose there exists E ∈ M, E 6= ∅ with µ(E) = 0; and suppose there exists F ⊂ Y such that F/∈ N . Then E × F ⊂ E × Y ,(µ × ν)(E × Y ) = 0, but, by Remark 1.99, E × F/∈ M × N . Therefore we will consider the completion (M × N )∗ in place of M × N . Theorem 1.101 (Fubini). Suppose that (X, M, µ) and (Y, N , ν) are complete σ-ﬁnite measure spaces. 40

(i) There exists a unique σ-ﬁnite measure on (X × Y, (M × N )∗), denoted with µ × ν and called product measure between µ and ν, such that (µ × ν)(E × F ) = µ(E) ν(F ) ∀ E ∈ M,F ∈ N , (where we deﬁne 0 ∞ = ∞ 0 = 0). ∗ (ii) If S ∈ (M1 × N ) then

Sy := {x ∈ X :(x, y) ∈ X × Y } ∈ N for ν-a.e. y ∈ Y,

Sx := {y ∈ Y :(x, y) ∈ X × Y } ∈ N for µ-a.e. x ∈ X,

y 7→ µ(Sy) is N -measurable, x 7→ µ(Sx) is M-measurable , Z Z Z (µ × ν)(S) = ν(Sx) dµ(x) = χS(x, y) dν(y) dµ(x) X X Y Z Z Z = µ(Sy) dν(y) = χS(x, y) dµ(x) dν(y) . Y Y X (iii) If u ∈ L1(X × Y, (M × N )∗, µ × ν) then y 7→ u(x, y) is ν-integrable for µ-a.e. x ∈ X,

x 7→ u(x, y) is µ-integrable for ν-a.e. y ∈ Y, Z Z Z u d(µ × ν) = u(x, y) dν(y) dµ(x) X×Y X Y Z Z = u(x, y) dµ(x) dν(y) . Y X Theorem 1.102 (Tonelli). Under the same assumptions of Theorem 1.101, if f : X × Y → [0, ∞] is (M × N )∗-measurable, then y 7→ u(x, y) is N -measurable for µ-a.e. x ∈ X,

x 7→ u(x, y) is M-measurable for ν-a.e. y ∈ Y, Z x 7→ u(x, y) dν(y) is M-measurable, Y Z y 7→ u(x, y) dµ(x) is N -measurable, X and Z Z Z Z Z u d(µ × ν) = u(x, y) dν(y) dµ(x) = u(x, y) dµ(x) dν(y) X×Y X Y Y X in the sense that either both expressions are inﬁnite or both are ﬁnite and equal. For the proofs of Theorems 1.101 and 1.102 we refer to [R1, Theorems 8.8 and 8.12] or [GZ, Theorem 6.46 and Corollary 6.47]. 41

Example 1.103 (Counterexample to Fubini-Tonelli’s theorem). Fubini and Tonelli’s theorems may fail when µ or ν is not σ-finite. Let X1 = X2 = [0, 1] , M := M1 ∩[0, 1] and N := P([0, 1]) be respectively the class of Lebesgue measurable sets and of all subsets in [0, 1], µ = L1 [0, 1] and ν = # be respectively the Lebesgue measure and 2 the counting measure in [0, 1]. Let u : [0, 1] → [0, ∞), u(x, y) := χD(x, y), where D := {(x, y) ∈ [0, 1]2 : x = y }. Then u is (M × N )-measurable, since D ∈ M × N . ∞ h 2 Indeed D = ∩h=1Qh with Qh := ∪i=1[(i − 1)/h, i/h] ∈ M × N . On the other hand Z Z u(x, y) dµ(x) = 0 ∀y ∈ [0, 1], u(x, y) dν(y) = 1 ∀x ∈ [0, 1], X Y so that Z Z Z Z u(x, y) dµ(x) dν(y) = 0, u(x, y) dν(y) = 1 . Y X X Y The failure of Theorems 1.101 (ii) and 1.102 is due to the fact that ν is not σ-additive. An interesting application of Fubini and Tonelli’s theorems is the following Cava- lieri’s principle. Proposition 1.104 (Cavalieri’s principle). Let (X, M) be a measure space, µ a positive measure on it and u : X → [0, ∞] be measurable. Let [0, ∞) 3 t 7→ µ({u > t}) denote the distribution function of u, that is, (1.75) µ({u > t} = µ ({x ∈ X : u(x) > t}) if t ∈ [0, ∞) . Let θ : [0, ∞) → [0, ∞) be (strictly) increasing such that θ(0) = 0, θ : [0,T ] → [0, ∞) is absolutely continuous for each T ∈ [0, ∞). Then Z Z ∞ (1.76) (θ ◦ u) dµ = θ0(t) µ({u > t}) dt . X 0 p In particular, if θ(t) = t with p > 1, then Z Z ∞ up dµ = p tp−1 µ({u > t}) dt . X 0 Proof. Let S := {x ∈ X : u(x) > 0} and consider the restriction measure of µ to S µ∗ := µ S. 1st step: Assume that µ∗ is not σ-finite. Then it is easy to see that both sides ∞ ¯ are ∞. Indeed, since S := ∪h=1{x ∈ X : u(x) > 1/h}, there exists an integer h such that µ({u > 1/h¯} = ∞. Thus Z Z 1 ¯ θ ◦ u dµ > θ ◦ u dµ > θ ¯ µ({u > 1/h}) = ∞ X {u> 1/h¯} h Z 1/h¯ Z ∞ ¯ 0 0 = µ({u > 1/h} θ (t) dt 6 θ (t) µ({u > t}) dt . 0 0 2nd step: Assume that µ∗ is σ-finite. Then, since Z Z Z ∞ Z ∞ θ ◦ u dµ = θ ◦ u dµ∗ and θ0(t) µ({u > t} dt = θ0(t) µ∗({u > t}) dt , X X 0 0 we can assume that µ itself is σ-finite. Let E := {(x, t) ∈ X × [0, ∞): u(x) > t} . 42

Et := {x ∈ X : u(x) > t} if t ∈ [0, ∞) . Then it can be proved that E ∈ M × B([0, ∞)) and the distribution function of u is then Z µ(Et) = χE(x, t) dµ(x) ∀t ∈ [0, ∞) . X Therefore, the right side of (1.76) is, by applying Fubini’s theorem with µ × L1, Z ∞ Z ∞ Z 0 0 µ(Et) θ (t) dt = χE(x, t) θ (t) dµ(x) dt 0 0 X (1.77) Z Z ∞ 0 = χE(x, t) θ (t) dt dµ(x) . X 0 For a given x ∈ X, observe that θ : [0,T ] → R is absolutely continuous for each 0 6 T < u(x) 6 ∞. Thus it follows that Z ∞ Z u(x) Z T 0 0 0 χE(x, t) θ (t) dt = θ (t) dt = lim θ (t) dt − (1.78) 0 0 T →u(x) 0 = lim θ(T ) = θ(u(x)) . T →u(x)− From (1.77) and (1.78), (1.76) follows. 1.7. Weak*-convergence of measures. Regularization of Radon measures in Rn. Weak*-convergence of measures. In this section we will assume that (X, d) is a l.c.s. metric space, the characteri- m m zation of the spaces of measures (Mloc(X)) and (M(X)) (see Corollary 1.78 and Theorem 1.83) as dual spaces induce on them a natural notion of weak*-convergence. Deﬁnition 1.105. Let (X, d) be a l.c.s. metric space. m (i) Let (νh)h and ν be measures in (Mloc(X)) ; we say that (νh)h locally weakly* converges ∗ to ν, and write νh * ν, if Z Z 0 lim u dνh = u dν ∀u ∈ Cc (X) . h→∞ X X m (ii) Let (νh)h and ν be measures in (M(X)) ; we say that (νh)h weakly* converges to ν if Z Z 0 lim u dνh = u dν ∀u ∈ C0(X) . h→∞ X X Remark 1.106. The weak*-convergence of ﬁnite Radon measures is called vague convergence, mainly in probability, and it is of considerable importance in applications.

Remark 1.107. Weak* convergence of (νh)h ⊂ M(X) to ν ∈ M(X) does not imply that limh→∞ νh(A) = ν(A), even when A = X as the following exercise shows.

Exercise: Let X = R and let νh := δh. Then prove that (νh)h weakly* converges to ν ≡ 0, but limh→∞ νh(R) = 1 6= 0 = ν(R). 43

Remark 1.108. The local weak* convergence is unique. More precisely m m ∗ Exercise: Given (νh)h ⊂ (Mloc(X)) and νi ∈ (M(X)) i = 1, 2. If νh*νi as h → ∞ for i = 1, 2, then ν1(E) = ν2(E) for each E ∈ Bcomp(X). (1) (m) (1) (m) m (Hint: If νh := νh , . . . , νh , νi := νi , . . . , νi : Bcomp(X) → R , let deﬁne 0 m the functionals Lh,Li :(Cc (X)) → R, if ψ = (ψ1, . . . , ψm), Z m Z m X (j) X (j) Lh(ψ) := ψj dνh ,Li(ψ) := ψj dνi . X j=1 X j=1 Prove that:

• both Lh and Li are linear continuous functionals according to Deﬁnition 1.69; 0 m • L1(ψ) = L2(ψ) for all ψ ∈ (Cc (X)) . Applying Riesz representation theorem 1.73, conclude that ν1 = ν2.) Proposition 1.109 (Locally weak* convergence vs. weak* convergence). Assume that (νh)h, ν ⊂ Mloc(X) . Then they are equivalent: ∗ (i) νh * ν and suph |νh|(X) < ∞; (ii)( νh)h, ν ⊂ M(X) and (νh)h weakly* converges to ν. . ∗ Proof. (ii)⇒ (i): it is trivial, by deﬁnition, that νh * ν. By the characterization of M(X) (see Theorem 1.83)

(1.79) |ν |(X) = kν k = kL k 0 0 h h νh (C0(X),k·k∞) and (1.80) 0 0 (νh)h weakly* converges to ν ⇐⇒ (Lνh )h weakly* converges to Lν in (C0(X), k·k∞) . 0 0 Thus, by (1.80),(Lνh )h is bounded in (C0(X), k · k∞) and, by (1.79), it follows that suph |νh|(X) < ∞. (i)⇒ (ii): By arguing as in the previuos implication, we have that the sequence 0 0 (Lνh )h is bounded in (C0(X), k · k∞) . By the sequential weak*-compactness of 0 0 bounded sets of (C0(X), k · k∞) and Theorem 1.83, we have that, up to a subsequence, 0 0 ∗ (1.81) (Lνh )h weakly* converges to Lν in (C0(X), k · k∞) . for some ν∗ ∈ M(X). By the assumptions and (1.81), we can infer that 0 Lν(u) = Lν∗ (u) ∀ u ∈ Cc (X) . 0 0 ∗ Since Cc (X) is dense in (C0(X), k · k∞), by the previous identity, we get that ν = ν and then the desired conclusion. Remark 1.110. The weak*-convergence of Radon measures if stable with respect to the push-forward of Radon measures. Indeed, let X, Y be l.c.s. metric spaces, f : X → Y continuous and proper: the continuity of f ensures that f −1(B) whenever 0 0 B ∈ B(Y ), and since f is proper the spaces Cc (Y ) and C0(Y ) are continuously 0 0 mapped in Cc (X) and C0(X) respectively by u 7→ u ◦ f. Thus, if the sequence (νh)h of Radon measures on X locally weakly* converges to the measure ν, then the 44 sequence (f#νh)h locally weakly* converges to f#ν, and the same statement holds for ﬁnite Radon measures and weak*-convergence. Example 1.111 (Blow-ups of a curve in Rn). A fundamental idea in GMT is the existence of tangent spaces to irregular submanifolds in terms of weak*-convergence of suitable Radon measures. This idea will be developed later and we now sketch it with an example. Let Γ be a C1 regular curve of Rn, that is Γ = γ([a, b]) for a C1 n 0 curve γ :[a, b] → R , injective with |γ (t)|= 6 0 for each t ∈ [a, b]. Given t0 ∈ (a, b), 0 the tangent space to Γ at x0 = γ(t0) is the line π = {s γ (t0): s ∈ R}. Consider now 1 Γ as a Radon measure, looking at ν = H Γ, and deﬁne the blow-ups νx0,r of ν at x0, setting 1 ν := (Φ ) (H1 Γ) , x0,r r x0,r # y − x with Φ (y) := 0 if y ∈ n. x0,r r R Exercise: Prove that Γ − x ν (E) = H1 0 (E) ∀ E ∈ B( n) . xo,r r R (Hint. Use (1.69) and (1.73).) ∗ Let us check that the property of π to be the tangent space implies that νx0,r * 1 0 H π. Indeed, by (1.74), if ϕ ∈ Cc (R), Z Z Z b 1 y − x0 1 1 γ(t) − γ(t0) 0 ϕ dνx0,r = ϕ dH (y) = ϕ |γ (t)| dt Rn r Γ r r a r Z (b−t0)/r γ(t0 + rs) − γ(t0) 0 = ϕ |γ (t0 + rs)| ds −(t0−a)/r r Z Z 0 0 1 + → ϕ (s γ (t0)) |γ (t0)| ds = ϕ dH , as r → 0 . R π

Other interesting examples of weak*-converging sequences of measures illustrating a wide variety of behaviours can be found in [Mag, Examples 4.20-4.23].

We now characterize the local weak*-convergence of positive Radon measures in terms of evaluation on sets and introduce a criterion about the narrow convergence of positive ﬁnite Radon measures. Let us ﬁrst introduce an useful criterion about foliations of Borel sets. Lemma 1.112 (Foliations of Borel sets for positive Radon measures). Let (X, d) be a l.c.s. metric space. If {Et}t∈I is a disjoint family of Borel sets in X, indexed over some set I, and µ is a positive Radon measure on (X, B(X)). Then µ(Et) > 0 for at most countably many t ∈ I. Proof. By Lemma 1.17, we can assume there exists an increasing sequence of compact ∞ sets (Kh)h of (X, d) such that X = ∪h=1Kh. Let Ih := {t ∈ I : µ(Et ∩ Kh) > 1/h}. Observe that ∞ (1.82) {t ∈ I : µ(Et) > 0} = ∪h=1Ih . 45

∞ Indeed, it is trivial that ∪h=1Ih ⊂ {t ∈ I : µ(Et) > 0}. Let us then prove the reverse inclusion. Assume that µ(Et) > 0 for some t ∈ I. Then there exists an integer h0 such that µ(Et) > 1/h0. On the other hand, since limh→∞ µ(Et ∩ Kh) = µ(Et), there exists an integer h > h0 such that µ(Et ∩ Kh) > 1/h0 > 1/h. Thus t ∈ Ih and (1.82) follows. Let us now show that Ih is ﬁnite with #(Ih) 6 hµ(Kh). Therefore, by (1.82), the proof will be accomplished. Let J ⊂ Ih be ﬁnite. Then X #(J) µ(K ) µ (∪ (E ∩ K )) µ (∪ (E ∩ K )) = µ((E ∩ K )) . h > t∈Ih t h > t∈J t h t h > h t∈J Theorem 1.113 (Characterization of the locally weak* convergence of positive Radon measures). Let (µh)h and µ be positive Radon measures on (X, B(X)).Then the following are equivalent. ∗ (i) µh * µ as h → ∞. (ii) If K compact and A open, then

(1.83) µ(K) > lim sup µh(K) , h→∞

(1.84) µ(A) lim inf µh(A) . 6 h→∞

(iii) If E ∈ Bcomp(X) with µ(∂E) = 0, then

µ(E) = lim µh(E) . h→∞ ∗ Moreover, if µh*µ as → ∞, then for every x ∈ sptµ there exists (xh)h ⊂ X with

(1.85) lim xh = x, xh ∈ sptµh ∀ h ∈ N . h→∞ Proof. (i) ⇒ (ii): Let K0 and A0 be respectively a compact and open set such that 0 0 0 K ⊂ A . By Urysohn’s lemma 1.65 there exists a function ϕ ∈ Cc (X) such that χK0 6 ϕ 6 χA0 , then Z Z 0 0 0 0 µh(K ) 6 ϕ dµh 6 µh(A ) ∀ h, µ(K ) 6 ϕ dµ 6 µ(A ) . X X By (i) and the previous inequality we have Z 0 0 (1.86) lim sup µh(K ) 6 ϕ dµ 6 µ(A ) , h→∞ X Z 0 0 (1.87) lim inf µh(A ) > ϕ dµ > µ(K ) . h→∞ X Setting K0 = K in (1.86) and passing to the inﬁmum on all open sets A0 ⊃ K, by Theorem 1.14 (i), (1.83) follows. Setting A0 = A in (1.87) and passing to the supremum on all compact sets K0 ⊂ A, by Theorem 1.14 (ii), (1.84) follows. (ii) ⇒ (iii): Notice that

◦ ◦ ◦ ¯ µ(E) 6 µ(E) 6 µ(E) = µ(E) + µ(∂E) = µ(E) , 46 then, since E¯ is compact, combining (ii) and the monotonicity of µ ◦ ◦ µ(E) = µ(E) lim inf µh(E) lim inf µh(E) 6 h→∞ 6 h→∞ ¯ ¯ 6 lim sup µh(E) 6 lim sup µh(E) 6 µ(E) = µ(E) . h→∞ h→∞

In particular it follows that µ(E) = lim infh→∞ µh(E) = lim suph→∞ µh(E), and then the desired conclusion. 0 (iii) ⇒ (i): Let ϕ ∈ Cc (X), ϕ > 0. By Lemma 1.112, there exists I ⊂ [0, ∞) such that L1([0, ∞) \ I) = 0 µ({ϕ = t}) = 0 ∀t ∈ I. Being ϕ continuous, ∂{ϕ > t} ⊆ {ϕ = t} ∀t ∈ [0, ∞) . Hence (iii) implies that

µ({ϕ > t}) = lim µh({ϕ > t}) ∀t ∈ I. h→∞

Let fh, f : [0, ∞) → R (h = 1, 2,... ), fh(t) := µh({ϕ > t}), and f(t) := µ({ϕ > t}). Then, being fh and f nondecreasing functions, fh and f are Borel measurable and 1 lim fh(t) = f(t) L -a.e. t ∈ [0, ∞] , h→∞

|fh(t)| µ(sptϕ) χ[0,sup n ϕ](t) ∀t ∈ [0, ∞) . 6 R By dominated convergence theorem and Cavalieri’s principle (1.76) with θ(t) = t, we have Z Z ∞ Z ∞ Z ϕ dµ = µ({ϕ > t}) dt = lim µh({ϕ > t}) dt = lim ϕ dµh . X 0 h→∞ 0 h→∞ X

If ϕ has a general sign, we can decompose it as ϕ = ϕ+ − ϕ− and apply the previous argument to ϕ+ and ϕ−. We ﬁnally prove (1.85) and let us prove that for every > 0 there exists h¯ ∈ N such that sptµh ∩B(x, ) 6= ∅. By contradiction, there exists 0 > 0 and an increasing sequence of integers (hk)h with limk→∞ hk = ∞ such that sptµhk ∩ B(x, 0) = ∅ for each k. By (1.84), it follows that

µ(B(x, 0)) lim inf µh (B(x, 0)) = 0 , 6 k→∞ k which contradicts the fact that x ∈ sptµ and then µ(B(x, 0)) > 0. Remark 1.114. (Limits points of support points and uniform lower bounds [Mag, ∗ Remark 4.28]) If µh*µ, xh ∈ sptµh for every h ∈ N, and xh → x, then it is not true, in general, that x ∈ sptµ. For instance, if X = R, the sequences 1 1 1 µ = 1 − δ + δ , x = . h h 1 h 1/h h h The implication becomes true as soon as some kind of uniform lower bound on the measure assigned by the µh around their support points is assumed. More precisely, let (µh)h be a sequence of positive Radon measures on X, such that, for every r > 0,

(1.88) lim sup inf {µh(B(x, r)) : x ∈ sptµh} > 0 . h→∞ 47

∗ Under this assumption, we claim that, if µh*µ, xh → x, and xh ∈ sptµh , then x ∈ sptµ. Indeed, let c(r) denote the left-hand side of (1.88). For every r > 0, let h0 ∈ N be such that B(xh, r) ⊂ B(x, 2r) for every h > h0. By (1.83), and if necessary extracting a subsequence so as to exploit (1.88),

µ(B(x, 2r)) > lim sup µh(B(x, 2r)) > lim sup µh(B(xh, r)) > c(r) > 0 . h→∞ h→∞ By the arbitrariness of r, we find that x ∈ sptµ. Proposition 1.115 (Characterization of the narrow convergence of positive Radon measures). Let (µh)h be a sequence of positive, finite Radon measures on (X, B(X)) and assume the existence of a positive, finite Radon measure µ such that

(1.89) lim µh(X) = µ(X) and lim inf µh(A) µ(A) h→∞ h→∞ > for every A ⊂ X open set. Then Z Z 0 (NC) lim u dµh = u dµ ∀ u ∈ Cb (X) h→∞ X X 0 where Cb (X) denotes the class of all bounded continuous function u : X → R. In particular (µh)h weakly* converges to µ. Moreover if (NC) holds so does (1.89), that is (NC) and (1.89) are equivalent.

0 Proof. Let u ∈ Cb (X). Possibly replacing u by αu + β for suitable α, β ∈ R, we can assume without loss of generality that 0 6 u 6 1. We ﬁrst show that Z Z (1.90) lim inf v dµh > v dµ , h→∞ X X for each continuous function v : X → [0, ∞). Indeed, by Cavalieri’s principle (1.76) with θ(t) = t and Fatou lemma, we infer Z Z ∞ Z ∞ lim inf v dµh = lim inf µh({v > t}) dt > lim inf µh({v > t}) dt h→∞ X h→∞ 0 0 h→∞ Z ∞ > µ({v > t}) dt . 0 Let us recall the following Exercise: Prove that, if (ah) and (bh)h are sequences of real numbers such that

a 6 lim inf ah, b 6 lim inf bh, lim sup(ah + bh) 6 a + b , h→∞ h→∞ h→∞ for some a, b ∈ R, then there exist limh→∞ ah = a and limh→∞ bh = b. Setting Z Z Z Z ah = u dµh, a = u dµ, bh = (1 − u) dµh, b = (1 − u) dµ , X X X X from (1.90) and the assumption limh→∞ µh(X) = µ(X), the proof is accomplished. Finally, if (NC) holds, by choosing as test function ϕ ≡ 1 in (NC), we get that

µ(X) = lim µh(X) . h→∞ 48

On the other hand, since (NC) also implies the local weak* convergence of (µh)h to µ, by Theorem 1.113, it follows that, for each open set A ⊂ X,

lim inf µh(A) µ(A) . h→∞ > Thus (1.89) follows.

Remark 1.116. Convergence (NC) is called narrow convergence of (µh)h to µ, and sometimes (by abuse of notation) weak-convergence of (µh)h to µ. It is a stronger convergence than the weak*-convergence: indeed, for instance, limh→∞ µh(X) = µ(X) is not granted for the weak*-convergence (recall the previous exercise). Note also that 0 0 0 Cc (X) ⊂ C0(X) ⊂ Cb (X) , 0 0 0 0 and Cc (X) = C0(X) = Cb (X) = C (X) if X is compact, and in this case the two notions of convergence coincide. On the other hand, for non-compact X the space 0 M(X) is not the dual of (Cb (X), k · k∞) as the following exercise shows. 0 Exercise: Let X = R and let F := {f ∈ C (R): ∃ f(∞) := lim|x|→∞ f(x) ∈ R}. 0 0 0 (i) Prove that F is a closed subspace of (Cb (R), k·k∞) and Cc (R) ⊂ C0(R) ⊂ F. (ii) Let L : F → R be the linear functional defined as L(f) := f(∞) and prove ˜ 0 0 that it can be extended to a functional L ∈ (Cb (R), k · k∞) by means of the Hahn-Banach theorem. (iii) Prove that there is no a finite Radon measure ν on R, that is an element ν ∈ M( ), such that L˜(f) = R f dν for each f ∈ C0( ). (Hint: Observe R R b R ˜ 0 that L(f) = 0 for each f ∈ Cc (R)). We now consider the local weak*-convergence of Radon vector measures and we point out some relationships with the local weak*-convergence of their total variation. Before let us introduce an alternative approximation to Theorem 1.14 for positive Radon measures, which will need in the proof. Lemma 1.117. Let (X, d) be a l.c.s. metric space, let µ be a positive Radon measure on (X, B(X)) and let E ∈ Bcomp(X) with µ(∂E) = 0. Then, for each > 0 there exist a compact set K and an open set A (which may be empty) such that ◦ A¯ ⊂ E ⊂ K and µ(K \ A) < . Remark 1.118. Lemma 1.117 is a refinement of the approximation of Radon measures on a l.c.s metric space by means of compact sets from below and open sets from above (see Theorem 1.14). Proof. Let us first observe that, since E¯ is compact, by Lemma 1.17, there is a relatively compact open set V such that (1.91) E¯ ⊂ V ⊂ V.¯ ◦ If h, m are integers and E 6= ∅ (otherwise choose A = ∅) , let us consider the sequences of sets ◦ 1 1 A := x ∈ E : d(x, ∂E) > ,K := x ∈ V¯ : d(x, E¯) . h h m 6 m 49

◦ ◦ ¯ ∞ Since (Ah)h is an increasing sequence of open sets, with Ah ⊂ E and E = ∪h=1Ah, it follows that ◦ lim µ(Ah) = µ(E) = µ(E) < ∞ . h→∞

Thus, if we take A = Ah for h large enough, then, µ(E \ A) < /2. In the same way, ∞ ¯ (1.92) (Km)m is a decreasing sequence of compact sets, ∩m=1Km = E, with ◦ (1.93) Km ⊃ E.

Indeed it is immediate, by construction, that (Km)m is a decreasing sequence of closed ∞ ¯ ¯ ¯ sets and, by (1.91), that ∩m=1Km = E. Since each Km ⊂ V and V is compact, Km is compact, too and then (1.92). On the other hand, if x0 ∈ E, then, by (1.91), ¯ x0 ∈ V and d(x0, E) = 0 < 1/m for each m. Since V is open and the function ◦ ¯ V 3 x 7→ d(x, E) is continuous, it follows that x0 ∈ Km for each m and then (1.93). Since µ(K1) < ∞, by (1.92), ¯ lim µ(Km) = µ(E) = µ(E) , m→∞ ◦ and, if we choose K = Km for m large enough, µ(K \ E) < /2 with K ⊃ E, by (1.93). m Theorem 1.119. Let (νh)h and ν be R -valued Radon vector measures, that is νh, ν : B(X) → Rm, and let µ a positive Radon measure on a l.c.s. metric space (X, d). ∗ (i) If νh*ν, then for every open set A ⊂ X

(1.94) |ν|(A) lim inf |νh|(A) . 6 h→∞ ∗ ∗ (ii) If νh*ν and |νh|*µ, then (1.95) |ν|(B) 6 µ(B) ∀ E ∈ B(X) .

Moreover, if E ∈ Bcomp(X) with µ(∂E) = 0, then

ν(E) = lim νh(E) . h→∞ ∗ (iii) If νh*ν and limh→∞ |νh|(X) = |ν|(X) < ∞, then (NC) holds with µh = |νh| ∗ and µ = |ν|. In particular |νh|*|ν|. 1 m Proof. (i): Let us recall that, by Riesz representation theorem 1.73, νh = (νh, . . . , νh ) = m wνh |νh|, ν = (ν1, . . . , νm) = wν |ν| with wνh , wν : X → R Borel measurable,

|wνh | = |wν| = 1, |νh|-a.e. and |ν|-a.e. in X, respectively, and, for each open set A ⊂ X,

Z 0 m (1.96) |ν|(A) = sup (wu, ψ)Rm d|ν| : ψ ∈ (Cc (A)) , kψk∞ 6 1 . X 0 By the assumptions, for each ϕ ∈ Cc (X) Z Z Z Z 0 ϕ wν d|ν| = ϕ dν = lim ϕ dνh = lim ϕ wνh d|νh| ∀ϕ ∈ Cc (X) . X X h→∞ X h→∞ X 50

0 m This implies that for each ψ ∈ (Cc (A)) with kψk∞ 6 1 Z Z m m (wν, ψ)R d|ν| = lim (wνh , ψ)R d|νh| 6 lim inf |νh|(A) X h→∞ X h→∞ 0 m Since ψ ∈ (Cc (A)) with kψk∞ 6 1 is arbitrary, by (1.96), (1.94) follows. (ii): Let A ⊂ X be a relatively compact open set and, At := {x ∈ A : d(x, ∂A) > t} 0 and let u ∈ Cc (A) such that χAt 6 u 6 χA. Then, by (1.94), we have Z Z |ν|(At) 6 lim inf |νh|(At) 6 lim inf u d|νh| = u dµ 6 µ(A) . h→∞ h→∞ X X + By letting t → 0 , we get |ν|(A) 6 µ(A), and since A is arbitrary, inequality (1.95) follows from Theorem 1.14. Let us now prove that ν(E) = limh→∞ νh(E) whenever E ∈ Bcomp(X) with µ(∂B) = 0. Given > 0, by Lemma 1.117, we ﬁnd an open set ◦ ¯ A and a compact set K such that A ⊂ E ⊂ K and µ(K \ A) 6 . Then, for every ◦ 0 u ∈ Cc (K), 0 6 u 6 1 with u = 1 on A, we ﬁnd Z Z

u dνh − νh(E) 6 |u − χE| d|νh| 6 |νh|(K \ A) , X X Z

u dν − ν(E) 6 |ν|(K \ A) 6 µ(K \ A) , X Z Z

lim u dνh − u dν = 0 . h→∞ X X ∗ Since |νh|*µ and K \ A is compact, by (1.83), we have lim suph→∞ |νh|(K \ A) 6 |ν|(K \ A) 6 µ(K \ A). Recalling µ(K \ A) 6 , we thus conclude that Z Z Z

lim sup |νh(E) − ν(E)| 6 lim sup νh(E) − u dνh + lim sup u dνh − u dν h→∞ h→∞ X h→∞ X X Z

+ lim sup u dν − ν(E) 6 2, ∀ > 0 . h→∞ X

(iii): Without loss of generality, we can assume that both |νh|(X) < ∞ for each m h and |ν|(X) < ∞, that is that (νh)h and ν are measures contained in (M(X)) . Thus, by claim (i), it follows that the assumptions of Proposition 1.115 are satisﬁed and then the proof is accomplished. Remark 1.120. A typical application of statement (ii) of Theorem 1.119 (or also of statement (iii) of Theorem 1.113) is the following: let us consider an increasing family ¯ (At)t of relatively compact open sets labelled on an interval I such that As ⊂ At, for s < t. Then,

Exercise: µ(∂At) = 0 except for countably many t ∈ I. (Hint: Let (Vi)i be an increasing sequence of relaatively compact open sets such that ∞ X = ∪i=1Vi (see Lemma 1.17). Since ∂At with t ∈ I are pairwise disjoint, by the additivity of µ, prove that , for given > 0 and i ∈ N the set

{t ∈ I : µ(∂At) > , At ⊂ Vi} . Then deduce the desired conclusion.) 51

1 Hence, by Theorem 1.119 (ii) νh(At) → ν(At) for L -a.e. t ∈ I. The classical De La ValléePoussin compactness criterion for finite Radon measures easily follows by the sequential weak*-compactness of bounded sets in a dual space of a separable normed vector space (see, for instance, [SC, Theorem 3.30]) and the characterization of the space of finite Radon measures (Theorem 1.83).

m Theorem 1.121 (Weak*-compactness). If (νh)h is a sequence of R - valued ﬁ- m nite Radon measures on the l.c.s. metric space X, that is (νh)h ⊂ (M(X)) , with suph |νh|(X) < ∞, then it has a weakly*- converging subsequence. Moreover, the map ν 7→ |ν| is lower semicontinuous with respect to the weak*-convergence, that is, assume that (νh)h weakly*-converges to ν, then

|ν|(X) lim inf |νh|(X) . 6 h→∞ . The previous theorem can be used to get immediately a corresponding result in the frame of local weak*-convergence.

m Corollary 1.122 (Local weak* compactness). Let (νh)h be a sequence of R - valued m Radon measures on the l.c.s. metric space X, (νh)h ⊂ (Mloc(X)) , such that

sup{|νh|(K): h ∈ N} < ∞ for every compact K ⊂ X; then it has a locally weakly*-converging subsequence.

Proof. Let (Vi)i be a sequence of relatively compact open sets such that ∞ X = ∪i=1Vi and Vi ⊂ Vi+1 for each i ∈ N . i m For given i, let (νh)h be the sequence of R -valued finite Radon measures defined as i ¯ νh := νh Vi . Since, by our assumption, i sup |νh|(X) = sup |νh|(Vi) < ∞ . h h By Theorem 1.121 and by means of a standard diagonal process, there exist a subse- i m quence (hk)k and finite Radon measure ν ∈ (M(X)) such that (νhk Vi)k weakly* converges to νi, that is Z Z Z i 0 (1.97) ϕ dνhk = ϕ d(νhk Vi) → ϕ dν for each ϕ ∈ C0(X), i ∈ N . Vi X X Moreover, by definition of Rm-valued finite Radon measure (see Definition 1.74 (ii)), m−1 thete exists a Borel measurable functions wi : X → S such that

i i i ν = wi|ν | with |ν | positive ﬁnite Radon measure on X, for each i ∈ N . Let us now prove that, for each i ∈ N, i i+1 ν Vi = ν Vi , that is, there exists a ﬁnte positive Radon measure on X such that i i+1 (1.98) |ν | Vi = |ν | Vi = µ and wi(x) = wi+1(x) µ-a.e. x ∈ Vi . 52

0 Fix i ∈ N and let us consider a function ϕ ∈ Cc (Vi) . Then we can infer that Z Z Z Z

ϕ dνhk Vi = ϕ dνhk = ϕ dνhk = ϕ d(νhk Vi+1) . X Vi Vi+1 X Passing to the limit as k → ∞ in the previous identity, by (1.97), we get Z Z Z i i i+1 ϕ wi d|ν | = ϕ dν = ϕ dν X X X Z i+1 0 = ϕ wi+1 d|ν | for each ϕ ∈ Cc (Vi), i ∈ N . X

Let wi = (wi,1, . . . , wi,m), then we can rewrite the previous identity as Z Z Z Z i i i+1 i+1 (1.99) ϕ wi,j d|ν | = ϕ wi,j d|ν | = ϕ wi+1,j d|ν | = ϕ wi+1,j d|ν | Vi X X Vi 0 for each ϕ ∈ Cc (Vi), i ∈ N, j = 1, . . . , m, which implies that, for each u = (u1, . . . , um) ∈ 0 m (Cc (Vi)) and for each i ∈ N, Z Z i i+1 (1.100) (wi, u)Rm d|ν | = (wi+1, u)Rm d|ν | X X By (1.100) and the representation of the total variation of a Rm-valued Radon measure (see (1.33))we get that i i+1 (1.101) |ν |(A) = |ν |(A) for each open set A ⊂ Vi . By the approxomiation with open sets for a positive Radon measure, we can infer that i i+1 (1.102) |ν | Vi = |ν | Vi = µ . Therefore, by (1.102), we can rewrite (1.99) as follows Z Z 0 (1.103) ϕ wi,j dµ = ϕ wi+1,j dµ for each ϕ ∈ Cc (Vi), i ∈ N, j = 1, . . . , m . Vi Vi By Remark 1.64 , for given i ∈ N, for each Borel set E ⊂ Vi there exists a sequence 0 (ϕh)h ⊂ Cc (Vi), such that 1 (1.104) ϕh → χE in L (Vi, µ) and |ϕh(x)| 6 1 for each x ∈ Vi . By (1.103), (1.104) and the Lebesgue dominated convergence theorem, we can infer that Z Z wi,j dµ = wi+1,j dµ for each Borel set E ⊂ Vi, i ∈ N, j = 1, . . . , m , . E E which is equivalent to

(1.105) wi(x) = wi+1(x) µ-a.e. x ∈ Vi . By (1.102) and (1.105), (1.98) follows. Let us now deﬁne a positive Radon measure µ : B(X) → [0, ∞] and a Borel measurable function w : X → Sm−1 as follows: i µ(E) := |ν |(E) if E ∈ B(Vi) for some i and w(x) := wi(x) if x ∈ Vi for some i . 53

By (1.98) both µ and w are well deﬁned. Let us deﬁne the Rm-valued Radon measure m ν : Bcomp(X) → R Z ν(E) = w dµ if E ∈ Bcomp(X) . E

Then by (1.97), it follows that (νhk )k locally weak* converges to ν and we accomplish the proof.

Regularization of Radon measures in Rn.

In this section we assume that X = Rn, endowed with the Euclidean metric, and we are going to deal with the approximation of a given Rm-valued Radon vector measure ν on Rn by means of a sequence of Rm-valued Radon vector measures ν = Ln on n h wh R ∞ n m ∗ (see (1.19)) with wh ∈ C (R , R ) and νh*ν. p n 0 n We saw that, given f ∈ L (R ) with 1 6 p < ∞, there exists (fh)h ⊂ Cc (R ) p n such that fh → f in L (R ) (see Theorem 1.63). Since the differential structure of Rn, we are going to improve this approximation, looking for an approximation by regular C∞-functions on Rn. A powerful tool for getting such a goal is the so-called approximation by convolution, which we will briefly recall here. Let us first recall the notion of support for a (Lebesgue) measurable function. Let us recall (see (1.6)) that given a function f : Rn → R, its support is the set (S) spt(f) := {x ∈ Ω: f(x) 6= 0 } . This definition is not suitable for a (Lebesgue) measurable function f : Rn → R. Indeed we would like that this notion satisfies the following property: n f1 = f2 a.e. in R ⇒ spt(f1) = spt(f2) , except for a negligible set. But this is not the case. Indeed

Example: Let f1 := χQ : R → R and f2 ≡ 0. Then it is clear that

f1 = f2 a.e. in R but spt(f1) = Q = R and spt(f2) = ∅ . Proposition 1.123 (Essential support of a function). Let f : Rn → R. Denote n Af := {ω ⊂ R : ω open set and f = 0 a.e. in ω} and let

Af := ∪ω∈Af ω .

Then Af is an open set and f = 0 a.e. in Af . The closed set n (ES) spte(f) := R \ Af is called the essential support of f in Rn. 54

Proof of Proposition 1.123. See [B, Proposition 4.17] n Remark 1.124. (i) From definition (ES), it follows that, if f1 = f2 a.e. in R , then spte(f1) = spte(f2). (ii) Definitions (S) and (ES) agree when the function is continuous. More precisely Exercise: If f : Rn → R is continuous, then n n R \ Af = {x ∈ R : f(x) 6= 0} . Definition 1.125 (Friedrichs’ mollifiers, 1944). A sequence of mollifiers is a sequence n of functions %h : R → R (h = 1, 2,... ) such that, for each h, ∞ n (Mo1) %h ∈ C (R );

(Mo2) spt(%h) ⊂ B(1/h); Z (Mo3) %h dx = 1 ; Rn n (Mo4) %h(x) > 0 ∀ x ∈ R . Example of mollifiers: It is quite simple constructing a sequence of mollifiers, starting from a given non vanishing function % : Rn → R satisfying ∞ n % ∈ Cc (R ) , spt(%) ⊂ B(1),% > 0 . For instance, let  1  exp if |x| < 1 %(x) := |x|2 − 1 .  0 if |x| > 1 ∞ n Then it is easy to see that % ∈ Cc (R ). Moreover we yield a sequence of mollifiers by defining

n n (1.106) %h(x) := c h %(h x) x ∈ R , h ∈ N and Z −1 c := % dx . Rn Remark 1.126. Observe that, without loss of generality, by (1.106), we can assume that a sequence of molliﬁers (%h)h satisﬁes the simmetry condition

n (1.107) %h(−x) = %h(x) ∀ x ∈ R , h ∈ N Notation: If A, B ⊂ Rn, A ± B denotes the set A ± B := {a ± b : a ∈ A, b ∈ B } Exercise: Prove that (i) if A is compact and B is closed, then A + B is closed; (ii) if A and B are compact so is A + B 55

1 n Proposition 1.127 (Definition and first mollifiers’ properties). Let f ∈ Lloc(R ) and n let (%h)h be a sequence of mollifiers. Define, for given h ∈ N and x ∈ R , Z n fh(x) = (%h ∗ f)(x) := %h(x − y) f(y) dy x ∈ R . Rn Then n (i) the function fh : R → R is well defined; n (ii) fh(x) = (%h ∗ f)(x) = (f ∗ %h)(x) for all x ∈ R and h ∈ N; 0 n (iii) fh ∈ C (R ) for each h; 0 n n (iv) if f ∈ C (R ), then %h ∗ f → f uniformly on compact sets of R , as h → ∞. th The function fh is called h - mollifier of f. Proof. See [B, Propositions 4.19 and 4,21] and [SC, Proposition 2.68 and Lemma 2.74] Remark 1.128. The symbol ∗ denotes the convolution product between two functions defined on the whole Rn. Notice also that the conclusions of Proposition 1.127 1 n 0 n still holds if f ∈ Lloc(R ) and % ≡ %h ∈ C (R ) satisfying (Mo2). Actually, it is possible to define the convolution product between two functions g ∈ Lp(Rn) with 1 n 1 6 p 6 ∞ and f ∈ L (R ) Z (g ∗ f)(x) := g(x − y) f(y) dy Rn and it holds that (see [GZ, Theorem 6.51]) p n (g ∗ f) ∈ L (R ) and kg ∗ fkLp(Rn) 6 kgkLp(Rn) kfkL1(Rn) . Theorem 1.129 (Friedrichs-Sobolev, approximation by convolution in Lp). Let f ∈ 1 n Lloc(R ) and (%h)h be a sequence of mollifiers. Then ∞ n (i) f ∗ %h ∈ C (R ) for each h ∈ N. p n (ii) kf ∗ %hkLp(Rn) 6 kfkLp(Rn) for each h ∈ N, f ∈ L (R ), for every p ∈ [1, ∞]. (iii) spt(f ∗ %h) ⊂ spte(f) + B(1/h) for each h ∈ N. p n ∞ n p n (iv) If f ∈ L (R ) with 1 6 p 6 ∞, then f ∗ %h ∈ C (R ) ∩ L (R ) for each p n h ∈ N, and f ∗ %h → f as h → ∞, in L (R ), provided that 1 6 p < ∞. Proof of Theorem 1.129. See [B, Proposition 4.20 and Theorem 4.22] and [SC, The- orem 2.70]. Historical notes: Mollifiers were introduced by K. Friedrichs in 1944, which are, according to P. Lax, a watershed in the modern theory of PDEs. However, S. Sobolev had used mollifiers in his epoch making 1938 paper [So] (the paper containing the proof of the Sobolev embedding theorem), as Friedrichs himself acknowledged in later papers.

Let us now come back to the approximation of a Radon vector measure. Let ν be m a R -valued Radon vector measure and (%h)h be a sequence of molliﬁers. Let us then n m deﬁne the sequence of functions ν ∗ %h : R → R (h ∈ N) as Z n (1.108) (ν ∗ %h)(x) := %h(x − y) dν(y) if x ∈ R . Rn 56

Theorem 1.130 (Approximation of Radon vector measures). Let ν = (ν1, . . . , νm) n be a Radon vector measure in R and let (%h)h be a sequence of molliﬁers which also satisﬁes simmetry condition (1.106). Then n m ∞ n m α (i) The functions ν ∗ %h : R → R belong to (C (R )) and ∇ (ν ∗ %h) = α n ν ∗ ∇ %h for any α ∈ N . (ii) If ν := Ln , the sequence of measures (ν ) locally weakly*converges in n h ν∗%h h h R to ν as h → ∞ and, for each E ∈ B(Rn), it holds the estimate Z |νh|(E) = |ν ∗ %h| dx 6 |ν|(I1/h(E)) . E

n (iii) The sequence of measures (|νh|)h locally weakly*converges in R to |ν| as h → ∞.

Proof. (i): The first statement can be easily proved as in the proof of Theorem 1.129 (i), by induction on the length of α by using a difference quotient argument and passing to the limit under the integral. (ii): Let us first note that, using Fubini’s theorem and simmetry condition (1.106), it is easily seen that

Z Z (1.109) (ν ∗ %h) v dx = (v ∗ %h) dν Rn Rn 1 n 0 n for each v ∈ L (R ). Thus, if u ∈ Cc (R ), by (1.109) and Proposition 1.127 (iv), Z Z Z Z u dνh = u (ν ∗ %h) dx = (u ∗ %h) dν → u dν as h → ∞ . Rn Rn Rn Rn n Let E ∈ B(R ) and let us estimate |νh|: by (1.20) and for Fubini’s theorem Z Z Z

|νh|(E) = |ν ∗ %h| dx = %h(x − y) dν(y) E E Rn Z Z Z Z 6 %h(x − y) d|ν|(y) dx = %h(x − y) dx d|ν|(y) E Rn Rn E Z Z 6 %h(x − y) dx d|ν|(y) 6 |ν|(I1/h(E)) . I1/h(E) E

(iii): Let At := U(t) if t ∈ I := (0, ∞). By Remark 1.120, we can ﬁnd an increasing n n ∞ sequence of open sets Ak b R such that R = ∪k=1Ak and |ν|(∂Ak) = 0 for each k ∈ N. As a consequence of (ii), ¯ lim sup |νh|(Ak) 6 |ν|(I0(Ak)) = |ν|(Ak) = |ν|(Ak) for each k . h→∞

On the other hand, Theorem 1.119 (i) implies that lim infh→∞ |νh|(A) > |ν|(A) for n any open set A ⊂ R . By Proposition 1.115, we infer that (|νh|)h weakly* converges to |ν| in Ak, and since k is arbitrary the statement follows. 57

2. Differentiation of Radon measures ([AFP, Ma]) Motivation: In this section we are going to introduce the main results about the differentiation of measures which will be used later in the rectifiability and in the study of sets of finite perimeter. One of the main goals is to prove the following result. Theorem 2.1 (of Lebesgue points). : Let µ be a positive Radon measure on (Rn, B(Rn)) 1 n and let f ∈ Lloc(R , µ). Then µ-a.e. x ∈ spt(µ) there exists 1 Z (LP) lim |f(y) − f(x)| dµ(y) = 0 . r→0 µ(B(x, r)) B(x,r) Definition 2.2. A point x ∈ Rn for which (LP) holds is called a Lebesgue point of f. 2.1. Covering theorems and Vitali-type covering property for measures on Rn. We begin to introduce two types of covering theorems in X = Rn. The difference between them is that the first ones (Vitali’s coverings) apply to a larger class of coverings and a narrower class of measures whereas in the second type the coverings (Besicovitch’s coverings) are more restricted but the measures can be very general; for example all Radon measures on Rn are included. In both cases we first prove a geometric result on collections of balls in Rn and then apply it to get a Vitali-type covering theorem for measures. Let us begin with some notions on coverings in a general metric space. To begin with, let us agree that by disjoint family of subsets of a metric space (X, d) we mean a family F such that E ∩ F = ∅ whenever E,F ∈ F and E 6= F ; we set also ∪F := ∪E∈F E. Definition 2.3. Let (X, d) be a metric space. (i) A family F of closed balls of (X, d) is a cover of a set A ⊂ X if A ⊆ ∪F . (ii) A family F of closed balls of (X, d) is a fine cover of A, or also that F covers A in the sense of Vitali, if it is a cover of A and, for each x ∈ A, (2.1) inf {d(B): B ∈ F, x ∈ B} = 0 .

Vitali covering theorem and the Lebesgue measure.

Theorem 2.4 (Vitali covering theorem). Let G be a family of closed balls in Rn with D = sup {d(B): B ∈ G} < ∞ . Then there exists a (pairwise) disjoint family F ⊆ G, which is at most countable, such that ˆ ∪B∈GB ⊂ ∪B∈F B. where Bˆ is an enlargement of B, that is Bˆ = 5B. Before the proof of Vitali’s covering theorem, let us recall the Hausdorﬀ Maximal Principle (see [GZ, Theorem 1.4]). 58

Theorem 2.5 (Hausdorﬀ Maximal Principle). If S is a family of sets (or a collection of families of sets) and if ∪{E : E ∈ E} ∈ S for any subfamily E of S totally ordered with respect to the inclusion, that is with the property that

E1 ⊂ E2 or E2 ⊂ E1 whenever E1,E2 ∈ E . Then there exists E∗ ∈ S, which is maximal in the sense that it is not a subset of any other member of S. Proof of Theorem 2.4. Let us deﬁne the sequence of subfamilies of G D D G := B ∈ G : < d(B) j = 1, 2,... j 2j 6 2j−1 Then it is trivial to see that ∞ (2.2) G = ∪j=1Gj ,

0 (2.3) Gj ∩ Gj0 = ∅ ∀j 6= j .

Let us define inductively a subfamily Fj ⊂ Gj as follows. Let F1 ⊂ G1 be a maximal subcolletion of pairwise disjoint elements if G1 6= ∅. For if not, let F1 = ∅. Exercise: Prove that such a family F1 exists by means of the Hausdorff Maximal Principle. Assuming that F1, F2,..., Fj−1 have been chosen, let Fj be a maximal pairwise disjoint family of the family ∗ 0 0 j−1 Gj := B ∈ Gj : B ∩ B = ∅, ∀B ∈ ∪i=1 Fi ∗ if Gj 6= ∅; for if not let Fj := ∅. Let ∞ F = ∪j=1Fj . From (2.2) and (2.3), F ⊂ G and F is a disjoint collection of balls. Moreover F is at ◦ most countable, since the pairwise disjoint family of open balls {B : B ∈ F} has to be at most countable (why?). Let us observe that, j (2.4) for fixed B ∈ Gj ∃ B1 ∈ ∪i=1Fi such that B1 ∩ B 6= ∅ . Indeed, for if not, the family ∗ Fj = Fj ∪ {B1} ∗ would be a disjoint family of Gj , thus contradicting the maximality of Fj. Moreover D D (2.5) d(B) = 2 < 2 d(B ) , 6 2j−1 2j 1 which implies that ˆ (2.6) B ⊂ B1 .

Indeed, if B = B(x, r), B1 = B(x1, r1), let z ∈ B and, since (2.4), there exists y ∈ B ∩ B1. Thus, by (2.5),

|z − x1| 6 |z − y| + |y − x1| 6 d(B) + r1 6 2 d(B1) + r1 6 4 r1 + r1 = 5r1 . and then (2.6) follows. 59

Remark 2.6. (i) We emphasize that the point of the lemma is that the subcol- lection consists of countable disjoint elements, a very important consideration since countable additivity plays a central role in measure theory. (ii) If D = sup {d(B): B ∈ G} = ∞, Vitali’s covering may fail. For instance Exercise: Let G := {B(h): h ∈ N}, then prove that there is no a pairwise ˆ disjoint subfamily F ⊂ G such that ∪B∈GB ⊂ ∪B∈F B. (iii) If the cover G is composed of open balls, the conclusion of Vitali’s covering theorem still holds. Now we are going to show, even though F could not cover the whole A, at least, it covers almost all of A with respect to the Lebesgue measure, that is, the so- called Vitali covering property holds for the Lebesgue measure. Theorem 2.7 (Vitali covering property for the Lebesgue measure). Let G be a family of closed balls in Rn, which is a fine cover of a (possibly non measurable) set A ⊂ Rn in Rn. Then there exists a disjoint subfamily F ⊂ G, at most countable, such that Ln (A \ ∪F) = 0 , where Ln denotes the n-dimensional Lebesgue outer measure. Proof. 1st step: Suppose A is bounded with 0 < Ln(A) < ∞, otherwise we are done. Since Ln is a Borel regular outer measure, by Corollary 1.11, there is an open n set U0 ⊂ R such that U0 ⊃ A and n −n n (2.7) L (U0) 6 (1 + 7 ) L (A) . Let G0 := {B ∈ G : B ⊂ U0, d(B) 6 1} . Being G fine, G0 is still a fine cover of A. Thus, by Vitali’s covering theorem 2.4, there exists a disjoint, at most countable, subfamily F0 ⊂ G0 ⊂ G such that ˆ A ⊂ ∪B∈GB ⊂ ∪B∈F0 B. Then −n n −n n −n X n X n (2.8) 6 L (A) < 5 L (A) 6 5 L (5B) = L (B) . B∈F0 B∈F0

From (2.8), there exists a ﬁnite family of balls F1 := {B1,...,Bk1 } ⊂ F0 ⊂ G0 such that

k1 −n n X n (2.9) 6 L (A) 6 L (Bi) . i=1 Deﬁne k1 A1 := A \ ∪F1 = A \ ∪i=1Bi . n If L (A1) = 0, we are done. Otherwise, from (2.7) and (2.9), we have that

k1 n n k1 n X n L (A1) 6 L (U0 \ ∪i=1Bi) = L (U0) − L (Bi) (2.10) i=1 −n −n n n 6 (1 + 7 − 6 ) L (A) = u L (A) , 60

−n −n n k1 where 0 < u := 1 + 7 − 6 < 1. Now A1 ⊂ R \ (∪i=1Bi) and therefore we can ﬁnd an open set U1 such that

n k1 A1 ⊂ U1 ⊂ R \ (∪i=1Bi) , n −n n L (U1) 6 (1 + 7 ) L (A1) .

Arguing as above there are disjoint balls Bk1+1,...,Bk2 in G such that Bi ⊂ U1 for i = k1 + 1, . . . , k2 and, if A := A \ ∪k2 B = A \ ∪k2 B , 2 1 i=k1+1 i i=1 i

n n (2.11) L (A2) 6 u L (A1) .

Thus, from (2.10) and (2.11), it follows that there exists a ﬁnite family F2 :=

{B1,...,Bk2 } ⊂ G of disjoint balls such that F1 ⊂ F2 and n 2 n L (A \ ∪F2) 6 u L (A) .

After m steps, we have that there exist m ﬁnite families Fi := {B1,...,Bki } ⊂ G (i = 1, . . . , m) of disjoint balls such that F1 ⊂ · · · ⊂ Fm

n m n (2.12) L (A \ ∪Fm) 6 u L (A) . n If L (A \ ∪Fm) = 0 for some m, the procedure stops and we are done. Otherwise we construct an increasing sequence (Fm)m of finite disjoint subfamilies of G such that (2.12) holds for each m ∈ N. Let us define ∞ F := ∪m=1Fm . then, from (2.12), it follows that n n m n L (A \ ∪F) 6 L (A \ ∪Fm) 6 u L (A) ∀ m ∈ N . Taking the limit as m → ∞ in the previous inequality, we complete the proof. n ∞ 2nd step: Assume A unbounded. We can write R = ∪i=1Qi where (Qi)i is a ◦ ◦ n sequence of closed cubes of R such that Qi ∩ Qj = ∅ if i 6= j. Applying the first step ◦ to A ∩ Qi, for given i, and noticing that ◦ n ∞ L A \ ∪i=1Qi = 0 we complete the proof. Remark 2.8. (i) A simple analysis of the proof of Theorem 2.7 yields that it still holds true for a fine cover G composed of open balls. (ii) All that we really used of the Lebesgue measure in the proof of Theorem 2.7 was the equality Ln(B(x, 5r)) = 5n Ln(B(x, r)) in fact only the inequality ” 6”. It is rather straightforward to modify the above proof to see that the theorem remain valid if Ln is replaced by any Radon measure µ on Rn such that for some τ ∈ (1, ∞),

µ(B(x, τr)) n (2.13) lim sup < ∞ µ − a.e. x ∈ R . r→0 µ(B(x, r)) 61

Moreover, the balls can be replaced by more general families of closed sets and Rn by more general spaces, see Federer [Fe, 2.8] for example. However, the above theorem is not valid even for all very nice Radon measures on Rn, as the following example shows.

Example 2.9 (Vitali covering property does not hold for all Radon measures in Rn [Ma]). Let µ be the Radon outer measure in R2 defined by 1 2 µ(E) := L ({x ∈ R :(x, 0) ∈ E}) if E ⊂ R . It is easy to see that µ = H1 A where H1 denotes the 1-dimensional Hausdorff measure on R2 and A := {(x, y) ∈ R2 : y = 0 }. The family of balls G := {B((x, y), y) : 0 < y < ∞} covers finely A. But for any countable subfamily F ⊂ G µ(A ∩ (∪F)) = 0 . Thus a Vitali covering property like the Lebesgue measure (see Theorem 2.7) cannot hold for a general Radon measure in Rn. Here A touches only the boundaries of the balls of G. By a slight modification, as suggested in [Ma], we could find a family G such that each point of A is an interior point of arbitrarily small balls of G and yet the conclusion of Theorem 2.7 fails. Exercise (suggested by R. Serapioni). Let µ be the the Radon measure as before with A := {(x, 0) : x ∈ [0, 1]}. Let G be the family of open balls

G := {Ux,n : x ∈ [0, 1], n ∈ N} with 1 1 U := U x, , r and r := + α e−n , x,n n n n n where α ∈ (0, 1] to be ﬁxed later. (i) G is a ﬁne cover√ of A; − n (ii) µ(A ∩ Ux,n) 6 3α e 2 for each x ∈ [0, 1] and n; (iii) for a given n, the number of disjoint balls Ux,n with x ∈ [0, 1] is at most n/2; (iv) let F ⊂ G be a disjoint, countable family , then, by (ii) and (iii), ∞ √ ∞ X n − n √ X − n 1 µ (∪F) 3α e 2 α n e 2 < 6 2 6 2 n=1 n=1 for α small enough. Thus, by (iv), for any disjoint subfamily F ⊂ G, 1 µ (A \ (∪F)) = µ(A) − µ (A ∩ (∪F)) µ(A) − µ (∪F) . > > 2 However, if we should require that each point of A is the centre (in fact, not too far from the centre would be enough) of arbitrarily small balls of G we would get the conclusion of Theorem 2.7. Next we shall develop a covering theorem of this type. 62

Besicovitch’s covering theorem and Radon measures on Rn. Again we shall ﬁrst introduce a theorem called Besicovitch’s covering theorem, which originated from Besicovitch [Be1] and [Be2]. Besicovitch’s covering theorem 2.10. There are integers P (n)and Q(n) depending only on n with the following properties. Let A be a bounded subset of Rn, and let G be a family of closed balls such that each point of A is the centre of some ball of G. (i) There is a ﬁnite or countable subfamily F ⊂ G which covers A and every point of Rn belongs to at most P (n) balls of F, that is, X χA 6 χB 6 P (n) . B∈F

(ii) There are subfamilies F1,..., FQ(n) ⊂ G covering A such that each Fi is disjoint, that is, Q(n) A ⊂ ∪i=1 (∪Fi) and 0 0 0 B ∩ B = ∅ for B,B ∈ Fi with B 6= B . Proof. See [Ma, Theorem 2.7]. We can now easily establish a Vitali-type covering theorem for arbitrary Radon measures on Rn. Theorem 2.11 (Vitali covering property for Radon measures). Let ϕ be a Radon o. m. in Rn, A ⊂ Rn (even not ϕ-measurable) and G a family of closed balls. Assume that G is a cover of A and (2.14) inf { r : B(x, r) ∈ G } = 0 ∀ x ∈ A. Then there is a disjoint subfamily F ⊂ G, at most countable, such that ϕ (A \ ∪F) = 0 . Proof. 1st step: Suppose ﬁrst A is bounded and we may assume 0 < ϕ(A) < ∞. By Theorem 1.14, there is an open set U0 such that A ⊂ U0 and −1 ϕ(U0) < (1 + (4Q(n)) )ϕ(A) , where Q(n) is as in Besicovitch’s covering theorem 2.10 . By that theorem we can ﬁnd subfamilies F1,..., FQ(n) ⊂ G such that each Fi is disjoint and Q(n) A ⊂ ∪i=1 ∪ Fi ⊂ U0 . Then Q(n) X ϕ(A) 6 ϕ (∪Fi) i=1 and consequently there is an i∗ with

ϕ(A) < Q(n)ϕ (∪Fi∗ ) . 0 Further, for some ﬁnite subfamily Fi∗ ⊂ Fi∗ we have 0 ϕ(A) < 2Q(n)ϕ ∪Fi∗ . 63

Letting 0 A1 = A \ ∪Fi∗ , 1 −1 we get with u = 1 − 4 Q(n) < 1 0 0 ϕ(A1) < ϕ U0 \ ∪Fi∗ = ϕ(U0) − ϕ ∪Fi∗ 1 1 < 1 + Q(n)−1 − Q(n)−1 ϕ(A) = u ϕ(A) 4 2 We can now continue by the same principle as in the proof of Theorem 2.7. 2nd step: Assume A unbounded. We may modify the last step of the proof of Theorem 2.7 making use of the fact that, by Lemma 1.112, ϕ(H) can be positive for at most countably many parallel hyperplanes H. More precisely, if i = 1, . . . , n and t ∈ R, denote by (i) n Ht := {x = (x1, . . . , xn) ∈ R : xi = t} , and n (i) o Ni := t ∈ R : ϕ Ht > 0 .

Then Ni is at most countable for each i = 1, . . . , n. Therefore, for each i = 1, . . . , n, (i) there exists a sequence (tk )k∈Z ⊂ R such that (i) 1 (i) k 6 tk < k + and ϕ H (i) = 0 ∀ k ∈ Z . 2 tk Let us deﬁne the closed rectangles

n (i) (i) Qk := Πi=1[tk−1, tk ] if k ∈ Z . Then, it is easy to see that

◦ ◦ n 0 R = ∪k∈ZQk, Qk ∩ Qk0 = ∅ if k 6= k and ◦ n ϕ R \ ∪k∈ZQk = 0 .

◦ Applying the ﬁrst step to A ∩ Qk we complete the proof. Remark 2.12. The above theorem still holds true for families of open balls if ϕ is the Lebesgue measure (in this case it reduces to the classical Vitali covering theorem 2.7); if ϕ is a general Radon measure, further conditions have to be imposed: for instance, we may require that for every x ∈ A and > 0 the cardinality of the balls of G centred at x with radius less than > 0 is more than countable, or that this property fails for a ϕ-negligible set of points. In this case, in fact, it is possible to select only those balls B such that ϕ(∂B) = 0 (thus getting again a ﬁne cover) and apply Theorem 2.11 to the cover given by the closure of the selected balls. However. it is interesting to note that Theorem 2.11 does not hold in its full generality for families of open balls, as the next example shows. 64

Example 2.13. Let Q = {qi : i ∈ N} and consider, respectively, the Radon (outer) measure ϕ on R and family of open intervals G deﬁned as ∞ X 1 ϕ := δ , G := {(a, b): a, b ∈ , a < b} . 2i qi Q i=1 Then G can be also meant as a family of open balls in R, which ﬁnely covers Q. On the other hand, for each countable, disjoint subfamily F ⊂ G it follows that

ϕ(Q \ ∪F) > ϕ(∪U∈F ∂U) > 0 .

Historical notes:([dG, I.3]) The most classical covering theorem in differentiation theory is that of Vitali [Vitali2], which has traditionally been the tool to obtain the Lebesgue differentiation theorem in Rn . In its original form the theorem of Vitali refers to closed cubic intervals and the Lebesgue measure. Later on Lebesgue [Le2] and others gave it a less rigid geometric form replacing cubes by other sets ”regular” with respect to cubes, keeping always the restriction to the Lebesgue measure. This restriction originates in the type of proof of the theorem, essentially that given by Banach [Ba], which requires that homothetic sets have comparable measures. Also Caratheodory’s proof [C] is based on this property although it is a little different. Besicovitch [Be1, Be2] and A.P. Morse [Mor] were the first in obtaining similar covering lemmas for more general measures in order to prove differentiability properties analogous to that of the Lebesgue theorem.

2.2. Derivatives of Radon measures on Rn. Lebesgue-Besicovitch diﬀeren- tiation theorem for Radon measures on Rn. In this section the environment metric space will be X = Rn and M = B(Rn). Notation: If x ∈ Rn, r > 0, ν and µ are positive Radon measures on Rn, then we interpret  ν(B(x, r)) ν(B(x, r))  if x ∈ spt(µ) := µ(B(x, r)) ; µ(B(x, r))  ∞ if x ∈ Rn \ spt(µ) R f dµ if f ∈ L1 ( n, µ), A ⊂ n bounded with µ(A) > 0, R f dµ := A , loc R R A µ(A) Derivatives of positive Radon measures. Let ν and ν be a positive Radon measure on Rn and assume that ν << µ. Then, 1 n from the Radon-Nikodym theorem for Radon measures, there exists w ∈ Lloc(R , µ), dν w 0 such that w = , that is, > dµ Z n ν(E) := w dµ ∀ E ∈ B(R ) . E Question: Z ν(B(x, r)) n (D) ∃ lim = lim w(y) dµ(y) = w(x) µ − a.e. x ∈ R ? r→0 µ(B(x, r)) r→0 B(x,r) 65

Deﬁnition 2.14. Let ν and µ be positive Radon measures on Rn. (i) The upper and lower derivatives of ν with respect to µ at a point x ∈ Rn are deﬁned by ν(B(x, r)) (2.15) Dµν(x) = lim sup ∈ [0, +∞]; r→0 µ(B(x, r))

ν(B(x, r)) (2.16) Dµν(x) = lim inf ∈ [0, +∞] . r→0 µ(B(x, r)) (ii) At a point x ∈ Rn where the limit exists, we deﬁne the derivative of ν with respect to µ by

(2.17) Dµν(x) = Dµν(x) = Dµν(x) . The basic differentiation result for positive Radon measures on Rn is contained in the following. Theorem 2.15 (Differentation for positive Radon measures). Let ν and µ be positive Radon measures on Rn. (i) The derivative Dµν(x) exists and is finite (that is Dµν(x) ∈ [0, ∞)) for µ-a.e. x ∈ Rn. n (ii) The function Dµν : R → [0, +∞] is Borel measurable, by defining Dµν = ∞ on the possible µ-negligible set where it does not exist. (iii) Let n (2.18) A := {x ∈ R : ∃ Dµν(x) ∈ [0, ∞)} . For all Borel sets B ⊂ Rn Z (2.19) Dµν dµ = ν(A ∩ B) 6 ν(B) , B with equality if ν << µ. In this case dν dν D ν(x) = (x) = ac (x) µ-a.e. x ∈ n . µ dµ dµ R dλ denoting the Radon-Nikodym derivative of λ with respect to µ. dµ n (iv) ν << µ if and only if Dµν(x) < ∞ ν-a.e. x ∈ R . As a corollary we obtain the following fundamental result. Theorem 2.16 (Lebesgue-Besicovitch differentiation theorem). Let µ be a positive n 1 n Radon measure on R and let f ∈ Lloc(R , µ). Then Z n ∃ lim f(y) dµ(y) = f(x) µ − a.e. x ∈ R , r→0 B(x,r) that is, by definition, there exists a µ-negligible set N ⊂ Rn (i.e. µ(N) = 0) such that Z n (?) ∃ lim f(y) dµ(y) = f(x) ∀ x ∈ R \ N. r→0 B(x,r) 66

1 n Remark 2.17. If f ∈ Lloc(R µ), denote by n n [f] := g : R → R : g measurable, g = f µ- a.e. in R . Then the previous theorem actually states that there exists a µ-negligible set N(f) such that Z n ∃ lim g(y) dµ(y) = f˜(x) ∀ x ∈ R \ N(f), g ∈ [f] , r→0 B(x,r) and n f˜(x) = f(x) ∀ x ∈ R \ N(f) . Thus the limit in (?) provides a way to deﬁne the value of f at x that is independent of the choice of representative in the equivalence class of f. Notice also that (?) can be written as Z n ∃ lim (f(y) − f(x)) dµ(y) = 0 µ − a.e. x ∈ R . r→0 B(x,r) Lebesgue-Besicovitch diﬀerentiation theorem yields at once the following result about the density of a point with respect to a set.

Corollary 2.18 (Density of a set). Let µ be a positive Radon measure on Rn and let E ⊂ Rn be a measurable set. Then ( µ(E ∩ B(x, r)) 1 for µ-a.e. x ∈ E ∃ lim = , r→0 µ(B(x, r)) 0 for µ-a.e. x ∈ Rn \ E that is, µ-a.e. x ∈ E is a point of density 1 for E and µ-a.e. x ∈ Rn \ E is a point of density 0 for E.

Proof of Theorem 2.16. It is not restrictive to assume that f > 0. Otherwise we can decompose f = f + − f − and to apply the same argument to f + and f −. Let us deﬁne the Radon measure Z n ν(B) := f dµ if B ∈ B(R ) . B By applying Theorem 2.15 (iii) we get that Z Z n Dµν(x) dµ(x) = ν(B) = f dµ(x) ∀B ∈ B(R ) , B B whence n Dµν(x) = f(x) µ-a.e. x ∈ R . Proof of Theorem 2.1. For each % ∈ Q, apply Theorem 2.16 to conclude that there is n a set N% ⊂ R with µ(N%) = 0 such that Z n (2.20) lim |f(y) − %| dµ(y) = |f(x) − %| ∀ x ∈ R \ N% . r→0 B(x,r) n Thus, with N = ∪%∈QN%, we have µ(N) = 0. Moreover, for x ∈ R \ N, % ∈ Q, since |f(y) − f(x)| 6 |f(y) − %| + |% − f(x)| , 67

(2.20) implies that Z n lim sup |f(y) − f(x)| dµ(y) 6 2 |f(x) − %| ∀ x ∈ R \ N,% ∈ Q . r→0 B(x,r) Since inf {|f(x) − %| : % ∈ Q} = 0 , the proof is complete. The proof of Theorem 2.15 relies on two preliminary results: the Vitali covering property for Radon measures on Rn (see Theorem 2.11), which yields the following comparison between ν and µ provided a pointwise estimates on the derivatives of ν with respect to µ. Lemma 2.19 (density estimates for diﬀerentiation of measures). Let ν and µ be positive Radon measures on Rn, let A ⊂ Rn be a Borel set and 0 < t < ∞. (i) If Dµν(x) 6 t for each x ∈ A, then ν(A) 6 t µ(A); (ii) If Dµν(x) > t for each x ∈ A, then ν(A) > t µ(A); n (iii) Dµν, Dµν : R → [0, ∞] are Borel measurable. In particular n (2.21) µ({x ∈ R : Dµν(x) = ∞ }) = 0 . Proof. (i) It is not restrictive to assume that A is bounded. By the approximation of Borel measures by open and closed sets (see Corollary 1.12), for every ε > 0 there exists a bounded open set U ⊃ A such that µ(U) < µ(A)+ ε. Moreover, by deﬁnition of lim inf, for each x ∈ A and > 0, there exists a sequence of positive real numbers (rh)h such that

(2.22) lim rh = 0 and ν(B(x, rh)) (t + ) µ(B(x, rh)),B(x, rh) ⊂ U ∀h. h→∞ 6 Fix ε > 0 and let us deﬁne n o G := B(x, r): x ∈ A, r ∈ (0, +∞) with B(x, r) ⊂ U, ν(B(x, r)) 6 (t+ε) µ(B(x, r)) ,. From (2.22), it is easy to see that G is a family of closed balls which covers A and satisﬁes (2.14). Applying the Vitali covering property for Radon measures on Rn (see Theorem 2.11), there exists a countable subfamily

F := {Bi : i ∈ N } ⊂ G such that ∞ Bi ∩ Bj = ∅ if i 6= j and ν (A \ ∪i=1Bi) = 0 . Thus ∞ ∞ ∞ X X ∞ (2.23) ν(A) 6 ν(∪i=1Bi) = ν(Bi) < (t + ε) µ(Bi) = (t + ε) µ(∪i=1Bi) i=1 i=1 6 (t + ε) µ(U) < (t + ε)(µ(A) + ε) . Letting ε → 0, the desired inequality follows. (ii) It is not restrictive to assume that A is bounded. By the approximation of Borel measures by open and closed sets, for every ε > 0 there exists an open set 68

U ⊃ A such that ν(U) < ν(A)+ ε. Moreover, by deﬁnition of lim sup, for each x ∈ A there exists a sequence of positive real numbers (rh)h such that

(2.24) lim rh = 0 and ν(B(x, rh)) (t − ) µ(B(x, rh)),B(x, rh) ⊂ U ∀h, . h→∞ > Fix ε > 0 and let us deﬁne n o G := B(x, r): x ∈ A, r ∈ (0, +∞) with B(x, r) ⊂ U, ν(B(x, r)) > (t−ε)µ(B(x, r)) . From (2.24), it is easy to see that G is a family of closed balls which covers A and satisﬁes (2.14). Applying the Vitali covering property for Radon measures on Rn (see Theorem 2.11), there exists a countable subfamily

F := {Bi : i ∈ N } ⊂ G . such that ∞ (2.25) Bi ∩ Bj = ∅ if i 6= j and µ (A \ ∪i=1Bi) = 0 . Thus, by (2.25) and (2.24), ∞ ∞ ∞ X X (2.26) (t − ε) µ(A) 6 (t − ε) µ(∪i=1Bi) = (t − ε) µ(Bi) 6 ν(Bi) i=1 i=1 ∞ = ν(∪i=1Bi) 6 ν(U) < (ν(A) + ε) . Letting ε → 0, the desired inequality follows. n (iii) Fix r > 0 and let gr : R → [0, ∞] be the function ν(B(x, r)) g (x) := if x ∈ n . r µ(B(x, r)) R Exercise: Prove that: (a) given r > 0, the function Rn 3 x 7→ ν(B(x, r)) (and also the function Rn 3 x 7→ µ(B(x, r))) is upper semicontinuous, that is n ν(B(x, r)) > lim sup ν(B(y, r)) ∀x ∈ R ; y→x n (b) given r > 0, the function gr : R → [0, ∞] is Borel measurable; (c) given x ∈ Rn, the function (0, ∞) 3 r 7→ ν(B(x, r)) (and also the function (0, ∞) 3 r 7→ µ(B(x, r))) is right-continuous, that is lim ν(B(x, s)) = ν(B(x, r)) ∀ r ∈ (0, ∞); s→r+ ! (d) Dµν(x) = limh→∞ inf gr(x) , Dµν(x) = limh→∞ sup gr(x) for r∈(0,1/h)∩ Q r∈(0,1/h)∩Q each x ∈ Rn. From (d) and (b), it follows that both Dµ and Dµ are Borel measurable. Let us now prove (2.21). For each h ∈ N let

Eh := {x ∈ B(h): Dµν(x) = ∞ } . By previous claim (ii), it follows that, for ﬁxed h, ν(B(h)) µ(E ) ∀ t ∈ [1, ∞) . h 6 t 69

Since ν(B(h)) < ∞, taking the limit as t → ∞ in the previoua inequality, we get that

µ(Eh) = 0 ∀ h ∈ N and then (2.21) follows. Proof of Theorem 2.15. (i) For 0 < R < ∞, 0 < s < t < ∞, let As,t,R := x ∈ B(0,R): Dµν(x) 6 s < t < Dµν(x) , At,R := x ∈ B(0,R): t 6 Dµν(x) . Then As,t,R and At,R are Borel sets and, by Lemma 2.19,

t µ(As,t,R) 6 ν(As,t,R) 6 s µ(As,t,R) < ∞ , u µ(Au,R) 6 ν(Au,R) 6 ν(B(0,R)) < ∞ . These inequalities yield

(2.27) µ(As,t,R) = 0 ∀ 0 < s < t < ∞ , and

(2.28) µ(∩u>0Au,R) = lim µ(Au,R) = 0 . u→∞ Notice that

∩u>0Au,R = ∩n∈NAn,R , and the ∩u>0Au,R is a Borel set. n Let denote by N1 the set of points x ∈ R such that @ Dµν(x) or Dµν(x) = ∞. Exercise: Prove that

N1 = ∪ {As,t,R : 0 < s < t < ∞, s, t ∈ Q, R > 0,R ∈ Q } ∪ {∩u>0Au,R : R > 0,R ∈ Q } .

Then, from (2.27) and (2.28), it follows that µ(N1) = 0, which settles (i). (ii) By Lemma 2.19 (iii), both Dµν and Dµν are Borel measurable functions. Thus, by deﬁnition, Dµν(x) is a Borel measurable function, too. n (iii) Let A be the set deﬁned in (2.18) and let N2 := {x ∈ R : ∃Dµν(x) = 0}. Observe that n (2.29) R \ A ⊆ N1 . Let us begin to prove that

(2.30) ν(N2) = 0 . For 0 < R < ∞, 0 < s < ∞, let ∗ As,R := {x ∈ B(0,R): Dµν(x) 6 s } . Then, by Lemma 2.19 (i), for given R > 0, it follows that ∗ ∗ ν(N2 ∩ B(0,R)) 6 ν(A,R) 6 µ(A,R) 6 µ(B(0,R)) ∀ > 0 . Letting → 0 in the previous inequality, we get

ν(N2 ∩ B(0,R)) = 0 ∀R > 0 , which establishes (2.30). Let us now prove the identity in (2.19). This amounts to prove the two inequalities 70

Z n (2.31) Dµν(x) dµ(x) 6 ν(A ∩ B) ∀ B ∈ B(R ) , B Z n (2.32) Dµν(x) dµ(x) > ν(A ∩ B) ∀ B ∈ B(R ) . B Fix B ∈ B(Rn) and choose 1 < t < ∞, let k k+1 Bk := x ∈ B : t 6 Dµν(x) < t k ∈ Z , and notice that 0 (2.33) (A \ N2) ∩ B = ∪k∈ZBk Bk ∩ Bk0 = ∅ if k 6= k , where A is the set deﬁned in (2.18). By (2.21), (2.28), Lemma 2.19 (i) and (2.30), Z Z ∞ Z ∞ X X k+1 Dµν(x) dµ(x) = Dµν(x) dµ(x) = Dµν(x) dµ(x) 6 t µ(Bk) B A∩B k=−∞ Bk k=−∞ ∞ X 6 t ν(Bk) 6 t ν((A \ N2) ∩ B) = t ν(A ∩ B) . k=−∞ Letting t → 1+ in the previous inequality, we establish (2.31). Let us now show (2.32). Choose 0 < t < 1, let k+1 k Bk := x ∈ B : t 6 Dµν(x) < t k ∈ Z , an notice that (2.33) still holds. Arguing as before, we get Z Z ∞ Z ∞ X X k+1 Dµν(x) dµ(x) = Dµν(x) dµ(x) = Dµν(x) dx > t µ(Bk) B A∩B k=−∞ Bk k=−∞ ∞ X > t ν(Bk) = t ν((A \ N2) ∩ B) = t ν(A ∩ B) . k=−∞ Letting t → 1−, in the previous inequality, we get (2.32) and then the equality in (2.19). The inequality in (2.19) trivially follows by the monotonicity of ν. If ν << µ, we have to prove that n ν(A ∩ B) = ν(B) ∀ B ∈ B(R ) . We need only to show that n ν(R \ A) = 0 . By (2.29), we have only to prove that

(2.34) ν(N1) = 0 . By previous point (i), and because of ν << µ, (2.34) follows. n (iv) If ν << µ, by previous claim (i), Dµν(x) µ-a.e. x ∈ R and then Dµν(x) ν-a.e. n n n x ∈ R , too. Suppose now that Dµν(x) < ∞ ν-a.e. x ∈ R and let B ∈ B(R ) with µ(B) = 0. Lemma 2.19 (i) gives ν x ∈ B : Dµν(x) 6 h 6 h µ(B) = 0 ∀ h ∈ N . 71

n Therefore, since ν({x ∈ R : Dµν(x) = ∞}) = 0, ∞ ν(B) = ν ∪h=1 x ∈ B : Dµν(x) 6 h ∞ X 6 ν x ∈ B : Dµν(x) 6 h = 0 . h=1 We are now going to deal with the problem of the Lebesgue decomposition of positive Radon measure ν with respect to a given Radon measure µ, by characterizing the absolutely continuous and singular parts in terms of derivatives of measures. From Theorem 2.15, the following result immediatly follows. Theorem 2.20 (Lebesgue decomposition in terms of derivatives of measures). Let ν n and µ be positive Radon measures on R . Let νac and νs denote respectively the absolutely continuous and singular parts of ν in the Lebesgue decomposition with respect to µ. Then, for each Borel set B, Z νac(B) = Dµν(x) dµ, νs(B) = ν S(B) = ν(S ∩ B) B where S is the µ-negligible Borel set n S = (R \ spt(µ)) ∪ {x ∈ spt(µ): Dµν(x) = ∞ }

Proof. Let us recall that, by deﬁnition, Dµν(x) = ∞ either if @Dµν(x) or if x∈ / spt(µ) and that A ⊂ spt(µ), where A is the set deﬁned in (2.18). Thus, by Theorem 2.15 (i), µ(S) = 0 and n R \ S = A. Moreover we can write (2.35) ν = ν A + ν S According to (RN) and by (2.19),

(2.36) ν A = µw << µ with w = Dµν , and, since ν A and ν S are mutually singular, by (2.35), (2.36)and the uniqueness of Lebesgue decomposition (see Theorem 1.32), we get the desired conclusion. As a byproduct of the previous theorem we get the following result (see also [SC, Lemma 1.51]).

Corollary 2.21. Let ν and µ be two positive Radon measures on Rn that are (mutually) singular, that is, ν ⊥ µ. Then n ∃ Dµν(x) = 0 µ-a.e. x ∈ R .

Proof. From Lebesgue decomposition theorem 1.32, it follows that νac ≡ 0 and ν = νs. Thus, from Theorem 2.20, it follows that Z n 0 = νac(B) = Dµν dµ for each B ∈ B(R ) . B n This implies that Dµν(x) = 0 µ-a.e. x ∈ R . 72

In several applications of the differentiation of measures, a differentiation result with respect to a more general class of sets rather than balls could very useful as, for instance, the case of the Lebesgue differentiation theorem for monotone functions (see [GZ, SC]). Therefore we are going to show that a more general class of sets can be considered in the derivative of measures, instead of balls.

Deﬁnition 2.22. Let µ be a positive Radon measure on Rn and let x ∈ Rn.A n sequence of Borel sets (Eh(x))h ⊂ B(R ) is called a (regular) diﬀerentiation basis at x for µ provided there is αx = α(x) > 0 with the following properties:

(i) there exists a sequence of balls (B(x, rh))h with rh → 0 such that

Eh(x) ⊂ B(x, rh) ∀ h ;

(ii) µ(Eh(x)) > αx µ(B(x, rh)) ∀ h .

Theorem 2.23 (Diﬀerentiation for positive Radon measures with respect to a differentiation basis). Let ν and µ be postive Radon measures on Rn, then there exists

ν(Eh(x)) dνac lim = (x) = Dµνac(x) µ-a.e. x ∈ spt(µ) , h→∞ µ(Eh(x)) dµ whenever (Eh(x))h is a diﬀerentiation basis of µ at x. Proof. Recall that, by Corollary 2.21, Theorems 2.20 and 2.15 (i),

(2.37) ν = νac + νs with νac << µ and νs ⊥ µ, ν (B(x, r)) (2.38) ∃ lim s = 0 µ-a.e. x ∈ spt(µ) , r→0 µ(B(x, r))

ν(B(x, r)) ν (B(x, r)) dν (2.39) ∃ lim = lim ac = ac (x) := w(x) µ-a.e. x ∈ spt(µ) . r→0 µ(B(x, r)) r→0 µ(B(x, r)) dµ

Let x ∈ spt(µ) be such that (2.38) holds and let (Eh(x))h be a differentiation basis at x of µ. Then νs(Eh(x)) 1 νs(B(x, rh)) 0 6 6 , µ(Eh(x)) αx µ(B(x, rh)) and, from (2.38), it follows that ν (E (x)) (2.40) ∃ lim s h = 0 . h→∞ µ(Eh(x)) To conclude the proof we need only to prove that ν (E (x)) (2.41) ∃ lim ac h = w(x) h→∞ µ(Eh(x)) when x ∈ spt(µ) is a Lebesgue point of w (see Definition 2.2).Observe that such a point x satisfies (2.39), too. Thus, (2.37), (2.40) and (2.41) conclude the proof. From Lebesgue points theorem, Z ∃ lim |w(y) − w(x)| dµ(y) = 0 µ-a.e. x ∈ spt(µ) . h→∞ B(x,rh) 73

(w(y) − w(x)) dµ(y) 6 |w(y) − w(x)| dyµ(y) → 0 . Eh(x) Eh(x) Then (2.41) holds. Derivatives of vector Radon measures. We are now going to deal with the problem of the Lebesgue decomposition of vector Radon measure ν with respect to a given Radon measure µ, by characterizing the absolutely continuous and singular parts in terms of derivatives of measures.

Theorem 2.24. Let ν and µ be respectively a Rm-valued Radon and a positive Radon measures on Rn. Then, for µ-a.e. x ∈ spt(µ),

ν(B(x, r)) m (2.42) ∃ w(x) := lim ∈ R . r→0 µ(B(x, r))

Moreover the Lebsesgue decomposition of ν with respect to µ is given by ν = µw + νs where νs(B) = ν S(B) = ν(S ∩ B) and S is the µ-negligible Borel set n S = (R \ spt(µ)) ∪ {x ∈ spt(µ): Dµ|ν|(x) = ∞ } Proof. Let us recall that, by the polar decomposition of ν with respect to |ν| (see Corollary 1.53) and the Lebsegue decomposition in terms of derivatives of measures (see Theorem 2.20) Z n (2.43) ν(B) = wν d|ν| ∀ B ∈ B(R ) , B Z n (2.44) |ν|(B) = Dµ|ν| dµ + ν S(B) ∀ B ∈ B(R ) , B n m where |ν| denotes the total variation of ν and wν : R → R is a Borel measurable n vector function with |wν| = 1 |ν|-a.e. in R . By combining (2.43) and (2.44), it follows that Z Z n (2.45) ν(B) = Dµ|ν| wν dµ + wν d(ν S) ∀ B ∈ B(R ) . B B In particular, by (2.45) and the the uniqueness of the Lebesgue decomposition of ν with respect to µ (see Theorem 1.52), we get Z Z n (2.46) νac(B) = Dµ|ν| wν dµ, νs(B) = wν d(ν S) ∀ B ∈ B(R ) . B B 74

In addition, since S is µ-negligible, by (2.45) and the Lebesgue-Besicovitch diﬀeren- tiation theorem 2.16, we can infer

ν(B(x, r)) m (2.47) ∃ w(x) := lim = Dµ|ν|(x) wν(x) ∈ R µ a.e. x ∈ spt(µ) . r→0 µ(B(x, r)) From (2.45), (2.46) and (2.47), we get the desired conclusion. 2.3. Extensions to metric spaces.

About Vitali’s covering theorem in metric spaces.

Vitali’s covering theorem still holds in any separable metric space (see, for instance, [AT, Theorem 2.2.3]). If (X, d) is a metric space boundedly compact, that is all bounded and closed sets of X are compact, a constructive proof of Vitali’s covering theorem, whithout the Hausdorﬀ Maximal Principle, can be given (see [Ma, Theorem 2.1]). If (X, d) is a metric space, using the Hausdorﬀ maximal principle, one can give much more general Vitali’s covering type-results ; for example families of balls can be replaced by many other families of sets (see, for instance, [Fe, 2.8.4-6]).

About Vitali’s covering property in metric measure spaces.

We call metric measure space a structure (X, d, µ) where (X, d) is a metric space and µ is a positive Radon measure on (X, B(X)). Given a metric measure space (X, d, µ) an important issue in the setting of analysis in metric space is to know whether Vitali’s covering property holds for the measure µ, that is, whether Theorem 2.11 still holds replacing metric measure space (Rn, | · |, µ) with (X, d, µ). More precisely we say that Vitali’s covering property holds for the metric measure space (X, d, µ) if, for each A ⊂ X measurable and bounded, for each family of closed balls G covering A and satisfying (2.14), there exists a disjoint subfamily F ⊂ G, at most countable, such that µ(A \ ∪F) = 0. When speaking of a closed ball B in (X, d), it will be understood B that it comes with a fixed center and radius (although these in general are not uniquely determined by B as a set, since neither center nor radius need be unique). Thus B = B(x, r) for some x ∈ X and some r > 0. Case of the doubling spaces. Vital’s covering property holds when the measure µ is supposed to be doubling on metric space (X, d), that is we assume that: • µ(X) > 0; • µ(B(x, r)) < ∞ ∀ x ∈ X, r > 0; • there exists a positive constant C > 0 such that (2.48) µ(B(x, 2r)) 6 C µ(B(x, r)) ∀ x ∈ X, r > 0 (see, for instance, [He, Theorem 1.6]). Indeed condition (2.48) can be weakened in asymptotic doubling condition (2.13) (see [Fe, 2.8]). Let us also recall that a metric space (X, d) is said to be doubling if there exists an integer C > 1 such that each closed ball with radius r > 0 can be covered with less than C balls with radius r/2. If µ is a doubling measure on a metric space (X, d), 75 then it is easy to see that (X, d) is doubling. On the other hand, not every doubling space carries a doubling measure (see [He, Chap.10]). Case of the Besicovich covering property. Vitali covering property also holds for a metric measure space (X, d, µ) with metric space (X, d) satisfying the so-called Besicovitch Covering Property (BCP) or also with (X, d) doubling metric space satisfying Weak Besicovitch Covering Property (WBCP) (see [LeR]). Definition 2.25 (Besicovitch Covering Property). We say that BCP holds for (X, d) if there exists an integer P > 1 with the following property. Let A be a bounded subset of (X, d) and let G be family of closed balls in (X, d) such that each point of A is the center of some ball of G. Then there is subfamily F ⊆ G, at most countable, whose balls cover A and such that every point in X belongs to at most P balls of F, that is, X χA 6 χB 6 P. B∈F Definition 2.26 (Weak Besicovitch Covering Property). We say that WBCP holds for (X, d) if there exists an integer N > 1 with the following property. Suppose there exist k points xi ∈ X and positive numbers ri > 0 (i = 1, . . . , k) such that k xi ∈/ B(xj, rj) if i 6= j and ∩i=1 B(xi, ri) 6= ∅ . Then k 6 N. The validity of BCP implies the one of WBCP. We stress that there exists metric spaces for which WBCP holds although BCP is not satisfied. However, when the metric is doubling, both covering properties turn out to be equivalent. An exaustive study of this topic is carried in [LeR]. An early study of metric spaces satisfying BCP was carried out by Federer [Fe, 2.8.9]. In particular he proved that BCP holds in compact Riemannian manifolds and in normed vector spaces of finite dimension. It is also well-known that BCP need not hold in sub-Riemannian structures (see, for instance, [LeR]). It is simple to show that, if a metric space (X, d) satisfies either a doubling or BCP condition, then (X, d) has finite topological dimension (in the sense of Lebesgue covering dimension) [LeR, 8.3]. It is also easy to prove that WBCP need not hold in an infinite dimensional space. For instance, let X be an infinite dimensional Hilbert space and let Bi := B(ei, 1) (i ∈ N), where {ei : i ∈ N } ⊂ X is a set of orthonormal vectors. Then ei ∈/ Bj if ∞ i 6= j and 0 ∈ ∩i=1Bi. Moreover there exist metric measure spaces (X, d, µ) with X separable Hilbert space and µ finite measure for which Vitali’s covering property does not hold [Ti].

About the Lebesgue diﬀerentiation theorem in metric measure spaces.

If µ is a locally finite Borel measure on a metric space (X, d), we say that the Lebesgue differentiation theorem holds on metric measure space (X, d, µ) if Z 1 ∃ lim f(y) dy = f(x) µ-a.e. x ∈ X, ∀ f ∈ Lloc(X, µ) . r→0 B(x,r) 76 where 1 n Lloc(X, µ) := g : X → R : g Borel measurable , ∀ x ∈ X ∃ rx > 0 such that Z o |f(y)| dµ(y) < ∞ . B(x,rx) It can be proved that the Lebesgue differentiation theorem holds for a metric measure space (X, d, µ) provided that it satisfies Vitali’s covering property (see, for instance, [He, Remark 1.13]). In particular, by the previous arguments, the Lebegue differentiation theorem holds either if (X, d, µ) is doubling or (X, d) satisfies BCP. An interesting characterization of metric measure spaces satisfying the Lebesgue differentiation theorem is due to D. Preiss [Pre]. Following his terminology, if (X, d) is a metric space, we say that d is finite dimensional on a subset Y ⊂ X if there exist C∗ ∈ [1, ∞) and r∗ ∈ (0, ∞] with the following property. Suppose there exist k points ∗ xi ∈ Y and positive numbers ri ∈ (0, r )(i = 1, . . . , k) such that k xi ∈/ B(xj, rj) if i 6= j and ∩i=1 B(xi, ri) 6= ∅ . ∗ Then k 6 C . We say d is σ-finite dimensional if X can be written as a countable union of subsets on which d is finite dimensional. Note that WBCP holds on (X, d) if and only if d is finite dimensional on X for some constant C∗ ∈ [1, ∞) and with r∗ = ∞. Theorem 2.27 ([Pre]). Let (X, d) be a complete separable metric space. The Lebesgue differentiation theorem holds on (X, d) for all locally finite Borel measures if and only if d is σ-finite dimensional. An interesting byproduct of the Lebesgue differentiation theorem is the following. Proposition 2.28. Let µ and ν be two locally finite Borel measures on a metric space (X, d) and assume that (i) the Lebesgue differentiation theorem holds for both measure spaces (X, d, µ) and (X, d, ν); (ii) µ(B(x, r)) = ν(B(x, r)) for each x ∈ X, 0 < r < ∞. Then µ(B) = ν(B) for each B ∈ B(Rn). An outline of the proof of Proposition 2.28 will be proposed in the proof of Theorem 3.30. The Lebesgue differentation theorem need not hold in an infinite dimensional metric space (see [Ti]).

3. An introduction to Hausdorff measures ([AFP, EG, Ma, Mag]).

Motivation: In this section we introduce Hausdorﬀ measures and dimension for measuring the metric size of quite general sets. They will be one of the basic means for studying geometric properties of sets and expressing results that these studies lead to. In particular they are very useful in this study because: • they measure both regular submanifolds and not regular subsets of Rn, such as fractal sets; • they do not depend on the parametrization of the submanifold. 77

3.1. Carathéodory’s construction and definition of Hausdorff measures on a metric space and their elementary properties; Hausdorff dimension. The main idea is the construction of (outer) measures in Rn which allow to measures subset of (roughly speaking) ”dimension” m < n. For instance, in R3, we are going to define three measures H1, H2 and H3 such that  H1(S) = length(S) if S is a curve  H2(S) = area(S) if S is a surface . H3(S) = volume(S) if S is a ball The basic definitions and first results on Hausdorff measures and dimension are due to Carathéodory [C2] in 1914 and Hausdorff [Ha] in 1919 and they can be introduced in the framework of a general metric space (X, d). We will start with a more general construction, called Caratheodory’s construction. It will yield also many other measures some of which will be briefly recalled.

Carath´eodory’s construction

Let (X, d) be a metric space, F a family of subsets of X and ζ : F → [0, ∞] a given nonnegative evaluation set function. We make the following two assumptions. ∞ (Ca1) For every δ ∈ (0, ∞], there is a sequence (Ei)i ⊂ F such that X = ∪i=1Ei and d(Ei) 6 δ for each i ∈ N. (Ca2) For every δ > 0, there is E ∈ F such that ζ(E) 6 δ and d(E) 6 δ. For 0 < δ 6 ∞ and A ⊂ X we deﬁne ( ∞ ) X ∞ (3.1) ψδ(A) := inf ζ(Ei): A ⊂ ∪i=1Ei, d(Ei) 6 δ, Ei ∈ F i=1 Remark 3.1. Assumption (Ca1) was only introduced to guarantee that such coverings always exist. The role of (Ca2) is to have ψδ(∅) = 0. It also allows us to use coverings (Ei)i∈I with I ﬁnite or countable without changing value of ψδ(A).

Remark 3.2. It is easy to see that ψδ is monotonic and subadditive so that it is an outer measure. Usually it is highly non-additive and not a Borel measure (see Exercise below). Evidently, ψδ(A) 6 ψ(A) whenever 0 < < δ 6 ∞ . hence we can deﬁne ψ = ψ(F, ζ) as follows

(3.2) ψ(A) := lim ψδ(A) = sup ψδ(A) for A ⊂ X. δ→0 δ>0

The measure-theoretic behaviour of ψ is much better than that of ψδ. Theorem 3.3. (i) ψ is a Borel outer measure. (ii) If F ⊂ B(X), then ψ is a Borel regular outer measure. 78

Proof. Proof. (i) The proof that ψ is an outer measure is straightforward and left to the reader. To show that ψ is a Borel outer measure, we apply Carath´eodory criterion (see Theorem 1.5 (iii)). Let A, B ⊂ X with d(A, B) > 0. Choose δ with 0 < δ < d(A, B)/2. If the sets E1,E2, · · · ∈ F cover A ∪ B and satisfy d(Ei) < δ, then none of them can meet both A and B. Hence ∞ X X X ζ(Ei) > ζ(Ei) + ζ(Ei) i=1 A∩Ei6=∅ B∩Ei6=∅ > ψδ(A) + ψδ(B) .

Taking the inﬁmum over all such coverings we have ψδ(A ∪ B) > ψδ(A) + ψδ(B). But the opposite inequality holds also as ψδ is an outer measure, and so ψδ(A ∪ B) = ψδ(A) + ψδ(B). Letting δ → 0, we obtain ψ(A ∪ B) = ψ(A) + ψ(B) as required. (ii) If A ⊂ X, choose for every i = 1, 2,... sets Ei,1,Ei,2, · · · ∈ F such that ∞ A ⊂ ∪j=1Ei,j, d(Ei,j) < 1/i and ∞ X ζ(Ei,j) < ψ1/i(A) + 1/i . j=1 ∞ ∞ Then B := ∩i=1(∪j=1Ei,j) is a Borel set such that A ⊂ B and ψ(A) = ψ(B). Thus ψ is a Borel regular outer measure.

Hausdorﬀ measures

Let (X, d) be separable metric space, 0 6 s < ∞, and choose (3.3) F := P(X) = {E : E ⊂ X} ,

s (3.4) ζ(E) = ζs(E) := αs d(E) 0 s with the interpretations 0 = 1 and d(∅) = 0, where αs is a geometric constant which depends only on s and the environment (X, d) and will be fixed later in the Euclidean context for normalization purposes of constants (see (3.11)). Exercise: Prove that (Ca1) and (Ca2) are satisfied under assumptions (3.3) and (3.4). The resulting measure ψ is called the s-dimensional Hausdorff measure and denoted by Hs . So s s s (3.5) H (A) := lim Hδ(A) = sup Hδ(A) . δ→0 δ> 0 s and Hδ is the s-dimensional Hausdorff pre-measure defined by ( ∞ ) s X s (3.6) Hδ(A) := inf αs d(Ei) : A ⊂ ∪i=1Ei, d(Ei) 6 δ i=1 s where, δ ∈ (0, ∞]. Let us observe that Hδ : P(X) → [0, ∞] is an outer measure but is not a Borel outer measure (see [Ma, Chap.4, Ex. 1]). n Exercise: Let X = R , 0 < s < ∞, 0 < δ 6 ∞. Prove that 79

s n s (i) Hδ : P(R ) → [0, ∞] is increasing, countably subadditive and Hδ(∅) = 0. In particular it is an outer measure. s (ii) Hδ is neither additive nor a Borel measure. Indeed it can be proved that, if n > 2, 0 6 s 6 1, 0 < δ < ∞ and U ≡ Uδ = U(0, δ/2) = U(δ/2), then s s ¯ s Hδ(U) = Hδ(U) = Hδ(∂U) . s s (iii) Hδ(A) 6 Hσ(A) for each 0 6 σ < δ 6 ∞, 0 6 s < ∞. (Hint: (ii) 1st step: Prove that, if n = 2 and s = 1 , then 1 (3.7) Hδ (∂U) = δ . Indeed, inequality 1 Hδ (∂U) 6 δ 1 is trivial by the the deﬁnition of Hδ (recall that α1 = 1). The reverse inequality is less trivial and the key point for its proof is the following property of a circumference: given m + 1 points P1,...,Pm+1 ∈ ∂U with m > 3 satisfying

P1,...,Pm are distinct,P1 = Pm+1, |Pi − Pi+1| < δ ∀ i = 1, . . . , m , then m X |Pi − Pi+1| > δ . i=1 Therefore, by (3.7), since 1 1 ¯ δ = Hδ (∂U) 6 Hδ (U) 6 δ , it follows that 1 1 ¯ (3.8) Hδ (∂U) = Hδ (U) = δ . 1 Let σ ∈ (0, δ) and notice that, by (3.8) and the subadditivity of Hσ 1 ¯ 1 σ = Hσ(Uσ) 6 Hσ(U) . Prove now that 1 1 (3.9) δ 6 lim inf Hσ(U) 6 Hδ (U) 6 δ . σ→δ− By (3.9) and (3.8), the conclusion of claim (ii) follows if n = 2 and s = 1. 2nd step: Let

n > 3, s = 1 and Γ := {x = (x1, . . . , xn) ∈ ∂U : x3 = ··· = xn = 0} . Then it follows that 1 1 Hδ (Γ) 6 Hδ (∂U) 6 δ . On the other hand, since Γ is isometric to the circle ∂U of R2, arguing as in the ﬁrst step, it follows that 1 Hδ (Γ) = δ . Therefore it follows that 1 Hδ (∂U) = δ . 80

Analogously, arguing as in the remaining cases of the ﬁrst step, the conclusion of claim (ii) follows if n > 2 and s = 1. 3rd step: Let n > 2, 0 < s < 1 . It is trivial that s s ¯ s s s Hδ(∂U) 6 Hδ(U) 6 αs δ and Hδ(U) 6 αs δ .

Observe now that, for each non negative sequence of real numbers (ai)i,

∞ !s ∞ X X s ai 6 ai . i=1 i=1 By the 2nd step, this implies that s 1 s −1 s −1 s ¯ s δ = Hδ (∂U) 6 αs Hδ(∂U) 6 αs Hδ(U) 6 δ , and s 1 s −1 s s δ = Hδ (U) 6 αs Hδ(U) 6 δ . Thus the conclusion of claim (ii) follows if n > 2 and 0 < s < 1.) s Remark 3.4. Observe that pre-measure Hδ does not ﬁt our project to deﬁne a measure which gives back the usual measure on submanifold of Rn. Indeed, let 2 X = R , s = 1 and Γ := {(x, x sin(1/x)) : 0 < x 6 1/π}. Then it is well-known that length(Γ) = ∞, but

1 Hδ (Γ) < ∞ ∀ 0 < δ < ∞ . Remark 3.5. We can consider in the previous procedure, in place of F = P(X), the family F of all closed balls of (X, d) and still the evaluation function in (3.4). s As before, we can define pre-measure Sδ := ψδ (see (3.1)) and the resulting measure ψ := limδ→0 ψδ (see (3.2)) is the so-called s-dimensional spherical Hausdorff denoted by Ss. Measures Hs and Ss can differ, but they are equivalent. Indeed it holds

s s s s (3.10) H (A) 6 S (A) 6 2 H (A) ∀ A ⊂ X.

The integral dimensional Hausdorff measures play a special role. Let us start from s = 0. It is easy to see that H0 agrees with the counting measure # on X defined in Example 1.2 (i) Next, for s = 1, H1 also has a concrete interpretation as a generalized length measure. In particular, for a rectifiable curve Γ in Rn, H1(Γ) can be shown to equal the length of Γ (see Theorem 3.25). For unrectifiable curves H1(Γ) = ∞. More generally, if m is an integer, 1 < m < n, and S is a sufficiently regular m-dimensional surface in Rn (for example, a C1 submanifold), then the restriction m H S gives the surface measure on S, for a suitable choice of αm, as we will see. This follows for example from the area formula, of which we will deal with in the next section. For s = n in Rn , (3.11) Ln = Hn , 81 Ln(B(1)) by choosing the constant α := in the definition of Hn. Observe that, by n 2n (3.11), it follows that

n n n n (3.12) H (B(x, r)) = 2 αn r ∀ x ∈ R , r > 0 . The proof of the equality (3.11) is rather complicated and based on the so-called isodiametric inequality

n n n L (A) 6 αn d(A) for A ⊂ R which we will deal with in the next section. But to see that Hn = c Ln with some positive and ﬁnite constant c is much easier and a proof is given in [Ma, 4.3] and we will give an other proof in Theorem 3.30. For any s > n, Hs turns out to be a trivial null measure on Rn. Indeed we will show that Hs(Rn) = 0 (see Theorem 3.11). We shall now derive some simple properties of Hausdorﬀ measures in a general separable metric space (X, d).

s Theorem 3.6. Let s ∈ [0, ∞), αs > 0 and ζ(E) := αs d(E) for E ⊂ X. If (i) F = {F ⊂ X : F closed } or (ii) F = {U ⊂ X : F open }, then ψ(F, ζ) = Hs, where ψ(F, ζ) is the set function deﬁned in (3.2). Proof. Assume that (i) holds. It is clear that, by deﬁnition,

s (3.13) H 6 ψ(F, ζ) . In order to prove the reverse inequality, let us first observe that, given E ⊂ X, then E ⊂ E¯ and d(E¯) = d(E) if E¯ denotes the closure of E. Assume that Hs(A) < ∞, s otherwise we are done. Then, by definition, Hδ(A) < ∞, for each δ ∈ (0, ∞]. Therefore, for fixed δ ∈ (0, ∞] and each > 0, there is a sequence of sets (Ei)i such ∞ ∞ ¯ ¯ that that A ⊂ ∪i=1Ei ⊂ ∪i=1Ei with d(Ei) = d(Ei) < δ, and ∞ ∞ X ¯ s X s s ψδ(F, ζ)(A) 6 αs d(Ei) = αs d(Ei) 6 Hδ(A) + . i=1 i=1 Passing to the limit as δ → 0 in the previous inequality and since is arbitrary, we get the desired inequality. Assume now that (ii) holds. Let us first observe that, for each E ⊂ X and σ > 0 , Iσ(E) := {x ∈ X : d(x, E) < σ } is an open set with

Iσ(E) ⊃ E and d(Iσ(E)) 6 d(E) + 2σ . Again, (3.13) is immediate. Let us prove the reverse inequailty. As before, we can assume that, for ﬁxed δ ∈ (0, ∞] and each > 0, there is a sequence of sets (Ei)i ∞ such that that A ⊂ ∪i=1Ei with d(Ei) < δ, and ∞ X s s (3.14) αs d(Ei) 6 Hδ(A) + . i=1 82

s For given i ∈ N and > 0, by the continuity of function [0, ∞) 3 σ 7→ (d(Ei) + 2σ) , there exists σi = σ(i, s, ) > 0 such that s s (3.15) (d(Ei) + 2σi) < d(Ei) + i . αs 2

Let Ui := Iσi (E) for i ∈ N. Therefore, by (3.14) and (3.15), it follows that ∞ ∞ X s X s ψδ(F, ζ)(A) 6 αs d(Ui) 6 αs (d(Ei) + 2σi) i=1 i=1 ∞ ∞ X X α d(E )s + Hs(A) + 2 . 6 s i 2i 6 δ i=1 i=1 Passing to the limit as δ → 0 in the previous inequality and since is arbitrary, we get the desired inequality. Combining Theorems 3.6 and 3.3, we get Corollary 3.7. Hs is a Borel regular outer measure. Notice that usually Hs is not a Radon outer measure since it need not be locally finite. For example, if s < n every non-empty open set in Rn has non σ-finite Hs measure. But taking any Hs-measurable set A in Rn with Hs(A) < ∞, the restriction Hs A is a Radon measure by Theorem 1.92 (ii). Often one is only interested in knowing which sets have null Hs-measure. For this s s it is enough to use any of the pre-measures Hδ, for example H∞. In fact we do not need any measure for defining the null Hs-measure sets. s Lemma 3.8 (H -null sets). Let A ⊂ X, 0 < s < ∞ and 0 < δ 6 ∞. Then the following conditions are equivalent: (i) Hs(A) = 0. s (ii) Hδ(A) = 0. (iii) ∀ > 0 ∃ E1,E2, · · · ⊂ X such that ∞ ∞ X s A ⊂ ∪i=1Ei and d(Ei) < . i=1 Proof. Implications (i)⇒ (ii) ⇒ (iii) are trivial. Let us prove implication (iii) ⇒ (i). Without loss of generality we can assume that s > 0, otherwise the conclusion is trivial. By assumptions, it follows that 1/s d(Ei) 6 δ() := ∀ i ∈ N . Therefore s Hδ()(A) 6 ∀ > 0. Passing to the limit as → 0 in the previous inequality, we get the desired conclusion. We will now compare measures Hs among them. Theorem 3.9. For 0 < s < t < ∞ and A ⊂ X, (i) Hs(A) < ∞ implies Ht(A) = 0, (ii) Ht(A) > 0 implies Hs(A) = ∞. 83

∞ P∞ s s Proof. (i) Let A ⊂ ∪i=1Ei with d(Ei) 6 δ and αs i=1 d(Ei) < Hδ(A) + 1 < ∞ . Then ∞ ∞ t X t X t−s s Hδ(A) < αt d(Ei) = αt d(Ei) d(Ei) i=1 i=1 ∞ X αt < α δt−s d(E )s δt−s (Hs(A) + 1) , t i 6 α δ i=1 s which gives (i) as δ → 0. (ii) This claim is only a restatement of (i).

Hausdorﬀ dimension

Theorem 3.9 enables the definition of an important notion in GMT, namely the Hausdorff dimension. Very roughly, according to [Fa, Introduction], ” a dimension provides a description of how much space a set fills. It is a measure of the prominence of the irregularities of a set when viewed at very small scales. A dimension contains much information about the geometrical properties of a set.” Definition 3.10. Let (X, d) be a separable metric space, the Hausdorff (or also metric) dimension of a set A ⊂ X Hdim(A) = sup {s : Hs(A) > 0} = sup {s : Hs(A) = ∞} = inf t : Ht(A) < ∞ = inf t : Ht(A) = 0 where we put sup ∅ = 0 and inf ∅ = ∞ whether someone of the previous sets may be empty. The Hausdorff dimension has the natural properties of monotonicity and stability with respect to countable unions:

(3.16) Hdim(A) 6 Hdim(B) for A ⊂ B ⊂ X,

∞ (3.17) Hdim(∪i=1Ai) = sup Hdim(Ai) for Ai ⊂ X, i = 1, 2,... i To state the deﬁnition in other words, Hdim(A) is the unique number (it may be ∞ in some metric spaces) for which (3.18) s < Hdim(A) ⇒ Hs(A) = ∞, t > Hdim(A) ⇒ Ht(A) = 0 . Remark 3.11. At the borderline case s = Hdim(A) we cannot have any general nontrivial information about the value Hs(A); all three cases Hs(A) = 0, s s 0 < H (A) < ∞, H (A) = ∞. If for some given A there is a s > 0 such that 0 < Hs(A) < ∞, then s must equal Hdim(A).

Remark 3.12. Since, we will see in the next section that Hdim(Rn) = n (see Corol- n lary 3.29). Hence 0 6 Hdim(A) 6 n for all A ⊂ R . One can prove that, for all s ∈ [0, n], Hdim(A) = s for some subset A of Rn (see Remark 3.35). 84

3.2. Recalls of some fundamental results on Lipschitz functions between Euclidean spaces and relationships with Hausdorff measures. Recalls of some fundamental results on Lipschitz functions between Euclidean spaces. Let us recall the general definition of Lipschitz function, which makes sense even for functions acting between metric spaces. Definition 3.13. Let (X, d) and (Y,%) be metric spaces, let E ⊂ X and let f : E ⊂ X → Y . (i) f is said to be Lipschitz or L-Lipschitz if there is L > 0 (3.19) %(f(x1), f(x2)) 6 L d(x1, x2) ∀ x1, x2 ∈ E. The smallest constant L such that (3.19) holds is called Lipschitz constant of f and denoted %(f(x1), f(x2)) Lip(f, E) := sup : x1, x2 ∈ E, x1 6= x2 ∈ [0, ∞) . d(x1, x2) (ii) f is said to be locally Lipschitz if for any compact subset of E there is L > 0 such that (3.19) holds for all x1, x2 in the compact. An important issue in GMT and, more generally, in Analysis is the extension of a C1 or Lipschitz function f : E ⊂ Rn → R to the whole Rn. Let us first recall a fundamental result due to Whitney about C1 function. Theorem 3.14 (Whitney’s extension theorem). Let C be a closed set in Rn. Let f : C → R and v : C → Rn be continuous functions. Define f(x) − f(y) − v(y) · (x − y) R(x, y) := ∀ x, y ∈ C, x 6= y . |x − y| Suppose that for all compact sets K ⊂ C (3.20) lim sup {|R(x, y)| : x, y ∈ K, 0 < |x − y| < r } = 0 . r→0 ˆ 1 n ˆ ˆ Then there is f ∈ C (R ) such that f|C = f and ∇f|C = v. Proof. The proof can be found in [EG, Sect. 6.5]. Remark 3.15. Let us observe that, by classical Taylor’s formula, condition (3.20) is satisfied if f ∈ C1(Rn). Thus Whiteney’s extension theorem is actually a characterization of the C1-extension of a function defined on a closed set, and is a partial converse to Taylor’s formula. A real valued L-Lipschitz function, defined on a subset E of a metric space (X, d), can always be extended to a L-Lipschitz function defined on the entire space X. Theorem 3.16 (Mc Shane’s extension theorem [McS]). Let (X, d) be a metric space ˆ ˆ and f : E ⊂ X → R be L-Lipschitz. Then there is f : X → R such that f|E = f and fˆ is L-Lipschitz. Proof. The proof is not difficult and can be found in the original Mc Shane’s paper [McS] as well as in many textbooks devoted to analysis in metric spaces (see, for instance, [AT, Theorem 3.1.2] and [He, He2]). 85

Let us recall that as an immediate consequence of Mc Shane’s extension theorem is the following extension theorem for Lipschitz maps with target an Euclidean space. Corollary 3.17. Let f : E → m, E ⊂ (X, d) be an L-Lipschitz function. Then √ R ˆ m ˆ there exists an mL-Lipschitz function f : X → R such that f|E = f.

Corollary 3.17 follows by applying√ Theorem 3.16 to the coordinate functions of f. The multiplicative constant m in the corollary is in fact redundant, but this is harder to prove.

Theorem 3.18 (Kirszbraun’s theorem). Let f : E → Rm, E ⊂ Rn, be an L-Lipschitz ˆ n m ˆ function. Then there exists an L-Lipschitz function f : R → R such that f|E = f. Proof. See, for instance, [He2, Theorem 2.5]. An other fundamental result concerning Lipschitz functions, acting between Eu- clidean spaces, concerns their diﬀerentiability a.e.

Theorem 3.19 (Rademacher’s theorem[Rad]). Let f : Rn → R be locally Lipschitz. Then f is differentiable (in classical sense) Ln-a.e., that is, n n ∃ ∇f(x) := (∂1f(x), . . . , ∂nf(x)) for L -a.e. x ∈ R and f(y) − f(x) − df(x)(y − x) (3.21) lim = 0 y→x |y − x| where df(x): Rn → R denotes the (linear) differential map of f at x defined by n df(x)(v) := ∇f(x) · v ∀ v ∈ R . ∞ n n Moreover ∇f ∈ (Lloc(R )) . Proof. See, for instance, [EG, Theorem 2, Sect. 3.1.2]. Remark 3.20. Let us point out that, in the 1-dimensional case, i.e. n = 1, Rademacher’s theorem is an immediate consequence of the fundamental theorem of calculus, since a locally Lipschitz function f : R → R is locally absolutely continuous, i.e. f|[a,b] ∈ AC([a, b]) for each a, b ∈ R with a < b (see, for instance, [SC]).

Rademacher’s theorem trivially extends to locally Lipschitz functions f = (f1, . . . , fm): Rn → Rm. In this case we get the existence, Ln-a.e. x ∈ Rn, of the Jacobian matrix of f at x , denoted Df(x) and deﬁned by   ∂1f1(x) . . . ∂nf1(x)  ......  (3.22) Df(x) :=   .  ......  ∂ f (x) . . . ∂ f (x) 1 m n m m×n Moreover (3.21) now holds with the (linear) diﬀerential map df(x): Rn → Rm n df(x)(v) := Df(x) · v ∀ v ∈ R , where the previous product has to be meant as product between m × n matrix Df(x) and the (column) vector v. 86

Whitney’s theorem together with Rademacher’s theorem yield an approximation of a Lipschitz function, which states that it coincides, up to a small set, with a C1 function.

Theorem 3.21 (Approximation of Lipschitz functions). Let f : Rn → R be a Lips- chitz function. Then for each > 0 there is a g ∈ C1(Rn) such that Ln ({x : f(x) 6= g(x)} ∪ {x : ∇f(x) 6= ∇g(x)}) < . In addition, there is a positive constant c = c(n) such that sup |∇g| 6 c Lip(f) . Rn Proof. See [EG, Sect. 6.6.1]. Let us now stress this simple relationship between Hausdorﬀ measures and Lipschitz maps.

Theorem 3.22 (Hausdorﬀ measures vs. Lipschitz maps). Let f : E ⊂ Rn → Rm be a Lipschitz map. Then s s s H (f(E)) 6 Lip(f) H (E) ∀ 0 6 s < ∞ . In particular Hdim(f(E)) 6 Hdim(E)

Proof. Let (Ei)i be a countable covering of E by sets with diameter less than δ. Then (f(Ei))i is a covering of f(E) with

diam(f(Fi)) 6 Lip(f)diam(Fi) 6 Lip(f)δ ∀ i .

Exploiting the arbitrariness of (Ei)i in the following inequalities, ∞ ∞ s X s s X s HLip(f)δ(f(E)) 6 αs diam(f(Ei)) 6 Lip(f) αs diam(Ei) , i=1 i=1 we get s s s HLip(f)δ(f(E)) 6 Lip(f) H (E) . We let δ → 0+ to get the desired inequality. Remark 3.23. By Theorem 3.22, we find that Hausdorff measures are decreased under projection over an affine subspace of Rn. Indeed, if H is an affine subspace of Rn and f : Rn → Rn is the projection of Rn over H, then Lip(f) = 1. The same happens, of course, if we project over a convex set.

n m Remark 3.24. We say that f : E ⊂ R → R (1 6 n 6 m ) is an isometry if |f(x) − f(y)| = |x − y| for every x, y ∈ E. n n m s s If s > 0, E ⊂ R , and f : E ⊂ R → R is an isometry, then H (f(E)) = H (E), as we may see either by applying Theorem 3.22 to f and to any extension g of f −1 with Lip(g) 6 1, or by the area formula (IAF). In particular, if π is an n-dimensional plane in Rm, then there exists an orthogonal injection P : Rn → Rm, such that π = P (Rn), that is, there exists P : Rn → Rm injective and satisfying n n (P (x),P (y))Rm = (x, y)Rn ∀ x, y ∈ R and P (R ) = π . 87

Indeed, let v1, . . . , vn ∈ π be a orthonormal basis of π with respect to the scalar product of Rm. Then n X n P (x1, . . . , xn) := xivi if (x1, . . . , xn) ∈ R i=1 turns out to be the desired function. In particlar, notice that P is an isometry It follows that n n H π = P#H . On the left-hand side, Hn stands for the n-dimensional Hausdorff measure on Rm, on the right-hand side, it denotes the n-dimensional Hausdorff measure on Rn (which in turn coincides with Ln; see Theorem 3.31). Indeed, notice that, if A ⊂ Rm, n −1 π ∩ A = P R ∩ P (A) . Since P : Rn → Rm is an isometry, we get n n n −1 H (π ∩ A) = H P R ∩ P (A) n n −1 n = H R ∩ P (A) = P#H (A) . Thus we get the desired identity. 3.3. Hausdorff measures in the Euclidean spaces; H1 and the classical notion of length in Rn; isodiametric inequality and identity Hn = Ln on Rn. In the following of the section, we will assume that n X = R equipped with the Euclidean distance , and we will fix constant αs in (3.6) in such a way s/2 ∗ s π (3.23) αs := 2 αs := s Γ 2 + 1 R ∞ −t t−1 where Γ(t) := 0 e x dx if t > 0 is the Euler Gamma function. ∗ Observe that, if s is equal to a positive integer n then αn agrees with the n- dimensional Lebesgue measure of a unit ball in Rn. We are going to study here the further properties of Hausdorff measures taking the structure of Rn into account. First, Hausdorff measures behave nicely under translations and dilations in Rn : for A ⊂ Rn, a ∈ Rn, 0 < t < ∞, (3.24) Hs(A + a) = Hs(A) where A + a := {x + a : x ∈ A},

(3.25) Hs(tA) = tsHs(A) where t A := {tx : x ∈ A} . These are readily veriﬁed from the deﬁnition.

Hausdorﬀ measures and length measure

A curve of Rn is a continuous function γ :[a, b] → Rn; its support is the set γ([a, b]) = Γ ⊂ Rn and, in this case, γ is called a parametrization of Γ. For the sake of simplicity, we can assume that a = 0 and b = a. Given a curve γ : [0, a] → Rn 88 and a subinterval [c, d] ⊆ [0, a], we define the length (or also variation) of γ over [c, d] as (3.26) ( m ) X l(γ;[c, d]) := sup |γ(ti) − γ(ti−1)| : t0 = c < t1 < ··· < tm = d ∈ [0, ∞] i=1 where the supremum is taken over all finite partitions {t0 = c < t1 < ··· < tm = d} of [c, d]. Moreoever let us denote l(γ) := l(γ;[a, b]) . Exercise: (i) l(γ;[b, c]) > |γ(b) − γ(c)|, whenever 0 6 b 6 c 6 a; (ii) l(γ;[b, c]) = l(γ;[b, d]) + l(γ;[d, c]), whenever 0 6 b 6 d 6 c 6 a. It is also well-known that, if γ : [0, a] → Rn is of class C1, then Z d (3.27) l(γ;[c, d]) = |γ0(t)| dt ∀[c, d] ⊂ [0, a] . c If l(γ; [0, a]) < ∞, the curve γ is said to be rectifiable. Whether l(γ; [0, a]) is finite or not, the following theorem holds true.

Theorem 3.25 (Classical length and H1). Let γ : [0, a] → Rn be a curve and denote Γ = γ([0, a]) its support. Then 1 H (Γ) 6 l(γ) and equality holds if γ : [0, a] ⊂ R → Rn is injective. Before the proof we need the following preliminary result and thanks to G.P. Leonardi for usefuel suggestions for the proof.

Lemma 3.26. Let γ : [0, a] → Rn be a rectiﬁable curve, that is l := l(γ; [0, a]) < ∞. Let v : [0, a] → [0, l] be the function deﬁned by (3.28) v(t) := l(γ; [0, t]) t ∈ [0, a] . Then v is a non decreasing continuous function. In particular v([0, a]) = [0, l]. Proof. The feature that v is non decreasing is straightforward. Let us prove that v is continuous. Since v is non decreasing, we have only to prove that lim v(s) = lim v(s) = v(t) ∀ t ∈ [0, a] . s→t+ s→t− Let us prove that (3.29) lim v(s) = v(t) ∀ t ∈ (0, a] . s→t−

Since l(γ; [0, a]) < ∞, for each > 0 there exists a partition of [0, a], t0 = 0 < t1 < ··· < tm = a, such that m X (3.30) l(γ; [0, a]) − |γ(ti) − γ(ti−1)| < . i=0 89

Moreover, without loss of generality, we can suppose that ti0 = t for some i0 =

1, . . . , m. Otherwise, since there exists i0 = 1, . . . , m such that ti0−1 < t < ti0 , we could enlarge the previuos partition by deﬁning a new partition  ti if 0 6 i 6 i0 − 1 ∗  ti := t if i = i0  ti+1 if i0 6 i 6 m and, because of m m+1 X X ∗ ∗ |γ(ti) − γ(ti−1)| 6 |γ(ti ) − γ(ti−1)| , i=0 i=0 we still have m+1 X ∗ ∗ l(γ; [0, a]) − |γ(ti ) − γ(ti−1)| i=0 m X 6 l(γ; [0, a]) − |γ(ti) − γ(ti−1)| < . i=0 Observe now, by claim (ii) of the previous exercise, we can write (3.30) as m X l(γ; [0, a]) − |γ(ti) − γ(ti−1)| i=0 m X = (l(γ;[ti−1, ti]) − |γ(ti) − γ(ti−1)|) < . i=0 which implies,

(3.31) l(γ;[ti0−1, t]) − |γ(t) − γ(ti0−1)| < .

Meanwhile, from (3.31), it follows that, for each ti0−1 6 s 6 t,

l(γ;[ti0−1, t]) − (|γ(ti0−1) − γ(s)| + |γ(s) − |γ(t)|)

= l(γ;[ti0−1, s]) + l(γ;[s, t]) − (|γ(ti0−1) − γ(s)| + |γ(s) − |γ(t)|)

l(γ;[ti0−1, t]) − |γ(t) − γ(ti0−1)| < , which implies, for each ti0−1 6 s 6 t, (3.32) v(t) − v(s) − |γ(s) − γ(t)| = l(γ;[s, t]) − |γ(s) − γ(t)| < . ¯ ¯ On the other hand, since γ is continuous, for each > 0 there is t = t() ∈ (ti0−1, t) such that for each t¯ < s < t (3.33) |γ(t) − γ(s)| < . Therefore, by (3.32) and (3.33), it follows that, for each > 0 there is t¯ < t such that v(t) − v(s) < 2 ∀ s ∈ (t,¯ t) , and (3.29) follows. To prove the right continuity, that is (3.34) lim v(s) = v(t) ∀ t ∈ [0, a) s→t+ 90

we can use the same procedure. Indeed, by (3.30), we can now suppose that ti0−1 = t for some i0 = 1, . . . , m. Arguing as before, we get that, for each t 6 s 6 ti0 , v(s) − v(t) − |γ(s) − γ(t)| = l(γ;[t, s]) − |γ(s) − γ(t)| < and then, by the continuity of γ, (3.34) follows. Proof of Theorem 3.25. We will follow the proof in [Mag, Theorem 3.8]. Let us begin with the following exercise. Exercise: Prove that H1 = L1 on R, as outer measures. 1 (Hint: Use the fact that L (E) 6 d(E) for each E ⊂ R.) The theorem is proved by Remark 3.24 and the previous exercise if Γ is a segment, Indeed, assume that Γ = γ([0, a]) with p − q γ(t) := p + t if 0 t a := |p − q| |p − q| 6 6 where p, q ∈ Rn with p 6= q. Since γ : [0, a] ⊂ R → Rn is an isometry, then H1(Γ) = H1 (γ([0, a]) = H1([0, a]) = L1([0, a]) = |p − q| = l(γ; [0, a]) . Set l = l(γ; [0, a]). We divide the proof into three steps. 1 n 1st step. Let us show that H (Γ) > |γ(a) − γ(0)|. Since the projection p : R → n n R of R onto the line defined by γ(0) and γ(a) satisfies Lip(p) 6 1, by Theorem 3.22 we have 1 1 H (p(Γ)) 6 H (Γ) . At the same time, p(Γ) must contain the segment S := {tγ(a) + (1 − t)γ(0) : 0 6 t 6 1}. Otherwise, Γ = γ([0, a]) would be disconnected, against the continuity of γ. 1 1 Thus H (p(Γ)) > H (S) = |γ(a) − γ(0)|. 1 2nd step. Let us prove that H (Γ) 6 l. If l = ∞, we are done. Thus we can assume that l < ∞ and we are going to construct a Lipschitz function γ∗ : [0, l] → Rn ∗ ∗ ∗ with Lip(γ ) 6 1 and Γ = γ ([0, l]). Indeed, by Theorem 3.22, the existence of γ will imply, as required, that 1 1 ∗ 1 H (Γ) = H (γ ([0, l]) 6 H ([0, l]) = l . First let us assume that γ is injective. To construct γ∗ (which is just the parametrization by arc length of γ, defined without using derivatives), we define Then v(0) = 0, v(a) = l and v is strictly increasing, that is, v(t) < v(s) if t < s, as γ is injective. In particular, v is continuous and invertible, with a continuous strictly increasing inverse w : [0, l] → [0, a]. Let then γ∗ : [0, l] → Rn be defined by γ∗(s) := γ(w(s)) s ∈ [0, l] . Then Exercise: Prove that (3.35) l(γ∗; [0, s]) = s ∀ s ∈ [0, l] (Hint: prove that l(γ∗; [0, s]) = l(γ; [0, w(s)]) = v(w(s)) = s). 91

∗ We easily find that Lip(γ ) 6 1, since, by properties (i) and (ii) in the exercise above, if [s1, s2] ⊂ [0, l], then ∗ ∗ ∗ ∗ ∗ |γ (s1) − γ (s2)| 6 l(γ ;[s1, s2]) = l(γ ; [0, s2]) − l(γ ; [0, s2]) = s2 − s1 . If γ is not injective, even if the construction is more difficult (see, for instance,[AT, Theorem 4.2.1]), we can still construct a parametrization of Γ, γ∗ : [0, l] → Rn with ∗ Lip(γ ) 6 1 and we can argue as before. 3rd step. Suppose now that γ is injective.If t0 = 0, . . . , tm = a is a competitor in the definition of l, then, setting Γh := γ([th−1, th]) (h = 1, . . . , m), we have Γ = m 1 1 ∪h=1Γh and, by the injectivity of γ, H (Γh ∩ Γh+1) = H ({γ(th)} = 0. We thus find 1 H (Γ) > l as, by step one, m m 1 X 1 X H (Γ) = H (Γh) > |γ(th) − γ(th−1)| . h=1 h=1 Remark 3.27. When γ : [0, a] → Rn is of class C1, it is immediately seen that (3.27) holds with c = 0 and d = a. In particular, by Theorem 3.25, if γ is injective and Γ = γ([0, a]), Z a H1(Γ) = |γ0(t)| dt . 0 This is the one-dimensional case of the area formula discussed in the previous section.

Hausdorﬀ measures and Lebesgue measure

We are going to compare the outer measures Ln and Hs on Rn. Let us ﬁrst estimate the values of Hs on the balls. Proposition 3.28. s s n (3.36) H (B(x, r)) = c(s, n) r x ∈ R , 0 < r < ∞ with c(s, n) positive and ﬁnite constant only when s = n; for s > n, c(s, n) = 0; for s < n, c(s, n) = ∞.

Corollary 3.29. (i) Hs is a (non trivial) Radon measure on Rn if and only s = n. (ii) Hdim(A) = n for each (nonemtpy) open set A ⊂ Rn. In particular Hdim(Rn) = n . Proof. (i) Is is an immeditae consequence of Proposition 3.28. (ii) By Proposition 3.28 and (3.18), it follows that Hdim(U(x, r)) = n for each x ∈ Rn and r > 0, where U(x, r) is an an open ball centered at x and with radius r > 0. Indeed, for ﬁxed r > 0, let (rh)h be a strictly increasing sequence of positive real numbers such that limh→∞ rh = r. Thus, we can write ∞ U(x, r) = ∪h=1B(x, rh) , ∞ and, by (3.17), we get the desired conclusion. Since by Lemma 1.16, A = ∪i=1Ui n with Ui (i ∈ N) open balls of R , by (3.17) we get the desired conclusion. 92

Proof of Proposition 3.28. Let us ﬁrst observe that, by (3.24) and (3.25), it follows that there exists c(s, n) := Hs(B(1)) ∈ [0, ∞] such that (3.36) holds. We have only to show that (3.37) 0 < c(n, n) = Hn(B(0, 1)) < ∞ . Indeed, from Theorem 3.9 and (3.37), it will follow that c(s, n) = ∞, if s < n and c(s, n) = 0 if s > n. Let us observe that, by (3.10), the proof of left-hand side inequality in (3.37) is equivalent to show that (3.38) Sn(B(1)) > 0 , where Sn denotes the n-dimensional spherical Hausdorﬀ measure on Rn. Let us n (i) (c) prelinimarly observe that, if B = B(x, r) is a closed ball of R and QB , QB denote, respectively, the inscribed and circumscribed n-dimensional (closed) cube to B, then

(i) (c) QB ⊂ B ⊂ QB , and their side length are 2r l Q(i) = √ and l Q(c) = 2r . B n B

(i) (c) In particular the diameters of QB and QB are

(i) (c) √ √ (3.39) d QB = 2r = d(B) and d QB = n 2r = n d(B) . Let us begin to prove (3.38). By definition of the n-dimensional pre-measure spherical Hausdorff measure and (3.39), we get, for each δ ∈ (0, ∞], (3.40) ( ∞ ) n X n ∞ Sδ (B(1)) = inf αn d(Bj) : B(1) ⊂ ∪j=1Bj,Bj closed ball, d(Bj) 6 δ j=1 ( ∞ ) n αn X (c) ∞ = inf d Q : B(1) ⊂ ∪ Bj,Bj closed ball, d(Bj) δ nn/2 Bj j=1 6 j=1 ( ∞ n ) X d (Qj) inf α √ : B(1) ⊂ ∪∞ Q ,Q closed cube > n n j=1 j j j=1 ( ∞ ) X n ∞ = inf αn L (Qj): B(1) ⊂ ∪j=1Qj,Qj closed cube j=1 n > αn L (B(1)) > 0 , where the last inequality follows according to one of possible definitions of n- dimensional Lebesgue measure (see, for instance, [GZ, Sect. 4.3]). Passing to the limit as (c) n δ → 0 in (3.40), (3.38) follows. Let Q∗ := QB(1) = [−1, 1] , then the right-hand side inequality in (3.37) will follow if we prove that

n (3.41) H (Q∗) < ∞ . 93

n n For each h ∈ N, let us divide Q∗ in h closed cubes Qk (k = 1, . . . , h ) with side length 2/h. Then, for each δ ∈ (0, ∞), by choosing h be such that √ n d(Q ) = < δ ∀ k = 1, . . . , hn , k 2h we get n n h n/2 h n/2 X αn n X 1 αn n Hn(Q ) α d(Q )n = = ∀ δ ∈ (0, ∞) . δ ∗ 6 n k 2n hn 2n k=1 k=1 By passing to the limit as δ → 0 in the previous inequality, (3.41) follows. By means of Proposition 3.28 and Theorem 2.15, we can infer the agreement, up to a constant, of outer measures Hn and Ln on Rn. Hn(B(1)) Theorem 3.30. Let c := ∗ . Then αn n n n (3.42) H (A) = c L (A) ∀ A ⊂ R . Proof. Assume that it holds n n n (3.43) H (B) = c L (B) ∀ B ∈ B(R ) . Let us prove that (3.42) also holds. Indeed, since Hn and Ln are regular Borel outer n measures, for each A ⊂ R there exist Borel sets Bi (i = 1, 2) such that n n n n (3.44) A ⊂ B1 and H (B1) = H (A),A ⊂ B2 and L (B2) = L (A) . By (3.43) and (3.44), it follows that n n n n H (A) = H (B1) = c L (B1) > c L (A) , n n n n c L (A) = c L (B2) = H (B2) > H (A) . 1 Thus (3.42) follows. Let us now prove (3.43) . Let µ := Ln, ν := Hn and λ := c 1 (µ + ν). Then, by Proposition 3.28, µ, ν and λ are positive Radon measures on 2 measure space (Rn, B(Rn)) and n (3.45) µ(B(x, r)) = ν(B(x, r)) = λ(B(x, r)) ∀ x ∈ R , r ∈ (0, ∞) . Moreover it is trivial that (3.46) µ << λ and ν << λ . Thus, by (2.19), Z Z n (3.47) µ(B) = Dλµ dλ and ν(B) = Dλν dλ ∀ B ∈ B(R ) . B B On the other hand, since n Dλµ(x) = Dλν(x) = 1 ∀ x ∈ R , (3.43) follows. We are now going to characterize the constant c in Theorem 3.30. Indeed we will prove that c = 1. 94

n n n n n n Theorem 3.31 (H ≡ L ). L (A) = Hδ (A) = H (A) for each A ⊂ R , 0 < δ 6 ∞. The proof of Theorem 3.31 is based on the isodiametric inequality, which, as said before, is rather complicated. Indeed it follows by Steiner symmetrization, which we will not introduce here. Isodiametric inequality states that Euclidean balls in Rn are the sets of maximum n-dimensional Lebesgue measure among all sets of a given diameter. Theorem 3.32 (Isodiametric inequality). n n n L (A) 6 αn d(A) for A ⊂ R . Proof. See [EG, Theorem 1, Sect. 2.2]. Remark 3.33. Notice that the isodiametric inequality is not trivial: a set of a given diameter is not necessarily contained in a ball of the same diameter. Exercise: Let A be an equilater triangle of the plane R2, with side length l. Prove that d(A) = l and there is no closed ball with diameter l containing A. Proof of Theorem 3.31. 1st step. Let us ﬁrst observe that, by Corollary 3.29 (i) , Hn is a Radon measure on Rn. 2nd step. Let us prove the inequality n n n (3.48) Hδ (A) 6 L (A) ∀ A ⊂ R , 0 < δ 6 ∞ . Let V ⊂ Rn be an open set such that A ⊂ V an let δ G := B(x, r): x ∈ A, r < ,B(x, r) ⊂ V . δ 2 By Vitali’s covering property for Radon measures (see Theorem 2.11) the exists a disjoint subfamily F ⊂ Gδ, at most countable, such that Hn(A \ ∪F) = 0 . By Lemma 3.8, it also follows that

n Hδ (A \ ∪F) = 0 . n Therofore, since Hδ is an outer measure, n n X n Hδ (A) 6 Hδ (∪F) 6 Hδ (B) B∈F X n n n 6 αn d(B) = L (∪F) 6 L (V ) . B∈F Taking the inﬁmum in the previous inequality, over all open sets V ⊃ A, we get (3.48). 3rd step. Let us prove the inequality n n n (3.49) Hδ (A) > L (A) ∀ A ⊂ R , 0 < δ 6 ∞ .

Let (Ai)i be a sequence of sets such that ∞ (3.50) A ⊂ ∪i=1Ai, d(Ai) < δ ∀ i . 95

Then, by the isodiametric inequality, ∞ ∞ n X n X n L (A) 6 L (Ai) 6 αn d(Ai) . i=1 i=1

Taking the inﬁmum in the previous inequality, over all sequences (Ai)i satisfying (3.50), we get (3.49).

Hausdorﬀ measures and Cantor sets

We have now introduced measures for measuring the size of very general sets. We give a look at some examples with which Hausdorff measures are convenient and useful. We begin with the most classical as the Cantor sets. Let us first introduce the Cantor sets in R. Let 0 < λ < 1/2 and define by steps the following intervals. Step k = 1: denote I0,1 := [0, 1] and let us delete the middle open interval of length (1 − 2λ) d(I0,1), that is interval (λ, 1 − λ). This yields 2 closed intervals

I1,1 =: [0, λ] and I1,2 =: [1 − λ, 1] . Let us deﬁne 2 C1(λ) := ∪j=1I1,j . We continue this process of selecting two subintervals of each already given interval.

Step k = 2: Let us delete from intervals I1,1 and I1,2 the open middle intervals of 2 length (1 − 2λ) d(I1,1) = (1 − 2λ) d(I1,2) = (1 − 2λ) λ. This yields 2 closed intervals which can denote from left to right as

I2,1,I2,2,I2,3,I2,4 . Let us deﬁne 4 C2(λ) := ∪j=1I2,j . Step k: if we have deﬁned the 2k−1 intervals

Ik−1,1,...,Ik−1,2k−1 , we define 2k intervals Ik,1,...,Ik,2k by deleting from the middle of each Ik−1,j an interval of length (1 − 2λ)d(Ik−1,j) = k−1 (1 − 2λ)λ . All the intervals Ik,j thus obtained have length k k d(Ik,j) = λ ∀ j = 1,..., 2 . Let us define 2k Ck(λ) := ∪j=1Ik,j . We define the limit set of this construction by ∞ (3.51) C(λ) := ∩k=1Ck(λ) . Then it is well-known that : • C(λ) is an uncountable compact set , ◦ • C(λ) = ∅, •H 1(C(λ)) = L1(C(λ)) = 0. 96

If λ = 1/3, C(1/3) is the celebrated Cantor middle-third set. We shall now study the Hausdorff measures and dimension of C(λ). As usual, it is much simpler to find upper bounds than lower bounds for the Hausdorff measures. This is due to the definition: a suitable chosen covering will give an upper estimate, but a lower estimate requires finding an infimum over arbitrary coverings. By using Remark 3.12, our goal is to find out a value s ∈ (0, 1) such that (3.52) 0 < Hs(C(λ)) < ∞ from which it will follow that α = Hdim(C(λ)). Question: How can we guess a value s satisfying (3.52)? A possible way is by using the self-similar structure of C(λ), of which we speak about. Let Si : R → R (i = 1, 2)

S1(x) := λ x, S2(x) := λ x + 1 − λ .

Then Si are two similarities of R and Exercise:

S1 (C(λ)) = C(λ) ∩ [0, λ],S2 (C(λ)) = C(λ) ∩ [1 − λ, 1] .

(Hint: Let us ﬁrst prove that C(λ) = S1(C(λ)) ∪ S2(C(λ)), from which the desired conclusion will follow.) Therefore, by the previous exercise, (3.24) and (3.25), it follows that Hs (C(λ)) = Hs (C(λ) ∩ [0, λ]) + Hs (C(λ) ∩ [1 − λ, 1]) s s = H (S1(C(λ))) + H (S2(C(λ))) = λs Hs (C(λ)) + λs Hs (C(λ)) = 2 λs Hs (C(λ)) . Thus, if (3.52) holds, then 2 λs = 1, or, equivalently, log 2 (3.53) s = 1 . log λ Theorem 3.34 (Hausdorﬀ dimension of the Cantor sets in R). Let s be the value in (3.53) and let αs be the constant in (3.23). Then s (i) H (C(λ)) 6 αs < ∞; s (ii) H (C(λ)) > αs > 0, s In particular H (C(λ)) = αs and log 2 Hdim(C(λ)) = 1 . log λ Remark 3.35. Note that Hdim(C(λ)) measures the sizes of the Cantor sets C(λ) in a natural way: when λ increases, the sizes of the deleted holes decrease and the sets C(λ) become larger, and also Hdim(C(λ)) increases. Notice also that when λ runs from 0 to 1/2, Hdim(C(λ)) takes all the values between 0 and 1. 97

Proof. (i) Notice that, by deﬁnition, 2k C(λ) ⊂ Ck(λ) := ∪j=1Ik,j∀ k and so, if δ = λk

2k s X s k ks s k Hλk (C(λ)) 6 αs d(Ik,j) = αs 2 λ = αs (2 λ ) = αs . j=1 Letting k → ∞ in the previous inequality, we get the desired inequality. (ii) This lower bound is harder and we recommend [Ma, 4.10]. Several diﬀerent Cantor-type sets can be constructed in Rn still for n = 1 that in higher dimensions n > 2 (see [Ma, 4.11-13] and [Fa] for a deeper analysis). Hausdorﬀ measures as Radon measures

It is immediate that Hk E induces a positive finite measure in Rn whenever E is Hk measurable and Hk(E) < ∞. Conversely, in many applications one needs to know whether a given measure µ is representable in terms of the Hausdorff measure, or at least needs to estimate the Hausdorff dimension of the set where µ is concentrated. In k ∗ k order to compare µ with H the natural idea is to look at the ratio µ(B(x, r))/αk r , and this motivates the following definition. Definition 3.36 (k-dimensional densities). Let µ be a positive Radon measure in an open set Ω ⊂ Rn and k > 0. The upper and lower k -dimensional densities of µ at x are respectively defined by

∗ µ(B(x, r)) µ(B(x, r)) Θ (µ, x) := lim sup , Θ∗k(µ, x) := lim inf k ∗ k + ∗ k r→0+ αk r r→0 αk r ∗ If Θk(µ, x) = Θ∗k(µ, x) their common value is denoted by Θk(µ, x) and this notation is also used for Rm -valued Radon measures µ whenever the densities of their component µi are defined, i.e. the i-th component of Θk(µ, x) is Θk(µi, x) for any i = 1, . . . , m. For any Borel set E ⊂ Rn we define also k k ∗ H (E ∩ B(x, r)) H (E ∩ B(x, r)) Θ (E, x) := lim sup , Θ∗k(E, x) := lim inf k ∗ k + ∗ k r→0+ αk r r→0 αk r and, if they agree, we denote the common value of these densities by Θk(E, x). ∗ ∗ k k Clearly Θk(E, x) = Θk(H E, x) and Θ∗k(E, x) = Θ∗k(H E, x). Using the left continuity of (0, ∞) 3 r 7→ µ(B(x, r)) it can be easily checked that all the densities ∗ are Borel functions of x. Now we see how the upper density Θk(µ, x) can be used to estimate from below and from above µ with Hk, which will turn out to be very useful in the topic of rectifiable sets. Theorem 3.37 (Estimates of the upper density of a Radon measure). Let Ω ⊂ Rn be an open set and µ a positive Radon measure in Ω. Then, for any t ∈ (0, ∞) and any Borel set B ⊂ Ω the following implications hold: ∗ k (3.54) Θk(µ, x) > t ∀ x ∈ B ⇒ µ > t H B,

∗ k k (3.55) Θk(µ, x) 6 t ∀ x ∈ B ⇒ µ 6 2 t H B. 98

Proof. See, for instance, [AFP, Theorem 2.56]. Two fundamental consequences of Theorem 3.37, very useful in the study of measure- theoretic property of sets, are the following.

Corollary 3.38. Let k ∈ [0, n] and assume that E ⊂ Rn is Hk-measurable and Hk(E) < ∞. Then k H (E ∩ B(x, r)) k n (3.56) ∃ Θk(E, x) = lim = 0 for H -a.e. x ∈ \ E ; + ∗ k R r→0 αk r

k −k H (E ∩ B(x, r)) k (3.57) 2 6 lim sup ∗ k 6 1 for H -a.e. x ∈ E. r→0+ αk r Remark 3.39. If k = n, let us recall that, by the result concerning the density of a set (see Corollary 2.18) n n H (E ∩ B(x, r)) L (E ∩ B(x, r)) n n lim = lim = χE(x) H -a.e. x ∈ . + ∗ n + n R r→0 αn r r→0 L (B(x, r) It is not the case when k ∈ (0, n), even though k is integer. Indeed it is possible to have Hk(E ∩ B(x, r)) lim sup ∗ k < 1 r→0+ αk r and Hk(E ∩ B(x, r)) lim inf = 0 + ∗ k r→0 αk r for Hk-a.e. x ∈ E even if 0 < Hk(E) < ∞, for a E not regular in measure-theoretic sense, that is if E is unrerctiﬁable (see Example 4.28). Proof of Corollary 3.38. (i) Let t > 0 and let n ∗ Bt := {x ∈ R \ E :Θk(E, x) > t} . k Then, by (3.54) with µ = H E and B = Bt, 1 Hk(B ) = (Hk B )(B ) µ(B ) t t t 6 t t 1 (Hk E)( n \ E) = 0 ∀ t > 0 . 6 t R Thus (3.56) follows. (ii) Let us ﬁrst prove the left inequality in (3.57), which amounts to prove that

k ∗ −k (3.58) H (B) = 0 if B := x ∈ E :Θk(E, x) < 2 . −k Let tj := 2 (1 − 1/j), if j > 2, and ∗ Bj := {x ∈ E :Θk(E, x) 6 tj} . k Then, by (3.55) with µ = H E and B = Bj, k k k H (Bj) = (H Bj)(Bj) 6 2 tj µ(Bj) k k = 2 tj H (Bj) ∀ j > 2 . 99

k k Since 2 tj < 1 for each j > 2, H (Bj) = 0 for each j, which implies ∞ k k ∞ X k H (B) = H (∪j=2Bj) 6 H (Bj) = 0 j=2 and then (3.58) follows. Finally the proof of the right inequality in (3.57) is similar. Indeed this amounts to prove that k ∗ (3.59) H (B) = 0 if B := {x ∈ E :Θk(E, x) > 1} .

Let tj := 1 + 1/j and ∗ Bj := {x ∈ E :Θk(E, x) > tj} . k Then, applying (3.54) , we get that H (Bj) = 0 for each j, which yields the desired conclusion.

3.4. Area and coarea formulas in Rn and some applications.

Some recalls of linear algebra

Before stating the two main results (Theorems 3.48 and 3.11) of this subsection, we recall some notations and results of linear algebra. Notation: If L : Rn → Rm is a linaar map, we identify L with its transformation matrix ML, that is the (m×n)- matrix which represents L with respect to the standard n m bases of R and R . Moreover, if n = m we define detL := detML. Definition 3.40. (i) Given L : Rn → Rm linear map, its norm kLk is defined by (3.60) kLk := sup {|L(v)| : |v| 6 1} (ii) A linear map O : Rn → Rm is orthogonal if n (Ox, Oy)Rm = (x, y)Rn ∀ x, y ∈ R . (iii) A linear map S : Rn → Rn is symmetric if n (Sx, y)Rn = (x, Sy)Rn ∀ x, y ∈ R . (iv) Let L : Rn → Rm be a linear map. The adjoint of L is the linear map LT : Rm → Rn defined by T n m (x, L y)Rn = (Lx, y)Rm ∀ x ∈ R , y ∈ R . (iv) Let L : Rn → Rm be a linear map. Then the rank of L, denoted rank(L) is the rank of its associated transformation matrix ML, that is, the maximun number of columns ( or rows) of ML linearly independent.

Remark 3.41. (i) If L : Rn → Rm is a linear map, the, L is Lipschitz function and kLk = Lip(L). n m (ii) If O : R → R is an orthogonal map, then it is injective and so n 6 m. Theorem 3.42 (Polar Decomposition). Let L : Rn → Rm be a linear function. Then 100

n n (i) If n 6 m there are a symmetric linear function S : R → R and an orthogonal linear function O : Rn → Rm such that L = O ◦ S. Moreover it holds |detS| = pdet(LT ◦ L) . m m (ii) If n > m there are a symmetric linear function S : R → R and an orthogonal linear function O : Rm → Rn such that L = S ◦ OT . Moreover it holds |detS| = pdet(L ◦ LT ) . Proof. See [EG, Sect. 3.2]. Definition 3.43 (Jacobian of a linear map). Assume L : Rn → Rm be a linear function. (i) If n 6 m and assume that L = O ◦ S as above. We define the Jacobian of L to be JL := |detS| = pdet(LT ◦ L) . T (ii) If n > m and assume that L = S ◦ O as above. We define the Jacobian of L to be JL := |detS| = pdet(L ◦ LT ) . Remark 3.44. (i) It follows from Theorem 3.42 that the definition of JL is independent of the particular choice of S and O. (ii) Clearly JL = JLT . n m Theorem 3.45 (Binet-Cauchy formula). Assume that n 6 m and let L : R → R be a linear map. Then s X 2 (3.61) JL = (detN)

N⊂ML where the sum is understood over each (n×n)-submatrix N of (m×n)- transformation matrix ML of L. Proof. See [EG, Sect. 3.2]. Remark 3.46. From the deﬁnition of jacobian and Binet-Cauchy formula (3.61), we can infer that, if L : Rn → Rm is a linear map, then JL = 0 if and only if rank(L) < min{n, m} .

Area formula We are going to prove that the integral n-dimensional Hausdorﬀ measure Hn with n = 1, 2, . . . , m turns out to be the classical surface n-measure ( or, also, n-volume) for regular n-dimensional submanifold of Rm. Let us recall that, by Theorems 3.25 and 3.31, we partially met this goal for cases k = 1, m. We are going to accomplish the task in the remaining cases by means of the area formula. 101

m A possible way for introducing a regular n-dimensional submanifold of R (n 6 m) is through a (regular) parametrization (see, for instance, [Fe, 3.1.19]). For instance, a n- regular submanifold Γ ⊂ Rm can be given as follows . Assume that Γ = f(A) where A ⊂ Rn is a regular bounded subset and f : A¯ ⊂ Rn → Rm, called parametrization of Γ, satisfies • f : A → Rm is injective; • f ∈ C1(A¯; Rm) and df : A → Rm has maximal rank. If this is the case, it well-known that the n- volume of Γ, is defined as Z n − volume(Γ) := Jf(x) dLn(x) , A where Jf(x) denotes the Jacobian of f at x. Jacobian Jf is the corrective factor relating the elements of volumes of the domain and image of f. Example: (Jacobian for a surface in R3) Suppose that n = 2 and m = 3, Γ = ¯ 2 ¯ f(A), with (s, t) ∈ A ⊂ R and f = f(s, t) = (f1(s, t), f2, (s, t), f3(s, t)) : A ⊂ 2 3 R → R satisfying the previous assumptions. Let ∂sf := (∂sf1, ∂sf2, ∂sf3), ∂tf := (∂tf1, ∂tf2, ∂tf3) and let Ni (i = 1, 2, 3) denote (all) the 2 × 2-submatrices of the 3 × 2 Jacobian matrix of f   ∂sf1(s, t) ∂tf1(s, t) Df(s, t) := ∂sf2(s, t) ∂tf2(s, t) ∂sf3(s, t) ∂tf3(s, t) Then it is well-known by differential geometry that the Jacobian of f at (u, v) equals

p 2 2 2 ¯ Jf(s, t) = |∂sf ∧ ∂tf|(s, t) = (detN1) + (detN2) + (detN3) (s, t)(s, t) ∈ A, where v ∧ w denotes the exterior product of vectors v, w ∈ R3. Problem: Hn(Γ) = n − volume(Γ)? A positive answer is a particular case of the area formula.

Definition 3.47. Assume f : Rn → Rm be a differentiable at x ∈ Rn. (i) If n 6 m, we define the Jacobian of f at x to be Jf(x) := Jdf(x) = pdet (Df(x)T · Df(x)) ; (ii) If n > m, we define the Jacobian of f at x to be Jf(x) := Jdf(x) = pdet (Df(x) · Df(x)T ); where Df(x) denote the m × n-Jacobian matrix of f at x.

n m According to the previous definition, if f : R → R is Lipschitz with n 6 m, we define as Jacobian of f the Borel measurable function Jf : Rn → [0, ∞] defined as (3.62) ( pdet (Df(x)T · Df(x)) if f is differentiable at x Jf(x) := if n m , ∞ if f is not differentiable at x 6 102 and (3.63) ( pdet (Df(x) · Df(x)T ) if f is differentiable at x Jf(x) := if n m . ∞ if f is not differentiable at x > Notice that, by Rademacher’s theorem, set {Jf < ∞} coincides with the set of points x ∈ Rn at which f is differentiable and it has full Lebesgue measure in Rn. n m Theorem 3.48 (Area formula). Let f : R → R be Lipschitz with n 6 m. Then for each Ln-measurable subset A ⊂ Rn Z Z (AF) Jf dLn = H0 A ∩ f −1(y) dHn(y) . A Rm Function Rm 3 y 7→ H0 (A ∩ f −1(y)) ∈ N ∪ {∞} is called multiplicity function of f. Remark 3.49. (i) Notice that, if y∈ / f(A), then H0 (A ∩ f −1(y)) = 0. Hence (AF) can be equivalently written Z Z (3.64) Jf dLn = H0 A ∩ f −1(y) dHn(y) . A f(A) n m (ii) It follows that, if n 6 m, f : R → R is Lipschitz and A is bounded, then, by (AF), 0 −1 n m H A ∩ f (y) < ∞ H -a.e. y ∈ R . Therefore f −1(y) is at most countable for Hn-a.e. y ∈ Rm. Area formula (AF) immediately yields a positive answer to the previous question.

n m Theorem 3.50 (Area formula for injective maps). Let n 6 m and let f : R → R be an injective Lipschitz function and A ⊂ Rn be a measurable set. Then Z (IAF) Hn(f(A)) = Jf dLn A and Hn f(Rn) is a Radon measure on Rm. We are now going to give an idea of the proof of the area formula for injective maps, that is Theorem 3.50. Proof of the general area formula, that is Theorem 3.48, can be obtained by using the are formula formula for injective maps (see [Mag, Theorem 8.9]). We preliminarly need the following fundamental results.

Lemma 3.51 (Measurability of Lipschitz functions images ). If n 6 m, E is a Lebesgue measurable set in Rn and f : Rn → Rm is a Lipschitz function, then f(E) is Hn-measurable in Rm. Proof. See [EG, Lemma 2, Sect. 3.3.1] or [Mag, Lemma 8.4]. n m Lemma 3.52 (Area formula for linear functions). Let n 6 m and let L : R → R be a linear function. Then for all A ⊂ Rn (3.65) Hn(L(A)) = JL Ln(A) 103

Remark 3.53. Notice that that an alternative deﬁnition of Jacobian JL for a linear n m n n map L : R → R (n 6 m) can be as value H (L(Q)) where Q = [0, 1] is the unit cube of Rn. Proof. See [EG, Lemma 1, Sect. 3.3.1] or [Mag, Theorem 8.5]. n Lemma 3.54 (Role of the singular set of a Lipschitz map). Let n 6 m and f : R → Rm be Lipschitz. Then n n H (f({x ∈ R : Jf(x) = 0})) = 0 . Proof. See [Mag, Theorem 8.7]. n m Lemma 3.55 (Lipschitz linearization). Let n 6 m and f : R → R be Lipschitz. Let us ﬁx (an arbitrary) t > 1. Denote n F := {x ∈ R : 0 < Jf(x) < ∞} , then there exists a countable disjoint family of Borel sets (Fh)h ∈ N such that ∞ (i) F = ∪h=1Fh;

(ii) f|Fh is injective; n n (iii) for each h, there exists a symmetric automorphism Sh : R → R such that −1 n m f|Fh ◦ Sh : Sh(Fh) ⊂ R → f(Fh) ⊂ R is a bi-Lipschitz map and the n following estimates hold: for every x, y ∈ Fh and v ∈ R −1 −1 (3.66) Lip f|Fh ◦ Sh 6 t, Lip Sh ◦ (f|Fh ) 6 t

1 (3.67) |S v| |Df(x)v| t |S v|, t h 6 6 h 1 (3.68) JS Jf(x) tn JS . tn h 6 6 h Proof. The primitive proof is in [Fe, 3.2.2]; see also [EG, Lemma 3, Sect. 3.3.1] and [Mag, Theorem 8.8]. Proof of Theorem 3.50. Step 1: Let us prove that (3.69) Hn(f(A)) = Hn(f(A ∩ F )) , where F is the set in Lemma 3.55. Let us recall that, by Theorem 3.22 and the agreement Ln = Hn on Rn (see Theorem 3.31) n n n n (3.70) H (f(E)) 6 Lip(f) L (E) ∀ E ⊂ R . Thus both sides of (IAF) are zero whenever Ln(A) = 0. By Rademacher’s theorem and Lemma 3.54, we get

n n n H (f(A ∩ F )) 6 H (f(A)) = H (f((A ∩ F ) ∪ (f(A \ F ))) n n 6 H (f(A ∩ F ) + H (f(A \ F )) n n n 6 H (f(A ∩ F ) + H (f({Jf = 0})) + H (f({Jf = ∞})) = Hn(f(A ∩ F )) , which shows (3.69). 104

Step 2: Let us prove (IAF). By the previous step, we can assume that A ⊂ F .

We now ﬁx t > 1 and consider the partition (Fh)h∈N of F given by Lemma 3.55. We see A as the union of the disjoint sets (Fh ∩ A)h, so that, by the global injectivity of f, we have that f(A) is the disjoint union of the sets (f(Fh ∩ A))h, which are Hn-measurable by Lemma 3.51. Therefore, by Theorem 3.22, the linear case of the area formula (3.65) and (3.66), we ﬁnd that

∞ ∞ n X n X n −1 H (f(A)) = H (f(A ∩ Fh)) = H (f|Fh ◦ Sh )(Sh(A ∩ Fh)) h=1 h=1 ∞ X −1 n n 6 Lip(f|Fh ◦ Sh ) H (Sh(A ∩ Fh)) h=1 (3.71) ∞ n X n 6 t JSh L (A ∩ Fh) h=1 ∞ Z Z 2n X 2n 6 t Jf(x) dx = t Jf(x) dx ∀ t > 1 . h=1 A∩Fh A In a similar way, by analogous argument Z ∞ Z ∞ X n X n Jf(x) dx = Jf(x) dx 6 t JSh L (A ∩ Fh) A h=1 A∩Fh h=1 ∞ n X n −1 (3.72) = t H (Sh ◦ (f|Fh ) )f(A ∩ Fh) h=1 ∞ 2n X n 2n n 6 t H (f(A ∩ Fh)) = t H (f(A)) ∀ t > 1 . h=1 We thus prove (IAF) by letting t → 1+ in (3.71) and (3.72). Step 3: Let us prove that Hn f(Rn) is a Radon measure. By Lemma 3.51, f(Rn) is Hn-measurable, while (IAF) implies Hn f(Rn) to be locally ﬁnite. By Theorem 1.94, Hn f(Rn) is a Radon measure. Some applications of the area formula:

m (1) (Length of a curve). Assume that n = 1, m > 1 and f : R → R is an injective Lipschitz function. In this case, the Jacobian matrix of f

 0  f1(t) 0 T  ...  1 Df(t) = f (t) =   L -a.e. t ∈ R ,  ...  f 0 (t) m m×1 and according to deﬁnition of the Jacobian of f

p 0 1 Jf(t) = det (Df(t)T · Df(t)) = |f (t)| L −a.e.t ∈ R . 105

Thus, by (IAF), for a, b ∈ R with a < b, Z b H1(Γ) = |f 0(t)| dt if Γ := f([a, b]) . a n (2) (Area of a graph). Assume that n > 1, m = n + 1 and u : R → R is a Lipschitz function. Let f : Rn → Rn+1 deﬁned as n f(x) := (x, u(x)) x ∈ R . Then  1 0 ...... 0   0 1 0 ... 0    n n Df(x) =  ......  L -a.e. x ∈ R ,    0 0 ... 0 1  ∂ u(x) ...... ∂ u(x) 1 n (n+1)×n and, by applying Binet-Cauchy formula (3.61), p p n n (3.73) Jf(x) = det (Df(x)T · Df(x)) = 1 + |∇u(x)|2 L -a.e. x ∈ R . Thus, if A ⊂ Rn is an open set and Γ = f(A) = graph(u; A) := {(x, u(x)) : x ∈ A}, Z (3.74) Hn(Γ) = p1 + |∇u(x)|2 dLn(x) . A Finally let us state a general formula for the change of variables.

n m Theorem 3.56 (Change of variables). Let n 6 m and f : R → R be Lipschitz. The for each Ln-integrable function g : Rn → R,   Z Z n X n (3.75) g(x) Jf(x) dL (x) =  g(x) dH (y) . n m R R x∈f −1(y) In particular, if f is injective, Z Z (3.76) g(x) Jf(x) dLn(x) = g(f −1(y)) dHn(y) . Rn f(Rn) Proof. See [EG, Theorem 2, Sect. 3.3.3]. Remark 3.57. Recall that f −1(y) is at most countable Hn-a.e. y ∈ Rm (see Remark 3.49 (ii)).

Coarea formula

We are now going to presente a far-reaching generalization of Fubini’s theorem (see Theorem 1.101), which is very useful in GMT and, more generally, in analysis. Let us begin with an example.

Example: Let n = 2 and Q = [0, 1]2 and let L : R2 → R be the linear map L(x1, x2) := x1, that is the (orthogonal) projection on the x1-axis. If t ∈ R, then −1 L (t) = {(t, x2): x2 ∈ R} , 106

−1 that is, L (t) is a straight line parallel to the x2-axis. It is trivial to see that, if we denote At := {s ∈ R :(t, s) ∈ A}, then 1 1 −1 L (At) = H (Q ∩ L (t)) ∀ t ∈ R , and, by Fubini’s theorem, we can write that, if n = 2, m = 1, Z Z 2 2 1 n−m −1 (3.77) H (Q) = L (Q) = L (At) dt = H (Q ∩ L (t)) dt . R R Let us now consider the linear map L(x1, x2) = −x1 + x2. If t ∈ R, then −1 L (t) = {(x1, x1 + t): x1 ∈ R} , that is L−1(t) is still a straight line, but it is not parallel to the coordinate axes. In particular identity (3.77) no more holds. Indeed, we can get, by a simple calculation, that (√ 2(1 − |t|) if |t| 1 H1(Q ∩ L−1(t)) = 6 0 if |t| > 1 and Z √ √ H1(Q ∩ L−1(t)) dt = 2 = 2 H2(Q) . R

n m Problem: Let n > m and let L : R → R be a linear function, ﬁnd out a non negative factor c(L) such that Z n−m −1 m n n (3.78) H (A ∩ L (y)) dL (y) = c(L) L (A) ∀ A ⊂ R . Rm n m Theorem 3.58 (Coarea for linear maps). Let n > m and let L : R → R be linear. Then (3.78) holds with c(L) := JL = pdet(L ◦ LT ) where LT : Rm → Rn denotes the adjoint linear map of L. Proof. See [EG, Lemma 1, Sect. 3.4.1]. n m Theorem 3.59 (Corea formula). Let n > m and let f : R → R be a Lipschitz function. The for each Ln-measurable set A ⊂ Rn Z Z Jf(x) dLn(x) = Hn−m(A ∩ f −1(y)) dLm(y) , A Rm where Jf is the Jacobian factor deﬁned in (3.63).

Proof. See [EG, Theorem 1, Sect. 3.4.1]. Remark 3.60. Applying the coarea formula to set A := {x ∈ Rn : Jf(x) = 0}, we get that n−m −1 m m (WMS) H {Jf = 0} ∩ f (y) = 0 L -a.e. y ∈ R . This is a weak variant of Morse-Sard’s theorem which asserts −1 m m (MS) {Jf = 0} ∩ f (y) = ∅ L -a.e. y ∈ R , provided that f ∈ Ck(Rn; Rm) for k = 1 + n − m. Observe, however, that (WMS) only requires that f be Lipschitz. 107

Theorem 3.61 (Change of variables formula). Under the same assumptions of the Coarea Formula. Then for each Ln-measurable function g : Rn → R n−m m m (i) g|f −1(y) is H -summable L -a.e. y ∈ R . (ii) Z n Z Z g(x) Jf(x) dLn(x) = g(x) dHn−m(x) dLm(y) , R Rm f −1(y) Proof. See [EG, Theorem 2, Sect. 3.4.3].

3.5. Extensions to metric spaces.

About Hausdorﬀ measures and length measure

Let (X, d) be a metric space. A curve of X is a continuous function γ :[a, b] → X; its support is the subset Γ = γ([a, b]) ⊂ X and, in this case, γ is called a parametrization of Γ. Given a curve γ :[a, b] → X and a subinterval [c, d] ⊆ [a, b], we deﬁne variation of γ on [c, d]

( m ) X Var(γ;[c, d)) := sup d(γ(ti), γ(ti−1)) : t0 = c < t1 < ··· < tm = d ∈ [0, ∞] . i=1 By analogy with the case where X = Rn and d is the Euclidean distance, the quantity Var(γ;[c, d)) represents the length (with respect to the metric d) of curve γ over [c, d]. Therehore, given a subinterval [c, d] ⊆ [a, b], we define the length of γ over [c, d] as (3.79) l(γ;[c, d]) := Var(γ;[c, d)) . Thus we can define as length of γ the quantity l(γ) := l(γ;[a, b]) . We say that γ is rectifiable if l(γ) < ∞. It is easy to see that, if γ :[a, b] → X is a curve, then

(3.80) l(γ;[c, d]) > d(γ(c), γ(d)), whenever a 6 c 6 d 6 a ; and

(3.81) l(γ;[a, b]) = l(γ;[a, c]) + l(γ;[c, b]), whenever a 6 c 6 b . Theorem 3.62 (Classical length and H1). Let γ :[a, b] → X be a curve and denote Γ = γ([a, b]) its support. Then 1 1 H (Γ) 6 S (Γ) 6 l(γ) and equality holds if γ is injective, where S1 denotes the 1-dimensional spherical Hausdorﬀ measure deﬁned in Remark 3.5. Proof. Proof’s strategy is similar to the one of Theorem 3.25 by means of suitable changes (see [AT, Theorem 4.4.2] and [SC, Theorem 2.29]). 108

About Hausdorﬀ dimension in a metric measure space

An useful criterion to estimate the Hausdorﬀ dimension of a metric measure space is the following. Theorem 3.63. Let (X, d, µ) be a metric measure space with (X, d) separable. Sup- pose that there exist constants c > 1 and Q > 0 such that 1 (3.82) rQ µ(B(x, r)) c rQ ∀ 0 < r < d(X) . c 6 6 Then there exists a constant c0 > 1 such that 1 (3.83) HQ(E) µ(E) c0 HQ(E) , c0 6 6 for each Borel set E ⊂ X. In particular Hdim(X) = Q.

Proof. See, for instance, [SC, Theorem 2.26]. Metric measure spaces where formula (3.82) holds are called Ahlfors regular of dimension Q. By (3.83), we can replace µ by the Hausdorﬀ Q-measure in an Q- regular space without essential loss of information.

About Lipschitz maps

Extension’s problem for Lipschitz maps acting between metric spaces is an issue of the current research in analysis in metric spaces (see [AT, Chap. 3], [He, Chap. 6] and [He2]). The proof of Kirszbraun’s theorem depends crucially on very euclidean properties of the domain space Rn. Hence its proof cannot be extended to general metric spaces. Diﬀerentiability for Lipschitz functions f : Rk → (X, d) has been studied. In particular Rademacher-type theorems have been obtained (see [AK]). A very important Rademacher type-result for real valued Lipschitz functions de- ﬁned on a metric measure space (X, d, µ), that is Lipschitz functions f :(X, d, µ) → R, has been obtained by Cheeger [Ch] and it has been a seminal paper for the de- velopement of analysis in metric measure spaces (see, for instance, [AG] and the references therein).

About Lipschitz maps and Hausdorﬀ measures

Notice that the conclusion of Theorem 3.22 holds for each Lipschitz map f : (X, d) → (Y,%). Notice also that for every isometric embedding f :(X, d) → (Y,%), that is a mapping satisfying %(f(x), f(y)) = d(x, y), by deﬁnition of Hausdorﬀ measure, it follows that Hs(f(A)) = Hs(A) ∀ A ⊂ X.

About area and coarea formulas 109

Area and coarea formulas have been obtained respectively for Lipschitz functions f : Rk → (X, d) and f :(X, d) → Rk (see [AK]).

4. Rectifiable sets and blow-ups of Radon measures ([AFP, Mag, Ma])

Motivation: the introduction of the notion of rectifiable set in Rn, which is a set smooth in a certain measure-theoretic sense. It is an extension of the idea of a rectifiable curve to higher dimensions and it has many of the desirable properties of smooth manifolds, including tangent spaces that are defined almost everywhere. Rectifiable sets are fundamental objects of study in geometric measure theory.

k-dimensional planes and the orthogonal group in Rn.

Let us begin to recall the notion of k-dimensional plane in Rn and some their properties. We simply mean by a k-dimensional plane π in Rn a k-dimensional subspace of Rn. We will denote by G(n, k) the class of k-dimensional planes of Rn, which is also called Grassmannian manifold of k dimensional subspaces of Rn. If π is a k-dimensional plane we denote by π⊥ the (n − k)-plane orthogonal to π, that is ⊥ n π := {x ∈ R :(x, y)Rn = 0 ∀ y ∈ π} . n Notice also that, if π ⊂ R is a k-plane and {v1, . . . , vk} is an orthonormal basis of π k n then the map Iπ : R → R k X k (4.1) Iπ(x) := xj vj if x ∈ R j=1 k deﬁnes an (injective) orthogonal map such that Iπ(R ) = π. n n n ⊥ n Given a k-dimensional plane π in R , we denote by Pπ : R → R and Pπ : R → Rn respectively the orthogonal projections of Rn onto π and π⊥ . In particular it ⊥ turns out that Pπ = Pπ⊥ . Tipically we identify π with Pπ and we endow G(n, k) by the distance

(4.2) |π1 − π2|G := kPπ1 − Pπ2 k = sup |Pπ1 (v) − Pπ2 (v)| if πi ∈ G(n, k) i = 1, 2 , v∈Sn−1 that is k · k denotes the norm in (3.60).

Exercise: Prove that

(4.3) (G(n, k), | · |G) is a compact metric space.

The orthogonal group O(n) of Rn consists of all linear orthogonal maps L : Rn → Rn according to Definition 3.40 (ii), that is linear maps preserving the inner product, n (L(x),L(y))Rn = (x, y)Rn ∀ x, y ∈ R , 110 or equivalently L is an isometry, n |L(x) − L(y)| = |x − y| ∀ x, y ∈ R . It is easy to see that O(n) turns out to be a group with composition as group law. The members of O(n) consist of rotations and rotations composed with a reflexion over some hyperplane. Another way to view them is to observe that they map orthonormal basis to orthonormal basis, and conversely given two orthonormal bases u1, . . . , un and n v1, . . . , vn of R one can define L ∈ O(n) by setting L(ui) = vi and extending linearly. One of the basic properties of O(n) is that it acts transitively on Sn−1: for any x, y ∈ Sn−1 there exists L ∈ O(n) such that L(x) = y. In the case n = 2, 0(2) is very simple. It consists of rotations around the origin and of rotations composed with the reflexion over the x-axis. By simple linear algebra the action of O(n) on G(n, k) is distance-preserving:

|L(π1) − L(π2)|G = |π1 − π2|G ∀ L ∈ O(n), πi ∈ G(n, m) i = 1, 2 .

Also O(n) transitively acts on G(n, k): for πi ∈ G(n, k), i = 1, 2, there is L ∈ O(n) such that L(π1) = π2, that is π1 and π2 are isometric. To see this, take orthonormal n bases for π1 and π2, complete them to orthonormal bases of R , and choose L ∈ O(n) which maps one of these onto the other. n In particular it follows, if B1 = B(1) denotes the unit closed ball in R centered at n 0 and π is a k-dimensional plane in R , then set π ∩ B1 is isometric to the unit closed ball in Rk and so, by Remark 3.24, k ∗ (4.4) H (π ∩ B1) = αk . k Analogously, set π ∩ ∂B1 is isometric to the unit sphere in R and so k (4.5) H (π ∩ ∂B1) = 0 ,

Regular surfaces and rectiﬁable sets.

Let us now recall the classical notion of regular surface in Rn. n Deﬁnition 4.1. Given k ∈ N, 1 6 k 6 n − 1, m > 1, we shall say that Γ ⊂ R is a k-dimensional (embedded) surface (or submanifold) of class Cm in Rn (or a Cm- hypersurface when k = n − 1) if for every x ∈ Γ there exist an open neighborhood V ⊂ Rn of x, an open set U ⊂ Rk and a bijection f : U → V ∩ Γ with f ∈ Cm(U) and df(x): Rk → Rn is injective for each x ∈ U. Each map f is called a coordinate mapping of Γ.

Remark 4.2. (i) It is well-known from standard calculus that df(x): Rk → Rn is injective if and only if Jf(x) > 0, where Jf(x) denotes the Jacobian of f at x (see Deﬁnition 3.47 and Remark 3.46). (ii) Notice that, in this way, Γ is relatively open in Rn, and can be covered by countably many images f(U), with f and U as above. The notion of rectiﬁable set is just a generalization of Remark 4.2 (ii) to the measure-theoretic setting.

n Deﬁnition 4.3. Let 1 6 k 6 n − 1 be integers and let Γ ⊂ R . 111

(i) We say that Γ is countably Hk-rectifiable if ∞ (CR) Γ = Γ0 ∪ (∪i=1fi(Ai)) , k k k n where H (Γ0) = 0, Ai ⊂ R are L -measurable and fi : Ai → R are Lipschitz functions for i = 1, 2,... . (ii) We say that Γ is locally Hk-rectifiable if Γ is countably Hk-rectifiable and Hk(Γ ∩ K) < ∞ for each compact K ⊂ Rn. (iii) We say that Γ is Hk-rectifiable if Γ is countably Hk-rectifiable and Hk(Γ) < ∞. Remark 4.4. Since Lipschitz functions defined on subsets of Rk can be extended to all of Rk, keeping a control of the Lipschitz constant (see Theorem 3.18), then we can equivalently say that Γ is countably k-rectifiable if it is Hk-measurable and ∞ (CR*) Γ ⊂ Γ0 ∪ (∪i=1Γi) , k k k n where H (Γ0) = 0, Γi := fi(R ) and fi : R → R are Lipschitz functions for i = 1, 2,... . Remark 4.5. Observe that the rectifiability is a metric notion since it depends on the notion of Lipschitz maps acting between metric spaces (see section IV.4). Historical notes: The concepts of rectifiability were first introduced for one-dimensional sets in R2 by Besicovitch [Be3] in 1928 and then subsequently studied in papers [Be4, Be5] for k = 1 and n = 2. Federer extended the study for general k and n in 1947 and those results are contained in his celebrated monograph [Fe]. Example 4.6 (Lipschitz k-graph). Let π be a k-plane and let φ : π → π⊥ be a Lipschitz function.We call graph of φ the set n Γ = graph(φ) := {z + φ(z): z ∈ π} ⊂ R . Let us define f : Rk → Rn k f(x) := Iπ(x) + φ(Iπ(x)) x ∈ R , k n where Iπ : R → R is the map defined in (4.1). It is easy to check that f is still Lipschitz and, from Theorem 3.22, we get that Γ is locally Hk-rectifiable. In particular any compact subset of Γ is Hk-rectifiable. Remark 4.7. Observe that that countable Hk- rectifiability (CR*) is equivalent to the seemingly stronger requirement that sets Γi (i = 1, 2,... ) is a Lipschitz k-graphs (see [AFP, Proposition 2.76]). Using Whitney’s extension theorem (see Theorem 3.14) to approximate Lipschitz functions by C1 functions, it could be possible to 1 show that Γi (i = 1, 2,... ) can be a C k-graphs (see [Fe, 3.1.16]). However, since Lipschitz functions are more flexible than C1 functions in many typical constructions of geometric measure theory (see Mc Shane’s extension theorem 3.16), the use of Lipschitz functions is preferred.

In this chapter we are going to study some geometric properties of locally Hk- rectifiable sets, such as the existence Hk-a.e. of a tangent plane in a suitable sense and a their characterization in terms of measure-theoretic properties. We will strictly follow the arguments in [Mag, Chap. 10]. Observe that, whenever Γ is a countably Hk-rectifiable, then Hk Γ is a regular Borel outer measure. However, from Theorem 1.94, Hk Γ is a Radon measure if 112 and only if Γ is locally Hk-rectifiable. Therefore, it is under the assumption of local Hk-rectifiability on Γ that we have a natural identification between Γ and a Radon measure µ. In turn, as seen in Example 1.111, this identification lies at the basis of the measure-theoretic formulation of the notion of tangent space. Indeed, if Γ is locally Hk-rectifiable and µ = Hk Γ, then for Hk-a.e. x ∈ Γ there exists a k-dimensional n k plane πx in R such that the blow-ups µx,r of µ at x weak* converge to H πx as r → 0+, that is Γ − x (4.6) Hk *∗ Hk π as r → 0+ . r x A crucial fact is that the converse also holds true: if µ is a Radon measure on Rn concentrated on a Borel set Γ and such that for every x ∈ Γ there exists a k- dimensional plane πx such that the k-dimensional blow- ups of µ have the property that (Φ ) µ (4.7) µ = x,r # *∗ Hk π , as r → 0+, x,r rk x k k n n then Γ is locally H rectifiable and µ = H Γ where Φx,r : R → R is the map defined by y − x Φ (y) := y ∈ n . x,r r R 4.1. Rectifiable sets of Rn and their decomposition in regular Lipschitz images. In this section we are going to replace the decomposition of a countably Hk- rectifiable set, provided in (CR), with a ”good decompostiton” composed or ”regular” k n Lipschitz parametrization maps fi : Ai ⊂ R → R . Let us begin to introduce the notion of regular parametrization Lipschitz map. Definition 4.8 (Regular Lipschitz image). Given a Lipschitz map f : Rk → Rn and a compact set E ⊂ Rk, we say that the pair (f, E) defines a regular Lipschitz image f(E) in Rn if (i) f is injective and differentiable on E, with Jf(x) > 0 for each x ∈ E; (ii) Lk-a.e. x ∈ E is a point of density 1 for E; (iii) Lk-a.e. x ∈ E is a Lebesgue point of Df. Example 4.9. A k-regular C1 surface Γ ⊂ Rn can be seen as countable union of regular Lipschitz images. Indeed, according to Definition 4.1 and Remark 4.2, Γ n can be covered by a countable union of relatively compact open sets (Vi)i of R for which there exists a countable family of coordinate mappings fi : Ui → Γ ∩ Vi, k 1 ¯ n which are bijective, with (Ui)i relatively compact open sets of R , fi ∈ C (Ui; R ), k n ¯ ¯ dfi(x): R → R is injective for each x ∈ Ui and Jfi(x) > 0 for each x ∈ Ui. ¯ n Moreover, since each map fi : Ui → R is a Lipschitz function, it can be extended, as a Lipschitz function, to the whole Rk (see Corollary 3.17) . Therefore the sequence ¯ of couples (fi,Ei), with Ei := Ui yield a countable union of regular Lipschitz images f(Ei). Remark 4.10. In particular, we immediately deduce from (ii) and (iii) of Definition 4.8 that |E ∩ B(x, r)| 1 Z lim = 1, lim |Jf(x) − Jf(y)| dy = 0 + ∗ k + k r→0 αk r r→0 r B(x,r) 113 for every x ∈ E. Indeed, if x is a Lebesgue point of the Jacobian matrix of f, Df, ∞ k k n then x is a Lebesgue point of Jf, since Df ∈ L (R ; Mn,k) and the map L(R ; R ) 3 k n L 7→ JL is continuous, where Mn,k and L(R ; R ) denote respectively the class of real matrices with n rows and k columns and the space of linear maps from Rk to Rn . We now show that we can always decompose a countably Hk- rectifiable set by means of (almost flat) regular Lipschitz images (see also Lemma 3.11). Theorem 4.11 (Decomposition of rectifiable sets). If Γ is countably Hk- rectifiable n n in R and t > 1, then there exist a Borel set Γ0 ⊂ R , countably many Lipschitz k n k maps fh : R → R and compact sets Eh ⊂ R such that ∞ k (4.8) Γ = Γ0 ∪ (∪h=1fh(Eh)) , H (Γ0) = 0 .

Each pair (fh,Eh) deﬁnes a regular Lipschitz image, with Lip(fh) 6 t and −1 (4.9) t |x − y| 6 |fh(x) − fh(y)| 6 t |x − y|,

−1 (4.10) t |v| 6 |Dfh(x)v| 6 t |v|,

−k k (4.11) t 6 Jfh(x) 6 t , k for every x, y ∈ Eh and v ∈ R . Before the proof of Theorem 4.8 we need the following result. Exercise: Let f : Rk → Rn be a Lipschitz map. Then there exists a Borel set D ⊂ Rk such that Lk(Rk \ D) = 0 and, for each x ∈ D, f is differentiable at x. (Hint: Use Radamacher’s theorem (see Theorem 3.19 and that Lk is a Borel regular o.m.) Proof of Theorem 4.8. Assume that ∞ ∗ (4.12) Γ = Γ0 ∪ (∪i=1fi (Ai)) , k k k ∗ n where H (Γ0) = 0, Ai ⊂ R are L -measurable and fi : Ai → R are Lipschitz functions for i ∈ N. Without loss of generality, by the Borel regularity of Lk, Theorem 3.22 and Kirszbraun’s theorem (see Theorem 3.18), we can also assume that, for each ∗ k n i, Ai is a Borel set and fi : R → R is a Lipschitz function. Let us divide the proof in three steps. ∗ 1st step: We can suppose that, for each i, fi is differentiable at each point x ∈ Ai and k ∗ (4.13) Ai ⊂ Fi := x ∈ R : 0 < Jfi (x) < ∞ . ˜ ∗ Indeed, let denote Ai the subset of points x ∈ Ai such that fi is differentiable at ˜ x. By the previous exercise we can assume that Ai is a Borel set for each i. Then, since, by Rademacher theorem ),

k ˜ L (Ai \ Ai) = 0 , it follows that, by Theorem 3.22, k ∗ ˜ (4.14) H fi Ai \ Ai = 0 . 114

By the area formula (see (AF)), it turns out that k ∗ ˜ (4.15) H fi Ai \ Fi = 0 for each i . ∗ ˜ Therefore, by (4.14) and (4.15), we are allowed to use in (4.12) functions fi : Ai∩Fi → Rn and we get the desired conclusion. 2nd step: By the previous step and applying the Lipschitz linearization to each ∗ function fi (see Lemma 3.55), we get that, for each t > 1 there exists a sequence of (i) disjoint Borel sets (Ah )h such that

∞ (i) (4.16) Ai = ∪h=1Ah ;

∗ (4.17) fi | (i) is injective ; Ah (i) k k for each h, there exists a symmetric automorphism Sh ≡ Sh : R → R such that ∗ −1 (i) k ∗ (i) n fi | (i) ◦ Sh : Sh(Ah ) ⊂ R → fi (Ah ) ⊂ R is a bi-Lipschitz map and the following Ah (i) n estimates hold: for every x, y ∈ Ah and v ∈ R ∗ −1 ∗ −1 (4.18) Lip fi | (i) ◦ Sh 6 t, Lip Sh ◦ (fi | (i) ) 6 t Ah Ah

1 (4.19) |S v| |Df ∗(x)v| t |S v|, t h 6 i 6 h 1 (4.20) JS Jf(x) tn JS . tn h 6 6 h Denote, for each i and h,

(i) (i) k (i) ∗ −1 (i) n Gh := Sh Ah ⊂ R , gh := fi | (i) ◦ Sh : Gh → R . Ah (i) k n Then, by Kirszbraun’s theorem , we can assume that gh : R → R is still Lipschitz (i) (i) k with Lip gh 6 t and, by Rademacher’s theorem, gh is differentiable at L -a.e. x ∈ Rk and (i) ∗ −1 k (i) Dgh (x) = Dfi (x) · Sh for L -a.e. x ∈ Gh . Therefore, by (4.16)- (4.20), and arguing again as in the first step, it follows that (i) (i) (4.21) (4.8)-(4.11) hold with fh ≡ gh , Eh ≡ Gh . (i) (i) 3rd step: Let us relabel sequences (gh )i,h and (Gh )i,h respectively by sequences ˜ ˜ ˜ (fh)h and (Eh)h. Therefore, by (4.21), we have that (4.8)-(4.11) hold with fh ≡ fh ˜ and Eh ≡ Eh. We have only to show that we can modify respectively the sequence ˜ ∗ of functions (fh)h and the one of sets (Eh)h by a sequence of functions (fh)h and one compact sets (Eh)h in order that they still satisfies (4.8)-(4.11) and couple (fh,Eh) induces a regular Lipschitz image for each h. For a given h, denote ˜(h) ˜ Em := Eh ∩ B(m) if m ∈ N . where B(m) denotes the closed ball of Rk centered at 0 with radius m. 115

By the approximation of Radon measures by compact sets from below (see Theorem (h) 1.14), for each h, m, there exists an increasing sequence of compact sets (Kmj )j such that (h) ˜(h) k ˜(h) (h) (4.22) Kmj ⊂ Em and L Em \ Kmj < 1/j ∀ j . In particular, since k ˜(h) ∞ (h) L Em \ ∪j=1Kmj = 0 , for each h, m , by the area formula k ˜ ˜(h) ∞ (h) (4.23) H fh Em \ ∪j=1Kmj = 0 for each h, m . Let us denote, for given h, m and j, (h) ˜ (h) (h) fmj := fh and Emj := Kmj ; (h) (h) by relabelling sequences (fmj )h,m,j and (Emj )h,m,j, we get sequences (fh)h and (Eh)h respectively which still satisfy (4.8)-(4.11). Let us now prove that, for each h, couple (fh,Eh) is a regular Lipschitz image. Observe that each Eh is compact and, by the density of a set (see Corollary 2.18) and Lebesgue-Besicovitch differentiation theorem k (see Theorem 2.16),we can assume that, for each h and L -a.e. x ∈ Eh k L (Eh ∩ B(x, r)) lim = 1, x is a Lebegue point of Dfh . + ∗ k r→0 αk r Thus conditions (ii) and (iii) of Definition 4.8 hold. Moreover, by (4.9) and (4.11), condition (i) of Definition 4.8 holds, too. 4.2. Approximate tangent planes to rectifiable sets. Theorem 4.11 allows us to prove the existence (in a measure-theoretic sense) of tangent spaces to rectifiable sets. Let us come back to the approximate tangent plane. Define y − x Φ : n → n as Φ (y) := , y ∈ n, x,r R R x,r r R so that, if µ is a Radon measure on Rn and E ⊂ Rn is a Borel set, then (Φ ) µ(E) µ(x + rE) (4.24) x,r # = . rk rk Theorem 4.12 (Existence of approximate tangent spaces). If Γ ⊂ Rn is a locally Hk-rectifiable set, then for Hk-a.e. x ∈ Γ there exists a unique k- dimensional plane + πx such that, as r → 0 , (Φ ) (Hk Γ) Γ − x (4.25) x,r # = Hk *∗ Hk π , rk r x that is Z Z 1 y − x k k 0 n (4.26) lim ϕ dH (y) = ϕ(y) dH (y) ∀ ϕ ∈ Cc (R ) . r→0+ k r Γ r πx 116

In particular k H (Γ ∩ B(x, r)) k (4.27) ∃ Θk(Γ, x) := lim = 1 for H -a.e. x ∈ Γ . + ∗ k r→0 αk r

Remark 4.13. Observe that, if (4.25) holds, then πx is unique. Indeed, if (4.25) holds for two k-planes πi (i = 1, 2), by the uniqueness of the local weak* convergence of measures (see Remark 1.108), then

k k (4.28) H π1 = H π2 . k Since spt(H πi) = πi (i = 1, 2), by (4.28), π1 = π2.

Definition 4.14 (Approximate tangent plane to a set). Let Γ ⊂ Rn be such that k n H (Γ∩K) < ∞ for each compact set K ⊂ R .A k-dimensional plane πx is said to be the approximate tangent plane to Γ at x, if (4.25) holds. Then we denote TxΓ := πx. Remark 4.15. We will see that the set of points x ∈ Γ such that (4.25) holds true depends only on the Radon measure µ = Hk Γ. It is a locally Hk-rectifiable set in Rn and is unchanged if we modify Γ on and by Hk-null sets (see Theorem 4.21). Lemma 4.16 (Approximate tangent plane to a regular Lipschitz image). Let f : Rk → Rn be a Lipschitz function and let (f, E) define a regular Lipschitz image in Rn. If Γ = f(E), then k k (4.29) Tf(z)Γ = df(z)(R ) L -a.e. z ∈ E. In particular −1 k k (4.30) ∃ TxΓ = df(f (x))(R ) H -a.e. x ∈ Γ . Before the proof of Lemma 4.16, let us point out the following Exercise: Let f : Rk → Rn be differentiable at z ∈ Rk and assume that Jf(z) > 0. Then there exist λ, s0 > 0 such that 0 0 0 (4.31) |f(z ) − f(z)| > λ |z − z| ∀ z ∈ B(z, s0) . (Hint: First observe that, since Jf(z) > 0, rank(df(z)) = k. This implies that k λ0 := min |df(z)(v)| : v ∈ R , |v| = 1 > 0.

On the other hand, since f is diﬀerentiable at z, one can prove that, if λ = λ0/2 there exits s0 > 0 such that (4.31) holds. )

0 n Proof. If ϕ ∈ Cc (R ), then by the integration with respect to a push- forward measure (see (1.69)) and the change of variables (see Theorem 3.56) we have Z Z Z 1 k 1 k 1 y − x k k ϕ d (Φx,r)#(H Γ) = k ϕ ◦ Φx,r dH = k ϕ dH (y) r Rn r Γ r Γ r Z 1 f(w) − f(z) k = k ϕ Jf(w) dL (w) r E r Z k = ur(w) dL (w) , Rk 117

k where ur : R → R is deﬁned as f(z + rw) − f(z) u (w) := χ (z + rw) ϕ Jf(z + rw) . r E r

Let (rh)h be an arbitary sequence of positive numbers satisfying limh→∞ rh = 0. Let z be a Lebesgue point of χE and Jf , and f is diﬀerentiable at z, it is easy to see that, up to a subsequence, for each R > 0 k lim ur (w) = u0(w) := ϕ (df(z)(w)) Jf(z) L -a.e. w ∈ B(R) . h→∞ h and then k k (4.32) lim urh (w) = u0(w) L -a.e. w ∈ R . h→∞ It is also easy to check that

k k k (4.33) |ur(w)| 6 sup |ϕ| Lip(f) L -a.e. w ∈ R , ∀ r > 0 . Rn We are going now to prove that there exist r0 and R0 > 0 such that

(4.34) spt(ur) ⊂ B(R0) ∀ r ∈ (0, r0) . Indeed, from (4.32)-(4.34), the dominated convergence theorem and the area formula, it will follow that Z Z Z 1 k k k lim ϕ ◦ Φx,r dH = lim ur (w) dL (w) = u0(w) dL (w) h→∞ k h h→∞ h rh Γ Rk Rk Z Z = ϕ (df(z)(w)) Jf(z) dLk(w) = ϕ dHk Rk df(z)(Rk) for each arbitrary sequence (rh)h of positive real numbers with limh→∞ rh = 0. Thus (4.29) is proved. Finally, let us show (4.34). By the previous exercise, there exist λ, s0 > 0 such that 0 0 0 (4.35) |f(z ) − f(z)| > λ |z − z| ∀ z ∈ B(z, s0) . Moreover, if R > 0 is such that spt(ϕ) ⊂ B(R), then

(4.36) |f(z + rw) − f(z)| 6 r R ∀ w ∈ spt(ur) . By the compactness of E and injectivity of f on E one has 0 0 inf {|f(z ) − f(z)| : z ∈ E \ U(z, s0)} = 0 > 0, so that, by (4.35), one gets (4.37) |f(z0) − f(z)| min λ, 0 |z − z0| = c |z − z0| ∀ z0 ∈ E. > d(E) 0

In this way, if w ∈ spt(ur) then z + rw ∈ E and, by (4.36), |f(z + rw) − f(z)| 6 Rr, so that, by (4.37) c0 r |w| 6 Rr. This proves spt(ur) ⊂ B(R0) with R0 = R/c0, which shows (4.34). Eventually, Let N ⊂ E denote the set of points z ∈ E such that z is k not a Lebesgue point of χE. Then L (N)= 0. Thus, by the area formula Hk(f(N)) = 0 . Thus, by (4.29), for each x ∈ Γ \ f(N), (4.32) holds. 118

∞ Proof of Theorem 4.12. 1st step. We decompose Γ = Γ0 ∪h=1 f(Eh) as in Theorem 4.11. If we let Γh = fh(Eh), then by Lemma 4.16 we ﬁnd that Z Z 1 k k 0 n k (4.38) lim ϕ ◦ Φx,r dH = ϕ dH ∀ ϕ ∈ Cc (R ), H − -a.e. x ∈ Γh , r→0+ k r Γh πx −1 k k n where we have set πx := dfh(fh (x))(R ). Since H Γ is a Radon measure on R , by Corollary 3.38, k H ((Γ \ Γh) ∩ B(x, r)) k (4.39) Θk(Γ \ Γh, x) = lim = 0 H -a.e. x ∈ Γh . + ∗ k r→0 αk r 0 n For a given ϕ ∈ Cc (R ), suppose that spt(ϕ) ⊂ B(R0). Then Z Z 1 k 1 y − x k k ϕ ◦ Φx,r dH = k ϕ dH (y) r Γ\Γh r Γ\Γh r Z 1 y − x k = k ϕ dH (y) r (Γ\Γh)∩B(x,R0r) r k H ((Γ \ Γh) ∩ B(x, R0r)) 6 sup |ϕ| k . Rn r Thus, by (4.39), it follows that Z 1 k lim ϕ ◦ Φx,r dH r→0+ k r Γ\Γh (4.40) Z 1 y − x k k = lim ϕ dH (y) = 0 H -a.e. x ∈ Γh . r→0+ k r Γ\Γh r From (4.38) and (4.40), (4.26) follows. k 2nd step. Let x ∈ Γ satisfy (4.25). Since, by (4.5), H (π ∩ ∂B1) = 0, by Theorem 1.113, we can infer that k −1 k ∗ k H Γ Φx,r(B1) H (Γ ∩ B(x, r)) αk = H (π ∩ B1) = lim = lim , r→0+ rk r→0+ rk and thus (4.27) follows. Let us also point out the following interesting locality result for the approximate tangent plane.

Proposition 4.17 (Locality of the approximate tangent plane). If Γi (i = 1, 2) are k n k locally H -rectiﬁable sets of R , then for H -a.e. x ∈ Γ1 ∩ Γ2

TxΓ1 = TxΓ2 . Proof. See [Mag, Proposition 10.5]. Example 4.18 (Approximate tangent space to a Lipschitz (n − 1)-graph). If φ : Rn−1 → R is a Lipschitz function, and we define f : Rn−1 → Rn as f(z) = (z, φ(z)), z ∈ Rn−1, then, as pointed out in Example 4.6, Γ := f(Rn−1) = graph(φ) is locally Hn−1-rectifiable and. Let us now show that for Ln−1-a.e. z ∈ Rn−1, if ν(z) := (−∇φ(z), 1), ⊥ n (4.41) Tf(z)Γ = ν(z) := {v ∈ R :(v, ν(z))Rn = 0} , 119 or, equivalently, for Hn−1-a.e. x ∈ Γ, −1 ⊥ (4.42) TxΓ = ν(f (x)) . In particular this implies that the classical tangent space to a Lipschitz (n − 1)- dimensional graph agrees Hn−1-a.e. with the approximate tangent space. Let us first observe that f is Lipschitz, injective and it actually satisfies 0 0 p 0 0 n−1 (4.43) |z − z | 6 |f(z) − f(z )| 6 1 + Lip(φ)2 |z − z | ∀ z, z ∈ R . Moreover, by (3.73), p n−1 n−1 (4.44) Jf(z) = 1 + |∇φ(z)|2 L -a.e. z ∈ R and, by the area formula, Z (4.45) Hn−1 Γ(E) = p1 + |∇φ(z)|2dLn−1 f −1(Γ∩E) for each Borel set E ⊂ Rn. n−1 Let A1 ⊂ R denote the set of points x satysfying: • φ is differentiable at x; • x is a Lesbegue point of ∂jφ for each j = 1, . . . , n − 1 (with respect to the (n − 1)-dimensional Lebesgue measure Ln−1); and let A2 ⊂ A1 denote the set of points of density 1 in A1 (with respect to the (n − 1)-dimensional Lebesgue measure Ln−1). By Rademacher’s and Lebesgue point theorems (see Theorems 3.19 and 2.1) and the density of a set (see Corollary 2.18) n−1 n−1 Ai (i = 1, 2) are measurable and they have full measure, that is L (R \ Ai) = 0 (i = 1, 2). Let us define

AR := A2 ∩ B(R) and ΓR := f(AR) if R > 0 . n−1 Since AR is measurable and L (AR) < ∞, by the approximation of Radon measures by means of compact sets (see Theorem 1.14), we can ﬁnd and increasing sequence of compact sets (Eh)h such that n−1 ∞ (4.46) Eh ⊂ AR ∀ h and L (AR \ ∪h=1Eh) = 0 . Let ∞ ΓR,0 := f(AR \ ∪h=1Eh), ΓR,h := f(Eh) . Then, by construction and (4.45), ∞ n−1 ΓR = ΓR,0 ∪ (∪h=1ΓR,h) and H (ΓR,0) = 0 ; by deﬁniton of AR, (4.43), (4.44), and (4.45)

ΓR,h = f(Eh) is a regular Lipschitz image for each h . Applying Lemma 4.16, for a given h, it follows that n−1 n−1 Tf(z)ΓR,h = df(z)(R ) L -a.e. z ∈ Eh. By standard linear algebra, we can infer that n−1 ⊥ n−1 df(z)(R ) = span {v1(z), . . . , vn−1(z)} = ν(z) L -a.e. z ∈ Eh , with v1(z) := e1 + ∂1φ(z) en, . . . , vn−1(z) := en−1 + ∂n−1φ(z) en , 120

n where e1, . . . , en denotes the standard basis of R . It is easy to see that vectors ⊥ vi(z) ∈ ν(z) for each z ∈ Eh, i = 1, . . . , n − 1 and they are a basis of the subspace ν(z)⊥. Thus, we have proved that

⊥ n−1 Tf(z)ΓR,h = ν(z) L -a.e. z ∈ Eh. or that is equivalent

−1 ⊥ n−1 (4.47) TxΓR,h = ν(f (x)) H -a.e. x ∈ ΓR,h .

By the locality of the approximate tangent plane (see Proposition 4.17) and (4.47), it follows that, for every h,

−1 ⊥ n−1 TxΓ = TxΓR,h = ν(f (x)) H -a.e. x ∈ ΓR,h = ΓR,h ∩ Γ , and then

−1 ⊥ n−1 (4.48) TxΓ = ν(f (x)) H -a.e. x ∈ ΓR = ΓR ∩ Γ .

By (4.48), since R > 0 is arbitrary, (4.42) follows. Finally, by (4.45), (4.42) and (4.41) are equivalent.

Let us point out the striking fact that (4.27) implies the rectiﬁability. Indeed

Theorem 4.19 (Besicovitch-Marstrand-Mattila). Let E a Borel set with Hk(E) < ∞. Then the following are equivalent: (i) E is Hk-rectiﬁable; k (ii) there exists Θk(E, x) = 1 for H -a.e. x ∈ E. Proof. Implication (i)⇒(ii) follows from Theorem 4.12. Implication (ii)⇒(ii) is much harder and can be found in [Ma, Theorem 17.6]. Remark 4.20. Preiss improved these results in [Pre2] proving that the existence of k Θk(E, x) ∈ (0, ∞) implies the H -rectiﬁabilty of E.

4.3. Blow-ups of Radon measures on Rn and rectiﬁability. We now prove a converse statement to Theorem 4.12, which plays an important role in GMT and, in particular, when studying the structure of sets of ﬁnite perimeter (see Chap. V).

Theorem 4.21 (Rectiﬁability by convergence of the blow-ups). If µ is a Radon measure on Rn, Γ is a Borel set in Rn, µ is concentrated on Γ (that is µ = µ Γ), n and, for every x ∈ Γ, there exists a k-dimensional plane πx in R such that (Φ ) µ x,r # *∗ Hk π as r → 0+, rk x then µ = Hk Γ and Γ is locally Hk-rectiﬁable.

The proof of Theorem 4.21 relies on a simple criterion for Hk rectiﬁability, very useful in GMT. 121

Deﬁnition 4.22 (Cones). If π is a k-plane, deﬁne the cone K(π, t) and opening t > 0, as n ⊥ K(π, t) : = y ∈ R : |Pπ (y)| 6 t |Pπ(y)| n √ o n 2 = y ∈ R : |y| 6 1 + t |Pπ(y)| (4.49) ( ) r 1 = y ∈ n : d(y, π) |y| R 6 1 + t2

Notice that K(π, t) is invariant by dilations, that is s K(π, t) = K(π, t) for each s > 0, K(π, t) reduces to π if t = 0, and K(π, t) \{0} ↑ Rn \ π⊥ as t ↑ ∞. Subsets of Lipschitz k-graphs can be easily characterised by cones. In fact Exercise: Let S ⊂ Rn. Then the following are equivalent: (i) there exist a k-dimensional plane π and an opening t > 0 such that S ⊂ x + K(π, t) for each x ∈ S; (ii) there exist a k-dimensional plane π , a unique function φ : E ⊂ π → π⊥ ⊂ Rn and a constant t > 0 such that φ is t-Lipschitz and graph(φ) := {z + φ(z): z ∈ E} = S. (Hint: Implication (ii)⇒ (i) is trivial. Implication (i)⇒ (ii) follows by noticing that: a) if x1, x2 ∈ S and Pπ(x1) = Pπ(x2) then x1 = x2; ⊥ b) for each z ∈ E := Pπ(S) ⊂ π there exists a unique y := φ(z) ∈ π such that z + y ∈ S and φ : E ⊂ π → π⊥ ⊂ Rn is a t-Lipschitz function. ) Theorem 4.23 (Rectifiability criterion). If Γ ⊂ Rn is a compact set, π is a k- dimensional plane in Rn, and there exist δ and t positive with (4.50) Γ ∩ B(x, δ) ⊂ x + K(π, t) ∀ x ∈ Γ , k k n then Γ is H -rectifiable, since there exist finitely many Lipschitz maps fh : R → R k (h = 1,...,N) and compact sets Fh ⊂ R with N Γ = ∪h=1fh(Fh) . Proof. Let z ∈ Γ, then notice that (4.51) Γ ∩ B(z, δ/2) ⊂ x + K(π, t) ∀ x ∈ Γ ∩ B(z, δ/2) . N Since Γ is compact, there exist x1, . . . , xN ∈ Γ such that Γ ⊂ ∪h=1B(xh, δ/2) and (4.51) holds with z = xh h = 1,...,N. By the previous exercise, for each h = 1,...,N there exists a t-Lipschitz function ˜ ⊥ n φh : Fh := Pπ (Γ ∩ B(xh, δ/2)) ⊂ π → π ⊂ R such that

(4.52) graph(φh) = Γ ∩ B(xh, δ/2) , and N N Γ = ∪h=1 (Γ ∩ B(xh, δ/2)) = ∪h=1graph(φh) , ˜ with Fh ⊂ π (h = 1,...,N) compact sets. Let us deﬁne, for h = 1,...,N, ˜ n gh : Fh ⊂ π → R , gh(z) := z + φh(z) . 122

Then gh is Lipschitz and, by Corollary 3.17, we can assume that gh is defined on the whole π. Let us now define k n fh := gh ◦ Iπ : R → R where Iπ is the map defined in (4.1), we get the desired conclusion by choosing −1 ˜ Fh := Iπ (Fh)(h = 1,...,N). Remark 4.24. As a consequence of (4.52), we also proved that a compact set Γ ⊂ Rn satisfying (4.50) is the union of finitely many Lipschitz k-graphs, according to Example 4.6 and then a Hk- rectifiable set. We will also need a technical lemma concerning inclusions between cones and balls.

Lemma 4.25. There is s0 ∈ (0, 1) such that for each two k-dimensional planes π and σ satisfying |π − σ|G < s0, it holds that n (4.53) B(w, s0|w|) ∩ K(π, 1) = ∅ ∀ w ∈ R \ K(σ, 2) . n Proof. Let us begin to observe that, since sets U(w, s0|w|), K(π, 1) and R \ K(σ, 2) are invariant by dilations, without loss of generality, we can prove (4.53) for each n−1 n w ∈ S ∩ (R \ K(σ, 2)). By contradiction, assume there are a sequence (sh)h ↓ 0, two sequences of k-dimensional planes (πh)h and (σh)h and two sequences (wh)h and n (zh)h in R satisfying, for each h,

(4.54) |πh − σh|G < sh , √ (4.55) zh ∈ B(wh, sh) ∩ K(πh, 1) ⇐⇒ |wh − zh| 6 sh and |zh| 6 2 |Pπh (zh)| , √ n−1 n (4.56) wh ∈ S ∩ (R \ K(σh, 2)) ⇐⇒ |wh| = 1 and |wh| > 5 |Pσh (wh)| . By compactness (recall (4.3)), we can assume that there exist two k-dimensional planes π∗ and σ∗, a pont w∗ ∈ Sn−1 such that ∗ ∗ (4.57) lim |πh − π |G = lim |σh − σ |G = 0 , h→∞ h→∞ (4.58) lim |w − w∗| = 0 . h→∞ Thus, by (4.54)-(4.58), we can infer that ∗ ∗ ∗ (4.59) π = σ and ∃ lim zh = w . h→∞

By the deﬁnition of distance | · |G and (4.60), the sequences of operators (Pπh )h and n−1 ∗ (Pσh )h uniformly converge on S to Pπ . In particular we also get that ∗ ∗ (4.60) lim |Pπ (zh) − Pπ∗ (w )| = lim |Pσ (wh) − Pπ∗ (w )| = 0 . h→∞ h h→∞ h By (4.60), we can pass to the limit as h → ∞ in the two inequalities in (4.55) and (4.57) and we get √ √ ∗ ∗ 5 |Pπ∗ (w )| 6 1 6 2 |Pπ∗ (w )| , and then a contradiction. An other useful tool in GMT is the following version of Severini-Egoroﬀ’s theorem 1.18 which applies to Radon measures on Rn and a family of convergent functions rather than a sequence of convergent functions. 123

Lemma 4.26 (Severini-Egoroﬀ’s theorem for a convergent family of functions). Let ϕ be a ﬁnite Radon outer measures on Rn, let Γ ⊂ Rn be ϕ-measurable and let fr :Γ → R (r > 0) be a family of functions such that

∃ lim fr(x) = f0(x) ϕ-a.e. x ∈ Γ . r→0+ Then, for each > 0, there exists a compact set Γ0 ⊂ Γ such that ϕ(Γ \ Γ0) < , and 0 + fr → f0 uniformly on Γ as r → 0 , i.e. lim sup |fr(x) − f(x)| = 0 . r→0+ x∈Γ0

Proof. Let us deﬁne the sequence of functions gh :Γ → R

gh(x) := sup {|fr(x) − f(x)| : 0 < r < 1/h} h = 1, 2,....

Observe that gh (h = 1, 2,... ) is well- deﬁned, ϕ- measurable and

lim gh(x) = 0 µ-a.e. x ∈ Γ . h→∞ Indeed, it is easy to see that

gh(x) := sup {|fr(x) − f(x)| : r ∈ Q, 0 < r < 1/h} . thus, since the family {fr : r > 0} is ϕ-measurable, also gh is ϕ-measurable. Given > 0, applying Severini-Egoroﬀ’s theorem 1.18 to sequence (gh)h we get the existsnce of a ϕ-measurable set E ⊂ Γ such that

(4.61) ϕ(Γ \ E) < /2 and gh → 0 uniformly on E as h → ∞ . On the other hand, by the approximation theorem for Radon measures (see Theorem 1.14), there is a compact set Γ0 ⊂ E such that (4.62) ϕ(E \ Γ0) < /2 . Therefore, by (4.61) and (4.62), we get the desired conclusion.

Proof of Theorem 4.21. Let s0 be the positive constant in Lemma 4.25. Then, by (4.3), there is ﬁnite family of k-dimensional planes σ1, . . . , σN such that s0 (4.63) min |σh − π|G 6 for each k-dimensional plane π . 16h6N 2 Let us now divide the proof in three steps. 1st step. We show that if Γ0 is a compact subset of Γ and assume that the limit relations µ(B(x, r)) lim = 1 , + ∗ k r→0 αk r µ (B(x, r) \ (x + K(π , 1))) lim x = 0 , + ∗ k r→0 αk r hold uniformly with respect to x ∈ Γ0, then Γ0 is Hk-rectiﬁable. Thus, assume that: for each > 0 there exists δ > 0 such that for each x ∈ Γ0, r ∈ (0, δ) µ(B(x, r)) (4.64) 1 − < ∗ k < 1 + , αk r 124

∗ k (4.65) µ (B(x, r) \ (x + K(πx, 1))) < αk r . Let n s o Γ0 := x ∈ Γ0 : |σ − π | 0 1 h N. h h x G 6 2 6 6 Then let us show that 0 0 (4.66) Γh ∩ B(x, δ) ⊂ x + K(σh, 2) ∀ x ∈ Γh, h = 1,...,N. 0 0 By contradiction, if x ∈ Γh and y ∈ B(x, δ) ∩ Γh \{x} but y∈ / (x + K(σh, 2)), that n is y − x ∈ R \ K(σh, 2), then, by (4.53), n (4.67) B(y, s0|y − x|) ⊂ R \ (x + K(πx, 1)) .

Since s0 ∈ (0, 1) B(y, s0|y − x|) ⊂ B(x, 2 |y − x|) by (4.67), it follows that

B(y, s0|y − x|) ⊂ B(x, 2 |y − x|) \ (x + K(πx, 1)) .

Applying (4.65) (at x with r = 2|y − x|) and (4.64) (at y with r = s0|y − x|) ∗ k k ∗ k k αk 2 |y − x| > (1 − ) αk s0 |y − x| a contradiction, as soon as is small enough with respect to k and s0. This proves 0 k (4.66). By the rectifiability criterion ( Theorem 4.23) , Γh is thus H -rectifiable for 0 N 0 0 k h = 1,...,N. By (4.63), Γ ⊂ ∪h=1Γh. Thus Γ is H -rectifiable. 2nd step. Let us prove that Γ is countable Hk-rectifiable. We have µ(B(x, r)) (4.68) lim = 1 , + ∗ k r→0 αk r

µ (B(x, r) \ (x + K(π , 1))) (4.69) lim x = 0 + ∗ k r→0 αk r k k for each x ∈ Γ. Indeed, since, by (4.5), H π(∂B1) = H (π ∩ ∂B1) = 0, therefore, by Theorem 1.113,

∗ k (Φx,r)#µ(B(x, r)) µ((B(x, r)) αk = H (πx ∩ B1) = lim = lim , r→0+ rk r→0+ rk that is (4.68). Let us now check (4.69) in a similar way. Notice that, if E := B1 \ K(πx, 1), then x + r E = B(x, r) \ (x + K(πx, 1)) , and

πx ∩ E = ∅, πx ∩ ∂E = {0} . Thus, since Hk π(∂E) = 0, by (4.24), (Φ ) µ(E) µ (B(x, r) \ (x + K(π , 1))) 0 = Hk(π ∩ E) = lim x,r # = lim x . + k + ∗ k r→0 r r→0 αk r Let us now prove tha, for each Borel set E ⊂ Rn k k (4.70) H Γ(E) 6 µ(E) = µ Γ(E) 6 2 H Γ(E) . 125

Indeed, by the estimates of the upper density of a Radon measure (see Theorem 3.37) since, by (4.68), ∗ 1 6 Θk(µ, x) = Θk(µ, x) 6 1 ∀ x ∈ Γ , (4.70) follows. Given R > 0, since µ(B(R)) < ∞, by applying Lemma 4.26 with ϕ = µ B(R), we can infer the existence of a compact set Γ1 ⊂ Γ such that the limit relations (4.68) and (4.69) hold uniformly on Γ1 with µ(Γ ∩ B(R)) µ((Γ ∩ B(R)) \ Γ ) < . 1 2

By an induction procedure, we can construct a disjoint sequence of compact sets (Γh)h such that µ(Γ ∩ B(R)) (4.71) µ ((Γ ∩ B(R) \ (∪m Γ )) < ∀ m = 1, 2,.... h=1 h 2m

(4.72) the limit relations (4.68) and (4.69) hold uniformly on Γh

k By step 1 and (4.72), it follows that Γh (h = 1, 2,...,) is H - rectifiable. Since, by ∞ k (4.71), µ((Γ∩B(R))\(∪h=1Γh)) = 0, by (4.70) we can also infer that H ((Γ∩B(R))\ ∞ (∪h=1Γh)) = 0. Thus, by definition, Γ ∩ B(R) is countably Hk- rectifiable for each R > 0 ,

∞ and since Γ = ∪h=1(Γ ∩ B(h)), we also get that Γ is countably Hk- rectifiable . Let us now prove that Γ is actually locally Hk- rectifiable. By (4.70), it follows that, since µ is a Radon o.m., Hk(Γ ∩ K) < ∞ for each compact set K ⊂ Rn. Therefore we get the desired conclusion. 3rd step. Let us prove that µ = Hk Γ on P(Rn). Since, by (4.70), Hk Γ is a Radon o.m., absolutely continuous with respect to µ, by the differentiation theorem for positive Radon measures (see Theorem 2.15), Hk Γ = w dµ with

µ(B(x, r)) k (4.73) w(x) = DHk Γµ(x) = lim H -a.e.x ∈ Γ . r→0+ Hk(Γ ∩ B(x, r)) By (4.26), (4.68) and (4.73), it follows that w(x) = 1 for Hk-a.e. x ∈ Γ. Thus Hk Γ = µ on the class of Borel sets. By Remark 1.15, it follows that they agree on n P(R ).

Purely unrectiﬁable sets

Definition 4.27 (Purely unrectifiable sets). Let E ⊂ Rn be a Borel set. We say that E is purely Hk-unrectifiable if Hk(E ∩ f(Rk)) = 0 for any Lipschitz function f : Rk → Rn. 126

Equivalently, we might say that E is purely Hk-unrectifiable if Hk(E ∩ F ) = 0 for any countably Hk-rectifiable set F . There exist examples of purely Hk-unrectifiable with Hausdorff dimension strictly greater than k (for k = 1 one example is the von Koch snowflake in the plane, whose Hausdorff dimension is log 4/ log 3, see [Ma, 4.13] and [Fa, Introduction]). In the next example we show how a purely H1- unrectifiable set in R2 with Hausdorff dimension 1 can be constructed. Example 4.28 ( An unrectifiable set). Let C := C(1/4) be the Cantor set defined in (3.51) with λ = 1/4 and let E := C × C ⊂ R2 (called the Cantor dust in the unit square). It is easy to see that E still presents a self-similar structure. Indeed, let

2k Ck := Ck(1/4) = ∪j=1Ik,j k = 1, 2,..., be the sets defined in (3.51) for C := C(1/4)’s definition. Since, by definition, ∞ C := ∩k=1Ck , then ∞ E := C × C = ∩h=1(Ck × Ck) . with 2k 2k Ck × Ck = ∪j=1Ik,j × ∪l=1Ik,l (4.74) 2k 2k = ∪i,l=1 (Ik,j × Ik,l) = ∪i,l=1Qk,jl k k where Qk,jl j, l = 1,..., 2 , is a family of 4 closed disjoint squares with side length 4−k for k = 1, 2,... . Then we claim that: √ √ 1 (i)0 < 3/ 5 6 H (E) 6 2. 1 1 H (E ∩ B((x, y), r) 1 (ii)Θ ∗(E, (x, y)) := lim inf 6 < 1 for each (x, y) ∈ E. r→0+ 2 r 2 h (iii) Let Ph denote the union of the boundaries of 4 squares Qh,jl for j, l = h 1 1 1,..., 2 . Then it holds that (H Ph)h weakly* converges to c H E on B(R2) as h → ∞ with c = 4/H1(E). From (i), it follows that Hdim(E) = 1 and, by Theorem 4.12 and (ii), that E is purely H1-unrectifiable. Claim (iii) shows that weak* convergence is an approximation too weak to preserve the concept of rectifiability. Indeed sets Ph are finite unions of Lipschitz curves and then H1-rectifiable, instead of E, which is purely H1- unrectifiable. Finally notice that E is totally disconnected, that is its connected components are points. Indeed it can be proved that, if Γ ⊂ Rn is closed and connected with H1(Γ) < ∞, then Γ is H1-rectifiable (see Theorem 4.31). Let us now prove (i),(ii) and (iii). √ −k Proof of (i): From (4.99), it follows that, if δk := 2 4 , √ H1 (E) 2 ∀ k δk 6 √ 1 and so inequality H (E) 6 2 follows. We have now to prove that

3 1 (4.75) √ 6 H (E) . 5 127

Let 2 π := (x, y) ∈ R : y = 2x 2 2 2 and let Pπ : R → R denote the orthogonal projection of R on straight line π. If T is the segment of π deﬁned by 3 T := (x, y) ∈ 2 : y = 2x, 0 x , R 6 6 5 then T turns out to be the orthogonal projection on π both of square Q0,1 and strip x x 3 S := (x, y) ∈ 2 : − y − + 0,1 R 2 6 6 2 2 that is

(4.76) Pπ (S0,1) = Pπ (Q0,1) = T, which is quite evident since the orthogonal subspace to π turns out to be n xo π⊥ = (x, y): y = − . 2 We also claim that

(4.77) Pπ(E) = T. Since 3 H1 (T ) = √ 5 2 2 and Pπ : R → R is a 1- Lipschitz map, by (4.77) and Propositon 3.22,

3 1 1 1 √ = H (T ) = H (Pπ(E)) 6 H (E) , 5 and (4.75) follows. Thus we have only to show (4.77). Thus it suﬃces to prove that

(4.78) Pπ(Ck × Ck) = T for each k = 1, 2,..., in order to show (4.77). Let us deﬁne, for k = 1, 2,... , the family of 4k substrips of S0,1 x h − 1 3 x h 3 S := (x, y) ∈ 2 : − + y − + h = 1, 2,..., 4k , k,h R 2 4k 2 6 6 2 4k 2 It is easy to prove that, 4k (4.79) S0,1 = ∪h=1Sk,h for each k = 1, 2,..., k and, for a given k = 1, 2,... , for each integer 1 6 h 6 4 there are unique integers k k 1 6 jh 6 2 and 1 6 lh 6 2 such that

(4.80) Qk,h := Qk,jhlh ⊂ Sk,h and Pπ(Qk,h) = Pπ(Sk,h) . k k Thus we can relabelling the family of 4 squares Qk,jl j, l = 1,..., 2 by means of k the family Qk,h, h = 1, 2,..., 4 , such that 4k (4.81) Ck × Ck = ∪h=1Qk,h , 128 1 (4.82) d(Q ,Q ) if h 6= m . k,h k,m > 2 4k−1 By (4.87), (4.79), (4.80) and (4.81), we can infer that

4k 4k Pπ(Ck × Ck) = Pπ ∪h=1Qk,h = ∪h=1Pπ(Qk,h)

4k 4k = ∪h=1Pπ(Sk,h) = Pπ ∪h=1Sk,h

= Pπ(S0,1) = T and (4.78) follows. Proof of (ii): Let us begin to point out the self-similar structure of E . Let λ = 1/4 2 2 and deﬁne the following four similarities Si : R → R (i = 1, 2, 3, 4)

S1(x, y) := (λ x, λ y),S2(x, y) := S1(x, y) + (1 − λ, 0)

S3(x, y) := S1(x, y) + (0, 1 − λ),S4(x, y) := S1(x, y) + (1 − λ, 1 − λ) . Since

(4.83) Si(Q0,1) = Q1,i for i = 1, 2, 3, 4 , it is easy to check that 4 4 (4.84) ∪i=1Si(Q0,1) = ∪i=1Q1,i = C1 × C1 and 4 (4.85) ∪i=1Si(Ck × Ck) = Ck+1 × Ck+1 for each k > 1 . In particular it follows that 4 (4.86) ∪i=1Si(E) = E, and

(4.87) Si(E) = E ∩ Q1,i for i = 1, 2, 3, 4 . Moreover, as usual in a self-similar uniform structure, we can infer that 1 (4.88) H1(E ∩ Q ) = H1(E) for k 1, i = 1, 2,..., 4k . k,i 4k > Indeed, by induction on k, (4.88) holds for k = 1 by (4.83) and (4.86). Assume that k+1 (4.88) for k, then let us prove it hold for k + 1. Fix an integer 1 6 s 6 4 . Then ∗ ∗ k it is clear that there exist integers 1 6 i 6 4 and 1 6 s 6 4 such that

Qk+1,h = Si∗ (Qk,s∗ ) . Therefore, by (4.83), (3.24), (3.25) and the inductive hypothesis, 1 1 1 H (E ∩ Qk+1,s) = H (E ∩ Si∗ (Qk,s∗ )) = H (E ∩ Q1,i∗ ∩ Si∗ (Qk,s∗ ))

1 1 1 1 1 = H (S ∗ (E) ∩ S ∗ (Q ∗ )) = H (E ∩ Q ∗ ) = H (E) . i i k,s 4 k,s 4k+1

Fix (x, y) ∈ E. By deﬁnition, (x, y) ∈ Ck × Ck for each k = 1, 2,... , then there exists a unique integers ¯ =s ¯(x, y) ∈ {1, 2,..., 4k} such that

(4.89) (x, y) ∈ Qk,s if and only if h =s ¯ , 129 √ −k and, if rk := 2 4 ,

(4.90) E ∩ Qk,s¯ = E ∩ B((x, y), rk) ∀ k = 1, 2,.... k Indeed because of (4.81) and (4.82), the family of cubes Qk,s s = 1, 2,..., 4 covering Ck × Ck are disjoint, which yields (4.89). Since d(Qk,s¯) 6 rk, it is immediate that

E ∩ Qk,s¯ ⊆ E ∩ B((x, y), rk) . The reverse inclusion follows by noticing that

B((x, y), rk) ∩ Qk,s = ∅ for each s 6=s ¯ , otherwise, √ 2 1 d(Q ,Q ) r = < with s 6=s ¯ , k,s¯ k,s 6 k 4k 2 4k−1 which contradicts (4.82). This proves (4.90). By (4.90), (4.88) and the previous claim (i), it follows that 1 1 1 H (E ∩ B((x, y), rk) H (E ∩ Qk,s¯) H (E) 1 = 6 √ 6 ∀ k , 2 rk 2 rk 2 2 2 which implies 1 1 H (E ∩ B((x, y), r) H (E ∩ B((x, y), rk) 1 lim inf 6 lim inf 6 , r→0+ 2 r k→∞ 2 rk 2 and the proof is accomplished. 1 2 1 2 Proof of (iii): Let µh := H Ph : B(R ) → [0, ∞] and λ := c H E : B(R ) → [0, ∞]. Then, since 2 µh(R ) = 4 ∀ h , by the compactness of weak* convergence (see Theorem 1.121)), it is not restrictive to assume that there exists a Radon measure µ on R2 such that

(4.91) (µh)h weakly* converges to µ . ∗ In particular we also get that µh* µ as h → ∞ and, by the characterization of the local weak* convergence (see Theorem 1.113),

(4.92) µ(K) > lim sup µh(K) for each compact set K, h→∞

(4.93) µ(A) lim inf µh(A) for each open set A. 6 h→∞ We are going to show that 2 (4.94) µ(F ) = λ(F ) for each Borel set F ⊂ R . Let us begin to show that µ is concentrated on E, that is 2 (4.95) µ(R \ E) = 0 and spt(µ) = E. Observe that, by deﬁnition, since Ph ⊂ Ck × Ck if h > k > 1, then

m 2 (4.96) Ph ∩ F = Ph ∩ (∩k=1Ck × Ck) ∩ F ∀ h > m > 1,F ⊂ R , k and, for each h > k > 1, s = 1,..., 4 1 1 k 1 (4.97) 4 = H (Ph) = H (Ph ∩ (Ck × Ck)) = 4 H (Ph ∩ Qk,s) 130

By (4.92), (4.93) and (4.96), it follows that, for each compact set K ⊂ R2 and integer m, ◦ ◦ 1 1 µ(K) 6 lim inf H Ph(K) 6 , lim sup H Ph(K) h→∞ h→∞ 1 m = lim sup H Ph (K ∩ (∩k=1Ck × Ck)) h→∞ m 6 µ (K ∩ (∩k=1Ck × Ck)) . Passing to the limit, when m → ∞, in the previous inequality, we get that ◦ 2 µ(K) 6 µ (K ∩ E) for each compact set K ⊂ R , and, since E is compact, this implies that 2 µ(R ) 6 µ(E) < ∞ and then (4.95) follows. Let us now prove that 1 (4.98) µ (B((x, y), r )) = µ (E ∩ B((x, y), r )) = for each k 1, (x, y) ∈ E. k k 4k−1 > The ﬁrst identity immediately follows by (4.95). By (4.89), (4.90) and (4.95), we can k infer that, for each (x, y) ∈ E, for each k there exists a unique 1 6 s¯ 6 4 such that

(4.99) µ (E ∩ B((x, y), rk)) = µ (E ∩ Qk,s¯) = µ (Qk,s¯) . By (4.92) and (4.97), it follows that 1 1−k µ (Qk,s¯) > lim sup H (Ph ∩ Qk,s¯) = 4 . h→∞ The reverse inequality can be obtained noticing that, by (4.82), we can fatten the ˜ k closed cube Qk,s¯ by an open cube Qk,s¯ in such a way, for each s = 1,..., 4 , ˜ ˜ Qk,s¯ ⊂ Qk,s¯ and Qk,s¯ ∩ Qk,s 6= ∅ if and only if s =s ¯ . Thus, by (4.95), (4.93) and (4.97), ˜ µ (Qk,s¯) = µ (E ∩ Qk,s¯) = µ E ∩ Qk,s¯ ˜ ˜ = µ Qk,s¯ lim inf µh Qk,s¯ 6 h→∞ 1−k = lim inf µh (Qk,s¯) = 4 . h→∞ By the two previous inequalities and (4.99), the second identity in (4.96) also follows. By (4.98), (4.88) and (4.90), we can infer that

µ (B((x, y), rk)) (4.100) 1−k = λ (B((x, y), rk)) = 4 ∀ (x, y) ∈ spt(µ) = E, k = 1, 2,.... Let us now prove that (4.101) µ << λ or, that is equivalent by the diﬀerentiation for positive measures (see Theorem 2.15),

µ (B((x, y), r)) 2 (4.102) Dλµ(x, y) := lim inf < ∞ µ-a.e. (x, y) ∈ R . r→0+ λ (B((x, y), r)) 131

By (4.100), we immediately get that

µ (B((x, y), r)) µ (B((x, y), rk)) lim inf 6 lim inf = 1 ∀ (x, y) ∈ E, r→0+ λ (B((x, y), r)) k→∞ λ (B((x, y), rk)) which implies, by (4.95), (4.102) and then (4.101). Since it is trivial that the sequence of sets 2 Ek(x, y) := B((x, y), rk) k = 1, 2,..., (x, y) ∈ R is a diﬀerentiation basis for λ, by applying Theorem 2.23 and (4.100)

µ(Ek(x, y)) ∃ 1 = lim = Dλµ(x, y) λ-a.e. (x, y) ∈ spt(λ) , k→∞ λ(Ek(x, y)) which implies (4.94).

4.4. Extensions to metric spaces. The notion of rectifiable set in a metric space was already introduced by Federer [Fe, 3.2.14]. Definition 4.29. A set Γ ⊂ (X, d) is said to be countably Hk-rectifiable if ∞ Γ = Γ0 ∪ (∪i=1fi(Ai)) , k k k k where H (Γ0) = 0, Ai ⊂ R are L -measurable and fi : Ai ⊂ (R , k · kRk ) → (X, d) (i = 1, 2,...,) are Lipschitz functions. Definition 4.30. A metric space (X, d) is said to be purely Hk- unrectifiable if for k each Lipschitz function f : A ⊂ (R , k · kRk ) → (X, d), Hk(f(A)) = 0 . A quite general structure rectifiability result can be obatined for the H1-rectifiability (see, for instance, [AT, Theorem 4.4.8]). Theorem 4.31. If (X, d) is complete, Γ ⊂ X is closed and connected, and H1(Γ) < ∞, then there exist countably many Lipschitz curves γh : [0, 1] → Γ (h = 1, 2,... ) such that 1 ∞ H (Γ \ ∪h=1γh([0, 1])) = 0 . In particular Γ is countably H1-rectifiable. k The study of higher H -rectifiability with k > 2 in a metric space is much harder. A systematic study of rectifiable sets in general metric spaces was made by Ambrosio and Kirchheim [AK] in 2000. However, the definitions they used are not always appropriate in some remarkable class of metric spaces such as the one called Carnot groups or also sub-Riemannian stratified groups. Indeed Ambrosio and Kirchheim proved the following result. Theorem 4.32. ([AK, Theorem 7.2]) The first Heisenberg group (H1, d) is purely k-unrectifiable for k = 2, 3, 4, for each invariant distance d. Therefore, taking the previous unrectifiability results into account, a new suitable notion of rectifiability in Carnot groups is needed, better fitting the new geometry. This study is still object of the current research and an account can be found in [SC2]. 132

5. An introduction to minimal surfaces and sets of finite perimeter. ([AFP, G, G2, Mag, MM])

Motivation: An introduction to the so-called Plateau’s problem for non-parametric minimal surfaces as far as the problem of existence, uniqueness and regularity is concerned. An introduction to the sets of ﬁnite perimeter and their relationships with the minimal surfaces.

5.1. Plateau problem: nonparametric minimal surfaces in Rn, area functional and its minimizers. Here we are going to deal with the problem of least area for the so-called non parametric hypersurfaces in Rn, that is hypersurfaces which are graphs of functions. More precisely we will consider an hypersurface S ⊂ Rn with

S = Su := {(z, u(z)) : z ∈ ω¯} where u ∈ C1(¯ω) and ω ⊂ Rn−1 is a bounded open set with smooth boundary. Non parametric minimal surfaces are a particular case of the general theory of (parametric) minimal surfaces, where also surfaces satisfying Definition 4.1 are allowd and they may be not graphs. A recent account of the development and open problems of this theory can be found in [Pe]. By (3.74), the (n − 1)-dimensional area of graph Su is given by Z n−1 p 2 (5.1) A (u) := H (Su) = 1 + |∇u| dx ω and this formula holds true for each funnction u ∈ Lip(ω). Thus we can define the classical area functional Z p A ≡ A (·, ω): Lip(ω) → [0, ∞), A (u) = A (u, ω) := 1 + |∇u|2 dx . ω This gives rise to the classical Plateau’s problem, that is to show the existence of an area minimizing hypersurface with a given boundary. More precisely, if we fix a boundary datum g : ∂ω → R we are concerned with the geometric variational problem of the calculus of variations, also called Dirichlet problem, (PP) min {A (u, ω): u ∈ Lip(ω, g)} , where Lip(ω, g) denotes the class of competitors functions Lip(ω, g) := {u ∈ Lip(¯ω): u = g on ∂ω} , which is supposed to be nonempty. As usual in the calculus of variations, the main two questions about (PP) concerns:

(EUPP) existence and uniqueness of a minimizer u0 of (PP), that is whether there is a function u0 ∈ Lip(ω, g) satisfying

A (u0) 6 A (u) ∀ u ∈ Lip(ω, g)

and whether u0 is unique; (RPP) regularity of minimizers, that is whether a minimizer is (locally) regular in ω, or also (globally) regular inω ¯ provided that both datum g and boundary ∂ω are regular. 133

Here we will mainly deal with the existence of minimizers, and we will only mention later some regularity results as far as minimal boundaries, which applies to problem (PP), too. Existence problem for (PP) essentially was studied by means of two strategies (see [G2, Introduction] for a more complete and interesting account of this issue). • The first strategy is by studying the associated Euler-Lagrange equation to area functional A . Namely ∞ Exercise: prove that, given ϕ ∈ C (ω) and u0 minimizer of (PP), let R 3 t 7→ a(t) := A (u0 + tϕ), then t = 0 is a minimum point of a and Z ∇u · ∇ϕ ∃ 0 = a0(0) = dz . p 2 ω 1 + |∇u| The previous identity yields the associated Euler-Lagrange equation to area functional A , called minimal surface equation and, in divergence form, it reads as ! ∇u (MSE) div = 0 in ω . p1 + |∇u|2 (MSE) is a nonlinear PDE equation, which has been deeply studied for two centuries and half long with regard to existence, uniqueness and regularity of its solutions. Then one transfers those results about solutions of (MSE) to minimizers of area functional (see [G, G2, MM]). • The second strategy consists in showing directly the existence of minimizers for the area functional without studying its associated Euler-Lagrange, that is the so-called direct methods of the calculus of variations. In the following we will follow this strategy first for problem (PP) and then for the generalized problem (PP) dealing with sets of finite perimeter.

Historical notes: Plateau’s problem originates much before with Lagrange who in 1762 studied it, derived the minimal surface equation (MSE) and gives a foundation of the calculus of variations. We have also to acknoweledge that Euler in 1741 already exibited an example of minimal surface by means of the catenoid (see Example 5.6). The historical development of the theory of minimal surfaces involves many of the greatest mathematicians of their time and an account can be found in [Pe].

5.2. Direct method of the calculus of variations and application to the existence of minimizers for the Plateau problem. One of the general results applying the direct methods in calculus of variations is the following generalized Weierstrass theorem for the existence of minimizers. Theorem 5.1 (Generalized Weierstrass theorem). Let (X, τ) be a topological space and let F : X → (−∞, +∞] be a functional such that (i) F is sequentially lower semicontinuous (slsc), i.e. for each x ∈ X and sequence (xh)h ⊂ X with limh→∞ xh = x, then

F (x) lim inf F (xh); 6 h→∞ 134

(i) F is sequentially coercive, that is for each t ∈ R there exists a sequentially compact set Kt ⊂ X such that

{x ∈ X : F (x) 6 t} ⊆ Kt .

Then ∃ minX F .

Proof. If F ≡ +∞, the conclusion is trivial, otherwise let (xh)h ⊂ X be such that

lim F (xh) = m := inf F < +∞ . h→∞ X ¯ If t > m, there exist h and a sequentially compact set Kt such that ¯ xh ∈ {x ∈ X : F (x) 6 t} ⊆ Kt ∀ h > h .

Up to a subsequence, we can suppose, by claim (ii), that there exist x ∈ Kt such that limh→∞ xh = x. Let us show F (x) = m , which will imply our conclusion. Indeed, by claim (i),

F (x) lim inf F (xh) = lim F (xh) = m = inf F, 6 h→∞ h→∞ X and we reach the desired conclusion. A classical application of the direct methods of the calculus of variations to problem (PP) can be given by the so-called bounded slope condition property for boundary datum g, which goes back to Hilbert, and we recall here.

Definition 5.2. We say that a function g : ∂ω → R satisfies the bounded slope condition with constant Q > 0 (Q-B.S.C. for short, or simply B.S.C. when the constant + Q does not play any role) if for every z0 ∈ ∂ω there exist two affine functions wz0 and − wz0 such that w− (z) g(z) w+ (z) ∀z ∈ ∂ω, z0 6 6 z0 − + wz0 (z0) = g(z0) = wz0 (z0) Lip(w− ) Q and Lip(w+ ) Q, z0 6 z0 6 where Lip(w) denotes the Lipschitz constant of w.

We also recall that a set ω ⊂ Rn−1 is said to be uniformly convex if there exist a positive constant C = C(ω) and, for each z0 ∈ ∂ω, a hyperplane Πz0 passing through z0 such that 2 |z − z0| 6 C d(z, Πz0 ) ∀z ∈ ∂ω, where d(z, Πz0 ) := inf{|z − w| | w ∈ Πz0 }. Remark 5.3. We collect here some facts on the B.S.C. a) If g : ∂ω → R satisfies the B.S.C. and is not affine, then ω has to be convex (see [G2, page 20]) and g is Lipschitz continuous on ∂ω. Moreover, if ∂ω has flat faces, then g has to be affine on them. This property seems to say that the B.S.C. is a quite restrictive assumption. Anyhow the following one, due to M. Miranda [Mi] (see also [G2, Theorem 1.1]), shows that the class of functions satisfying the B.S.C. on a uniformly convex set is quite large. 135

b) Let ω ⊂ Rn be open, bounded and uniformly convex; then every g ∈ C1,1(Rn) satisﬁes the B.S.C. on ∂ω.

Theorem 5.4. Let ω be a bounded open set in Rn−1 and assume that g : ∂ω → R satisﬁes B.S.C. with constant Q > 0. Then problem (PP) has a unique minimizer u0 ∈ Lip(ω, g). Moreover u0 satisﬁes the estimate

Lip(u0) 6 Q. Proof. See, for instance, [G2, Theorem 1.2]. Remark 5.5. In Theorem 5.4 we obtained existence and uniqueness of minimizers under special assumptions of convexity on the domain ω. If we want to weaken these conditions, so as to treat more general domains ω, we could use new comparison functions, more general than the aﬃne functions of the B.S.C. Indeed it is possible to prove, by means of the barriers method, the existence and uniqueness of a minimizer for (PP) in Lip(ω, g), provided that bounded open set ω has a boundary of class C2, with non-negative mean curvature and g is of class C2 (see [G2, Theorem 1.6]). Let us recall that if ω is an open bounded set of class C2 and convex, then its boundary has non-negative mean curvature (see [G2, Page 29]). The condition on non-negative mean curvature on the boundary is almost necessary. In fact, one can prove that if the mean curvature is negative at some point of boundary ∂ω, then there exists a regular function g for which the area functional has no minimum in Lip(ω, g) (see [G2, Theorem 1.7]). We will show by means of an example such an eventuality.

Example 5.6 (Non-existence for Plateau’s problem). Let n = 3 and let ω ⊂ R2 be the annulus 2 (5.2) ω := {z = (x, y) ∈ R : % < |z| < R} with 0 < % < R given. Consider problem (PP) with boundary datum 0 if |z| = R (5.3) g(z) = M if |z| = %, with M > 0. We will show that this problem admits no minimizer when M is large enough. We begin by proving that, if a minimizer exists, then there exists a rotationally invariant one. To this aim, it is enough to prove that for any u ∈ Lip(ω) we have Z Z p 2 p 2 (5.4) 1 + |∇u˜|2 dL 6 1 + |∇u|2 dL ω ω 2 where, after setting Rθ to be the rotation in R of an angle θ, that is the linear 2 2 isometry Rθ : R → R

Rθ(x, y) := (cos θ x − sin θ y, sin θ x + cos θ y) , we deﬁne the rotationally symmetric functionu ˜ : ω → R by Z 2π Z 2π (5.5)u ˜(z) := (u ◦ Rθ)(z) dθ = u(|z| cos θ, |z| sin θ) dθ . 0 0 Indeed, when u ∈ Lip(ω) one has 136

Exercise: 2 (5.6) ∇(u ◦ Rθ) = R−θ ◦ (∇u) ◦ Rθ L -a.e. in ω . 2 By (5.6) and since Rθ and R−θ are isometries, it follows that, for L -a.e. z ∈ ω, Z 2π

|∇u˜(z)| = ∇(u ◦ Rθ)(z) dθ 0 Z 2π

= (R−θ ◦ ∇u ◦ Rθ)(z) dθ

(5.7) 0 Z 2π 6 |R−θ ◦ ∇u ◦ Rθ| (z) dθ 0 Z 2π = |∇u| (z) dθ = |∇u(z)| . 0 Thus, by (5.7), (5.4) is proved . Moreover, it is not diﬃcult to show that

(5.8)u ˜|∂ω = g for any u ∈ Lip(ω, g), that is that alsou ˜ ∈ Lip(ω, g). We are now going to exclude the existence of rotationally invariant minimizers. Let u(z) = u(x, y) = v px2 + y2 where

v ∈ Lip∗(%, R) := {v ∈ Lip([%, R]) : v(%) = M and v(R) = 0} . Then it is easy to set that u ∈ Lip(ω, g) and |∇u(z)| = |v0 (|z|)| L2-a.e. z ∈ ω . Thus, we obtain Z Z R (5.9) p1 + |∇u|2 dL2 = 2π rp1 + v0(r)2 dr =: L(v) . ω % We are going to show that for M 1 the functional L does not admit minimizers in the one-dimensional class Lip∗(%, R). Suppose now that v is a minimizer in Lip∗(%, R). Exercise: Prove that function r v0(r) [%, R] 3 r 7→ is absolutely continuous 1 + v0(r)2 and d r v0(r) (5.10) = 0 L1-a.e. r ∈ [%, R] . dr 1 + v0(r)2 (Hint: Let us consider the Euler-Lagrange equation associated to L: let ϕ ∈ 1 0 Cc ((%, R)) be given and let l(t) := L(v + t ϕ) if t ∈ R; then l (0) = 0 and deduce the desired conclusion.) By (5.10), it follows that r v0(r) (5.11) = c ∀ r ∈ [%, R] , p1 + v0(r)2 137 for a suitable c ∈ R. In particular for r ∈ [%, R]

r v0(r)

|c| = p 6 |r| 1 + v0(r)2 0 and so |c| 6 %. From (5.11) and taking into account sgn v = sgn c we obtain c sgn c (5.12) v0(r) = √ = L1-a.e. r ∈ [%, R] . r2 − c2 p(r/c)2 − 1 By (5.12), v has to be monotone, and since v(%) = M > v(R) = 0, v is not increasing. Thus C < 0, and, without loss of generality, by replacing c with −c,the solution is given by Z R dσ Z R/c c dσ v(r) = = √ p 2 2 (5.13) r (σ/c) − 1 r/c σ − 1 = c (arccosh(R/c) − arccosh(r/c)) if % 6 r 6 R. where, by deﬁnition, arccosh : [1, ∞) → [0, ∞) is the inverse function of hyperbolic cosine function et + e−t cosh : [0, ∞) → [1, ∞), cosh(t) := 2 and it can be explicitely represented as √ arccosh(s) = log s + s2 − 1 if s > 1 .

Moreover the constant 0 < c 6 % has to be chosen in order that v(%) = M. Observe that we can equivalently represent v as √ R + R2 − c2 (5.14) v(r) = c log √ if % 6 r 6 R, r + r2 − c2 and √ √ ! R + R2 − c2 R + R2 − c2 (5.15) M = v(%) = c log p 6 sup c log p . % + %2 − c2 c∈(0,%] % + %2 − c2

Exercise: Prove that √ ! R + R2 − c2 R + pR2 − %2 (5.16) sup c log p = % log =: M0(%, R) . c∈(0,%] % + %2 − c2 % √ R + R2 − c2 (Hint: Prove that the function (0,%] 3 c 7→ c log is nondecreasing.) % + p%2 − c2

By (5.14), (5.15) and (5.16), it follows that problem (PP) can be solved only if M < M0, when ω is the open set in (5.2) and g is the boundary datum in (5.3) . In the limit case M = M0 we have c = % and, p % |∇u(x, y)| = v0 x2 + y2 = px2 + y2 − %2 138 becomes infinite on the internal circumference and then u is not admissible because u∈ / Lip(ω). However a minimizer exists in a larger class of competitors, namely in the Sobolev space W 1,1(ω), as proved in the following exercise. Exercise: Let us define the space of functions ∗ AC (%, R) := {v ∈ AC([%, R]) : v(%) = M0 and v(R) = 0} . (i) Prove that functional L in (5.9) is well-defined on AC∗(%, R), that is, L : AC∗(%, R) → [0, ∞). (ii) Let v0 be the function in (5.14) with c = %. Prove that v0 is a minimizer of ∗ ∗ functional L : AC (%, R) → [0, ∞), that is, v0 ∈ AC (%, R) and ∗ L(v0) 6 L(v) ∀ v ∈ AC (%, R) . (iii) Let

(5.17) uh(z) := vh(|z|), u0(z) := v0(|z|) if z ∈ ω . where R + pR2 − %2 + 1/h vh(r) := ch % log if % 6 r 6 R r + pr2 − %2 + 1/h

and the constant ch is chosen in such a way that

vh(%) = M0 . Prove that 1 (5.18) (uh)h ⊂ C (¯ω) ∩ Lip(ω, g),

1 2 (5.19) uh → u0 uniformly onω, ¯ ∇uh → w in (L (ω)) where z w(z) := v0 (|z|) if % < |z| < R . 0 |z| Moreover

(5.20) L(v0) = lim L(vh) = lim A (uh, ω) = inf {A (u, ω): u ∈ Lip(ω, g)} , h→∞ h→∞

where g is the boundary datum in (5.3) with M = M0. (iv) Prove that 1,1 0 u0 ∈ W (ω) ∩ C (¯ω) , 0 1 that is, by deﬁnition, u0 ∈ C (¯ω), u0 ∈ L (ω) and there exists its weak gradient

Du0(z) = w(z) if % < |z| < R , 1 2 1 with Du0 ∈ (L (ω)) . Moreover u0 ∈/ C (¯ω) as well u0 ∈/ Lip(ω), even if Z p 2 L(v0) = 1 + |Du0| dx . ω

If M > M0 there is no solution to problem (PP). In this case, one can prove that, by means of the direct methods and by replacing the space of Lipschitz continuous functions with the one of bounded variation functions , the minimal surface is given by the graph of the solution u corresponding to the limit value M0, plus the 139 portion of the vertical cylinder having for base the internal circumference of radius %, that lies between the levels M0 and M (see [G, Chapter 14]). Finally let us notice that, in the case M < M0, the function v in (5.13) is the inverse function of v − b r : [0,M] → [%, R], r(v) := c cosh c where b := c arccosh(R/c), which is an arc of catenary joining the points (0,R) and (M,%) in the plane v, r. By rotation about the v-axis, it generates a surface called catenoid, graph of u, which is a surface of revolution of minimal area. Surfaces of revolution of minimal area was an issue very studied in the history of calculus of variations, starting from Euler. An account of this fascinating development can be found in [GH, Chap. 5, Sect. 2.4, Example 5].

Historical notes: [MM] Variational problems concerning manifolds, one or more dimensional, immersed in an Euclidean space are among the most classical ones. We mean that they have been considered since Bernoulli’s time and have not obtained a general satisfactory treatment until the the 1950s. At the start of the last century very interesting new ideas about variational problems for surfaces, are contained in the thesis of Lebesgue ”Integrale, Longueur, Aire”. Of the same period of time, are the interesting papers of Tonelli about the length of the curves. In the 1930s appeared the relevant series of papers by Douglas and Radò about the Plateau Problem, together with some interesting contributions of Tonelli concerning variational problems with two independent variables. The school of Tonelli, particularly Cesari, worked at the problem of a definition of the surface area, good from the variational point of view. But it was only in the 1950s that new definitions of surfaces were introduced and used for a general treatment of classical variational problems like the isoperimetric property of the sphere and the Plateau Problem. In the new approaches, like Reifen- berg surfaces, Federer-Fleming integral currents, De Giorgi perimeters and Almgren varifolds, ideas from the Modern Algebra, General Measure Theory and Distribution Theory are used together with the classical arguments from Differential Geometry and Real Variable Functions Theory. 5.3. Sets of finite perimeter, space of bounded variation functions and their main properties; sets of minimal boundary. We are going to introduce an alternative approach to the theory minimal surfaces, which applies to hypersurfaces of Rn, that is surfaces of topological dimension n−1 and which extend the one studied before for non-parametric minimal surfaces. The main idea is that a hypersurface in n n R is meant as boundary of a set E ⊂ R whose characteristic function χE has bounded variation, namely E is a set of finite perimeter. The notion goes back to De Giorgi, who introduced it in the pioneering papers [DG1, DG2], strongly inspired by some previous ideas of Caccioppoli (see [A2] for an interesting account of Euclidean sets of finite perimeter). Caccioppoli’s primitive idea, then refined by De Giorgi through sets of finite perimeter, considered oriented hypersurfaces, which (at least locally) are boundaries of sets, and exploited techniques of measure theory. Let us begin to stress some benefits for the introduction of sets of finite perimeter by means of two least-area problems. 140

Problem 1 (Plateau’s problem for general domains). We can give a formulation of Plateau’ problem (PP) in terms of boundary of a set in Rn+1 as follows. Let ω ⊂ Rn be a bounded open set with regular boundary and let denote with Ω and Ω¯ respectively the open and closed cylinders (5.21) Ω := ω × R, Ω¯ :=ω ¯ × R . n+1 If v :ω ¯ → R, let Ev ⊂ R the (closed) subgraph induced by v, that is ¯ (5.22) Ev := (z, xn+1) ∈ Ω: xn+1 6 v(z) n+1 and let Sv ⊂ R be the graph of v, that is ¯ (5.23) Sv := (z, xn+1) ∈ Ω: xn+1 = v(z) . Let u, g :ω ¯ → R be Lipschitz continuous functions. Then it is easy to see that

∂Eu ∩ Ω = Su ∩ Ω and we can mean the boundary condition u = g on ∂ω as n+1 (5.24) Eu = Eg in R \ Ω . Therefore Plateau’s problem (PP) can be formulated and extended in terms of sets as follows n min {H (∂Eu ∩ Ω) : u ∈ X,Eu satisfies (5.24) } where X is a suitable set of functions to be chosen. The benefit of this formulation is the chance to get existence for more general regular domains ω than the ones strict convex allowed in Theorem 5.4 and Remark 5.4 (see also Example 5.6). Indeed this is the case since, by the relaxation method, one can find out that suitable class X of competitors is the the space of functions of bounded variations on ω ( or, equivalently, sets Eu of finite perimeter in cylinder Ω) and then, by a direct methods of the calculus of variation, to get the existence (see [G, Sect. 14.4]). Problem 2 (Isoperimetric problem). The classical isoperimetric problem is the most celebrated problem of least-area and, likely, one of the earliest problem in this argument: it asks to find out a plane figure of the largest possible area whose boundary has a specified length. It can be extended to higher dimensions as follows: if E ⊂ Rn and |E| := Ln(E), Hn−1(∂E) , respectively denote the n-dimensional volume of E and the (n − 1)-dimensional area (we will call also perimeter) of ∂E, to find out a set Eiso with maximum n-volume, among sets E ∈ X with |E| < ∞ and fixed perimeter Hn−1(∂E) = c, where X is a suitable class of admissible sets for the n-volume and perimeter functions, stable by translations an dilations, to be chosen (for instance, X could be the class of sets with C1 regular boundary). It is also well-known that the isoperimetric problem is equivalent to the so-called isovolumetric problem, that is to find out a set Eiso with minimum perimeter, among sets E ∈ X with fixed volume |E| = c. Indeed, by means of the isoperimetric inequality, that is the inequality (n−1)/n n (n−1)/n n−1 (II) min |E| , |R \ E| 6 C H (∂E) 141 which holds for a suitable positive constant C > 0 and for each set E ∈ X, the isoperimetric problem is equivalent to the isovolumetric problem (IP) m = min Hn−1(∂E): E ∈ X, |E| = 1 since the set functions X 3 E 7→ |E|(n−1)/n and X 3 E 7→ Hn−1(∂E) are homogeneous of degree n − 1. Moreover, if problem (IP) has a solution, the value Ciso = 1/m is the minimum constant C for which (II) turns to be true. It is also well-known that balls are the only solutions of problem (IP) when X is 1 the class of sets with C boundary. Thus we can explicitely calculate the value of Ciso and it turns out to be n ∗ 1/(1−n) (5.25) Ciso = (n αn) , ∗ where αn := |B(0, 1)|. Observe that the isoperimetric constant (5.25) could depend on the class X. However De Giorgi [DG3] proved the minimality of balls in the largest admissible class X, namely the sets of finite perimeter.

An other important feature that class X of competitors in problems 1 and 2 has to satisfy is the sequential compactness with respect to a suitable topology τ, in order to apply the direct methods of the calculus of variations. It is quite clear that neither the class of sets with boundary of class C1 nor the one of class Lipschitz are appropriate for this goal. For instance, let us assume that (Eh)h is the following sequence

Eh = Euh , 2 where ω ⊂ R is the domain in (5.2) and (uh)h is the sequence of functions deﬁned in (5.17). Then, by the previous exercise, it turns out that (uh)h is a minimizing sequence for Plateau’s problem (PP) in the class Lip(ω, g) and, if Ω := ω × R, Z Z 2 p 2 p 2 lim H (∂Eh ∩ Ω) = lim 1 + |∇uh| dx = 1 + |Du0| dx h→∞ h→∞ ω ω 1 but u0 is not neither in C (¯ω) nor in Lip(ω), even though (uh)h uniformly converges to u0. The fundamental result to take into account for the introduction of sets of ﬁnite perimeter is the Gauss-Green theorem, which holds for sets with regular boundary.

Theorem 5.7 (Gauss-Green). Assume that E ⊂ Rn is an open set with its boundary 1 1 n n 1 n n ∂E of class C and let g = (g1, . . . , gn) ∈ Cc (R , R ) ≡ (Cc (R )) . Then Z Z n n−1 (GG) div(g) dL = (NE, g)Rn dH , E ∂E n−1 where NE : ∂E → S denotes the (continuous) outward unit normal to E and n X ∂ gi div(g)(x) := (x) x ∈ n . ∂x R i=1 i Proof. See, for instance, [Mag, Theorem 9.3]. 142

Remark 5.8. Notice that if E is an open set with C1 regular boundary ∂E, then its n−1 outward normal boundary NE : ∂E → S can be extended to a continuous function ˜ n n ˜ n NE : R → R satisfying |NE(x)| 6 1 for each x ∈ R (see [Mag, 9.2]). Gauss-Green formula (GG) can be read in the sense of distributions by means of measure theory as follows: Z Z n n−1 1 n n (5.26) χE div(g) dL = − (−NE, g)Rn d(H ∂E) ∀ g ∈ Cc (R , R ) . Rn Rn By (5.26) and measure theory, we can infer the following suggestions in order to define the class of sets of finite perimeter: 1 n • let g = ϕ ei with ϕ ∈ Cc (R ), where {e1, . . . , en} denotes the standard basis n (1) (n) of R , then, by (5.26) it follows that, if NE = (NE ,...,NE ), Z Z n (i) n−1 1 n χE ∂iϕ dL = − ϕ (−NE ) d(H ∂E) ∀ ϕ ∈ Cc (R ), i = 1, . . . , n , Rn Rn n that is, characteristic function χE admits a weak gradient DχE in R repre- n−1 n n sented by finite Radon vector measure νE = −NE H ∂E : B(R ) → R ; • by (5.26) it follows that Z n n−1 1 n n (5.27) χE div(g) dL 6 H (∂E) kgk∞ ∀ g ∈ Cc (R , R ) , Rn

where kgk∞ := supRn |g|. In particular, by the previous inequality and the 1 n n 1 n n 0 n n density of Cc (R , R ) ≡ (Cc (R )) in ((Cc (R )) , k · k∞), it follows that the 1 n n 1 n n linear functional LE : Cc (R , R ) ≡ (Cc (R )) → R Z n LE(g) := χE div(g) dL Rn ˜ 0 n n can be extended to a linear functional LE :(Cc (R )) → R, which is also continuous according to Definition 1.69. Taking the previous suggestions into account, we can introduce the space of functions with bounded variation on Rn, which is a bigger class of sets of finite perimeter. Definition 5.9. Let Ω ⊆ Rn be an open set. (i) If u ∈ L1(Ω) we call variation of u on Ω the value

Z n 1 n (5.28) |Du|(Ω) := sup u(x)divg dL : g ∈ Cc (Ω, R ), |g(x)| 6 1 ∈ [0, ∞]. Ω (ii) We say that u ∈ L1(Ω) has bounded variation in Ω if |Du|(Ω) < ∞. The space BV (Ω) is the set of functions u ∈ L1(Ω) with bounded variation in Ω. The 0 space BVloc(Ω) is the set of functions u :Ω → R such that u|Ω0 ∈ BV (Ω ) for each 0 open set Ω b Ω.

Exercise: Let Ω ⊂ Rn be an open set and let u ∈ C1(Ω), Then Z |Du|(Ω) = |∇u| dx . Ω 143

(Hint: By (GG) Z Z n n 1 n u(x)divg dL = (g, ∇u)Rn dL ∀ g ∈ Cc (Ω, R )) . Ω Ω

Theorem 5.10 (Structure of BV functions). Let Ω ⊂ Rn be an open set and let u ∈ BVloc(Ω). Then there exist a unique Radon measure µu : B(Ω) → [0, ∞] and a (1) (n) n−1 Borel measurable vector function wu = (wu , . . . , wu ):Ω → S such that

(5.29) µu(A) = |Du|(A) for each open set A ⊂ Ω, Z Z n (5.30) u divg dL = (wu, g)Rn dµu, Ω Ω 1 n for all g ∈ Cc (Ω, R ). Moreover (1) (n) n Du = (D1,...,Dn) := −(wu , . . . , wu ) µu : Bcomp(Ω) → R is a Radon vector measure such that Z Z Z n (i) 1 (5.31) u ∂iϕ dL = ϕ wu dµu := − ϕ dDiu ∀ ϕ ∈ Cc (Ω), i = 1, . . . , , n . Ω Ω Ω 1 Viceversa if (5.31) holds for some u ∈ L (Ω), a Radon measure µ ≡ µu and functions (i) 1 wi ≡ wu ∈ Lloc(Ω, µ) (i = 1, . . . , n), then u ∈ BVloc(Ω) and (5.29) and (5.30) respectively holds with v Z u n uX 2 (5.32) µu(B) := t wi dµ if B ∈ B(Ω) , B i=1

 (w , . . . , w )  1 n (x) if 0 < Pn w2(x) < ∞ pPn 2 i=1 i (5.33) wu(x) := i=1 wi . 0 otherwise

Notation: In the following, if u ∈ BVloc(Ω), since (5.29), we will identify variation |Du| (5.28) with measure µu.

Proof. 1st step: Let us prove there exist a unique Radon measure µu : B(Ω) → [0, ∞] n−1 and a Borel vector function wu :Ω → S such that (5.29) and (5.30) hold. 1 n n 1 n n Let Lu : Cc (R , R ) ≡ (Cc (R )) → R be the linear functional Z n Lu(g) := u div(g) dL . Rn 0 Arguing as in (5.27), it follows that, for each open set Ω b Ω, 0 1 0 n (5.34) |Lu(g)| 6 |Du|(Ω ) kgk∞ ∀ g ∈ Cc (Ω , R ) . Let K ⊂ Ω be a compact. Then there exists an open set Ω0 such that 0 K ⊂ Ω b Ω . 0 n Let g ∈ (Cc (Ω)) with spt(g) ⊂ K and let

(5.35) gh(x) := ((g1 ∗ %h)(x),..., (gn ∗ %h)(x)) if x ∈ Ω , 144

n where (%h)h is a sequence of mollifiers in R . Then, it is easy to see that 1 g ∈ C1(Ω0, n) if < d(K, ∂Ω0) , h c R h 0 |gh(x)| 6 kgk∞ ∀ x ∈ Ω , h , lim kgh − gk∞ = 0 . h→∞ By (5.34), sequence (L(gh))h ⊂ R is a Cauchy sequence. Thus we can extend ˜ 0 n functional Lu to a linear functional Lu :(Cc (Ω)) → R as follows ˜ Lu(g) := lim Lu(gh) h→∞ and the limit is independent of the choice of the sequence (gh)h converging to g. Moreover n o ˜ 0 n 0 sup Lu(g): g ∈ (Cc (Ω)) , |g| 6 1, spt(g) ⊂ K 6 |Du|(Ω ) < ∞ . ˜ 0 n Therefore linear functional Lu :(Cc (Ω)) → R is continuous according to Definition 1.69. Applying the Riesz representation theorem 1.73 and defining µ := µ , w := w , u L˜u u L˜u (5.29) and (5.30) follow. 2nd step: To prove (5.31), it is sufficient to choose g := ϕ ei (i = 1, . . . , n) and use (5.30). 3rd step: Suppose that (5.31) holds for some u ∈ L1(Ω), a Radon measure (i) 1 µ ≡ µu and functions wi ≡ wu ∈ Lloc(Ω, µ)(i = 1, . . . , n), then let us prove that u ∈ BVloc(Ω) and (5.29) and (5.30) respectively hold with µu and wu given by, respectively, 1 n (5.32) and (5.33). By assumptions, for each g = (g1, . . . , gn) ∈ Cc (Ω, R ), if w = (w1, . . . , wn), Z Z n n Z n X n X n u divg dL = u ∂igi dL = u ∂igi dL Ω Ω i=1 i=1 Ω n Z Z n (5.36) X X = wi gi dµ = wi gi dµ i=1 Ω Ω i=1 Z = (w, g)Rn dµ . Ω 0 1 0 n 1 n Let us fix an open set Ω b Ω and let g ∈ Cc (Ω , R ). Since w ∈ (Lloc(Ω)) , it follows that Z Z (5.37) (w, g)Rn dµ 6 kgk∞ |w| dµ < ∞ . Ω Ω0 Thus, by (5.36) and (5.37), we obtain that Z 0 |Du|(Ω ) 6 |w| dµ < ∞ , Ω0 0 for each open set Ω b Ω, that is, u ∈ BVloc(Ω). By (5.30) and (5.36), it follows that Z Z (5.38) (wu, g)Rn dµu = (w, g)Rn dµ , Ω Ω 145

1 n for each g ∈ Cc (Ω, R ). By using the approximation by convolution deﬁned in (5.35), 0 n one can prove that (5.38) still holds for each g ∈ Cc (Ω, R ). The approximation by p continuous functions in L (see Remark 1.64) enables us to use functions g = wu χΩ0 w 0 and g = χ 0 in (5.38), for each open set Ω ⊂ Ω, which yields inequalities |w| Ω Z Z 0 0 µu(Ω ) 6 |w| dµ and µu(Ω ) > |w| dµ , Ω0 Ω0 that is Z 0 0 µu(Ω ) = |w| dµ for each open set Ω ⊂ Ω , Ω0 Thus (5.32) follows by the approximation in measure of Borel sets by means of open sets from above. By (5.32) and (5.38), (5.33) follows. BV functions of one variable ([AFP, Sect. 3.2]) ????.

Deﬁnition 5.11. A (Lebesgue) measurable set E ⊂ Rn is of locally ﬁnite perimeter n in an open set Ω ⊂ R (or is a Caccioppoli set) if the characteristic function χE ∈ BVloc(Ω). In this case we call the perimeter of E the measure

(5.39) |∂E| := |DχE| and we call the (generalized inward unit) normal to ∂E in Ω the vector

(5.40) νE(x) := −wχE (x). Remark 5.12. Observe that, by (5.40) and (5.30), it follows that, if E ⊂ Rn is a set of locally ﬁnite perimeter in an open set Ω ⊂ Rn, then it satisﬁes, in sense of distributions, the following generalized Gauss-Green formula Z Z n 1 n (5.41) div(g) dL = − (νE, g)Rn d|∂E| ∀ g ∈ Cc (Ω, R ) . E Ω n−1 Thus vector function νE :Ω → S acts as the inward normal to E in the classical Gauss-Green formula (see (5.26)).

Theorem 5.13. If E ⊂ Rn is an open set with C1 boundary , then E is a set of locally ﬁnite perimeter in Rn and for each open set Ω ⊂ Rn (5.42) |∂E|(Ω) = Hn−1(∂E ∩ Ω) . Proof. By deﬁnition of perimeter and (GG) (5.43) Z n 1 n |∂E|(Ω) := sup divg dL : g ∈ Cc (Ω, R ), |g| 6 1 E Z n−1 1 n n−1 = sup (NE, g)Rn dH : g ∈ Cc (Ω, R ), |g| 6 1 6 H (∂E ∩ Ω) ∂E∩Ω n−1 where NE : ∂E → S is the outward unit normal to E. To prove the reverse inequality, let us observe that, by Remark 5.8, there exists a continuous function ˜ n n NE : R → R with ˜ ˜ n (5.44) NE|∂E = NE and |NE(x)| 6 1 ∀ x ∈ R . 146

1 Let ϕ ∈ Cc (Ω) with 0 6 ϕ 6 1 and let ˜ n gh(x) := (ϕNE) ∗ %h (x) x ∈ R , n where (%h)h is a sequence of mollifiers in R and gh is the convolution product in (5.35). It is easy to see that, by (5.44), 1 n gh ∈ Cc (Ω, R ) for h large , and ˜ n n gh → ϕNE uniformly on R , |gh(x)| 6 ϕ(x) ∀ x ∈ R . 1 Therefore, for given ϕ ∈ Cc (Ω) with 0 6 ϕ 6 1, by the dominated convergence theorem, Z n−1 1 n sup (NE, g)Rn dH : g ∈ Cc (Ω, R ), |g(x)| 6 1 ∂E∩Ω (5.45) Z Z n−1 n−1 > lim (NE, gh)Rn dH = ϕ dH . h→∞ ∂E∩Ω ∂E∩Ω 1 By (5.43) and (5.45), taking the supremum on all ϕ ∈ Cc (Ω) with 0 6 ϕ 6 1, it follows that n−1 |∂E|(Ω) > H (∂E ∩ Ω) . Remark 5.14. Notice that (5.42) may not hold true if E is a set of finite perimeter n but its boundary ∂E is no more regular. Indeed there exists a set E ⊂ R (n > 2) of finite perimeter in Rn, that is |∂E|(Rn) < ∞, but 0 < Hn(∂E) = Ln(∂E) < ∞ (see, for instance,[Mag, Example 12.25]). In particular Hn−1(∂E) = ∞ . Thus a set of finite perimeter may have a wild topological boundary. Remark 5.15. The perimeter is invariant under translations, that is n n (5.46) |∂(p + E)|(p + A) = |∂E|(A), ∀p ∈ R and for any Borel set A ⊂ R , if p + E := {p + x : x ∈ E} . Indeed the differential operator div is invariant under translations and the n-dimensional Lebesgue measure Ln is invariant under traslations. Moreover the perimeter is homogeneous of degree n − 1 with respect to the dilations , that is n−1 n (5.47) |∂(λE)|(λA) = λ |∂E|(A), ∀ λ > 0 and for any Borel set A ⊂ R , if λE := {λ x : x ∈ E} . Also this fact is elementary and can be proved by changing variables in formula (5.28). Let us recall some simple properties of sets of (locally) finite perimeter, Proposition 5.16. L Ω ⊂ Rn be an open set and let E and F be measurable subsets of Rn. Then (i) spt(|∂E|) ⊂ ∂E; (ii) |∂E|(Ω) = |∂(Rn \ E)|(Ω); 147

(iii) (locality of the perimeter measure) |∂E|(Ω) = |∂(E ∩ Ω)|(Ω); (iv) |∂(E ∪ F )|(Ω) + |∂(E ∩ F )|(Ω) 6 |∂E|(Ω) + |∂F |(Ω). Proof. See [AFP, Proposition 3.38]. The direct methods of the calculus of variations apply to the sets of ﬁnite perimeter and, more generally, to the space of bounded variations since they satisﬁes the two important properties of semicontinuity and compactness.

1 Theorem 5.17 (Lower semicontinuity in BV). Let u, uh ∈ Lloc(Ω), h ∈ N and 1 suppose that uh → u in Lloc(Ω), that is 1 uh → u in L (K) as h → ∞, for each compact set K ⊂ Ω . (i) Then lim inf |Duh|(Ω) |Du|(Ω) . h→∞ > (ii) Assume also that 0 0 sup {|Duh|(Ω ): h ∈ N} < ∞ for each open set Ω b Ω . ∗ Then u ∈ BVloc(Ω) and Duh*Du in Ω, that is, Z Z 0 ϕ dDuh → ϕ dDu ∀ ϕ ∈ Cc (Ω) . Ω Ω Proof. See [AFP, Propositions 3.6 and 3.16].

Theorem 5.18 (Compactness in BV ). Every sequence (uh)h ⊂ BVloc(Ω) satisfying Z 0 0 (5.48) sup |uh|dx + |Duh|(Ω ): h ∈ N < ∞ for each open set Ω b Ω , Ω0 1 admits a subsequence (uhk )k converging in Lloc(Ω) to u ∈ BVloc(Ω). If Ω is a bounded 1 0 open set with C boundary, (uh)h ⊂ BV (Ω) and (5.48) holds with Ω ≡ Ω, then u ∈ BV (Ω) and 1 uhk → u in L (Ω) as k → ∞ . Proof. See [AFP, Theorem 3.23]. The previous results for the space of bounded variations simply yield the following ones for sets of ﬁnite perimeter.

n Deﬁnition 5.19. Given (Lebesgue) measurable sets (Eh)h and E in R and an open set Ω ⊂ Rn, loc (i) we say that (Eh)h locally converges to E in Ω, and write Eh→E in Ω, if 1 χEh → χE in Lloc(Ω), which amounts to

|K ∩ (Eh∆E)| → 0 as h → ∞, for each compact K ⊂ Ω .

(ii) We say that (Eh)h converges to E in Ω, and write Eh → E in Ω, if χEh → χE in L1(Ω), which amounts to

|Ω ∩ (Eh∆E)| → 0 as h → ∞ .

Corollary 5.20 (Lower semicontinuity for sets of ﬁnite perimeter). Let (Eh)h and E n loc be measurable sets of R and suppose Eh→E. 148

(i) Then lim inf |∂Eh|(Ω)) |∂E|(Ω)) . h→∞ > (ii) Assume also that 0 0 sup {|∂Eh|(Ω ): h ∈ N} < ∞ for each open set Ω b Ω . ∗ Then E is a set of locally ﬁnite perimeter in Ω and νEh |∂Eh|*νE |∂E| in Ω, that is, Z Z 0 ϕ νEh d|∂Eh| → ϕ νE d|∂E| ∀ ϕ ∈ Cc (Ω) . Ω Ω where νF denotes the generalized inward normal to F , if F is a set of locally ﬁnite perimeter. Proof. The proof is immediate by Theorem 5.17 .

Corollary 5.21 (Compactness for sets of ﬁnite perimeter). Let (Eh)h be a sequence of sets with locally ﬁnite perimeter in an open set Ω ⊂ Rn, satisfying 0 0 0 (5.49) sup {|Ω ∩ Eh| + |∂Eh|(Ω ): h ∈ N} < ∞ for each open set Ω b Ω .

Then there exist a subsequence (Ehk )k and a set E with locally finite perimer in Ω such that loc Eh→E in Ω . 1 If Ω is a bounded open set with C boundary, (Eh)h is a sequence of sets with finite perimeter in Ω and (5.49) holds with Ω0 ≡ Ω, then E is a set with finite perimeter in Ω and

Ehk → E. Proof. The proof easily follows from Theorem 5.18 and the following exercise. 1 Exercise: If uh = χEh , u ∈ Lloc(Ω) and 1 χEh → u in Lloc(Ω) , n then u = χE L -a.e. in Ω, for some measurable set E ⊂ Ω. Minimal boundaries De Giorgi [DGCP, Theorem 1.1, Chap. II] (see also [G] and [Mag, Proposition 12.29]) proved the existence of minimizers for a Plateau type-problem concerning a geometric variational problem dealing with sets of finite perimeter. Theorem 5.22 (Existence of minimal boundaries àla De Giorgi, 1960). Let A ⊂ Rn be a bounded set and let L ⊂ Rn be a measurable set with |∂L|(Rn) < ∞. Then there exists a solution of the minimization problem n n (5.50) min {|∂F |(R ): F ⊂ R measurable,F \ A = L \ A} Remark 5.23. In some sense the set L determines the boundary value of F . Roughly speaking, prescribing that F \A = L\A we impose L∩∂A as a ”boundary condition” for the admissible sets F . At the same time, the set A, being the region where L can be modified to minimize perimeter, may act as an obstacle. In general, we do not expect uniqueness of minimizers for this problem (see [Mag, Sect. 12.5]). 149

Proof of Theorem 5.22. 1st step: Let us denote by n n X := {F ⊂ R : F has locally ﬁnite perimeter in R ,F \ A = L \ A} Since L ∈ X, minimization problem (5.50) is equivalent to the problem

n (5.51) min {|∂F |(R ): F ∈ X} 2nd step: We are going to apply the direct methods of the calculus of variations for showing the existence of problem (5.51). More precisely we will apply generalized Weierstrass theorem 5.1 to functional n P : X → [0, ∞],P (F ) := |∂F |(R ) and we endow the class of competitors X by topology τ = loc of local convergence of sets in Rn introduced in Deﬁnition 5.19 (i). Thus we have to prove that (5.52) P is sequentially lower semicontinuous ; (5.53) P is sequentially coercive. Property (5.52) immediately follows by the lower semicontinuity for sets of ﬁnite perimeter (see Corollary 5.20 (i)). Let us prove (5.53). Fix t ∈ (0, ∞) and let

(5.54) (Eh)h ⊂ {F ∈ X : P (F ) 6 t} .

We have to prove there exist a subsequence (Ehk )k and a set E ∈ X such that loc (5.55) Ehk →E as k → ∞ . Observe that Eh = (Eh ∩ A) ∪ (Eh \ A) ⊆ A ∪ L ∀ h . 0 n This implies that, for each open set Ω b R 0 0 (5.56) |Eh ∩ Ω | 6 |(A ∪ L) ∩ Ω | ∀ h . By (5.54) and (5.56), it follows that (5.49) is fulﬁlled with Ω = Rn. Thus, by the compactness of sets of ﬁnite perimeter (see Corollary 3.11), (5.55) follows. IN PROGRESS! 150

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INFORMATION ABOUT SOME QUOTED MATHEMATICIANS

Biographical and scientiﬁc information more detailed may ﬁnd at the website http:///www-history.mcs.st-andrews.ac.uk/

• ALAOGLU Leonidas (1914, Red Deer, Alberta, Canada-1981,): Alaoglu was a Canadian-American mathematician, most famous for his widely-cited result called Alaoglu’s theorem on the weak-star compactness of the closed unit ball in the dual of a normed space, also known as the Banach-Bourbaki-Alaoglu theorem. • BAIRE René-Louis(1874, Paris, France- 1932, Chambéry, France): Baire worked on the theory of functions and the concept of a limit. He is best known for the Baire category theorem, a result he proved in his 1899 thesis. • BANACH Stefan (1892, Kraków,Austria-Hungary (now Poland) - 1945 in Lvov, (now Ukraine)): Banach founded modern functional analysis and made major contributions to the theory of topological vector spaces. In addition, he contributed to measure theory, integration, and orthogonal series. • BESICOVITCH Abram S. (1891, Berdyansk, Russia - 1970, Cambridge, Eng- land): He was a Russian mathematician, who worked mainly in England. He worked mainly on combinatorial methods and questions in real analysis, and gave important contributions in topics such as the Kakeya needle problem, the Hausdorff-Besicovitch dimension and the rectifiability in the plane. • BOREL Emil F.E.J. (1871, Saint Affrique, Aveyron, Midi-Pyrénées,France - 1956, Paris): Borel created the first effective theory of the measure of sets of points, beginning of the modern theory of functions of a real variable. • BOURBAKI Nicolas: Nicolas Bourbaki is the pseudonym of a group of (mainly) French mathematicians who published an authoritative account of contemporary mathematics. • CACCIOPPOLI Renato (1904, Napoli- 1959, Napoli) was an Italian mathematician, known for his deep contributions to mathematical analysis, including the theory of functions of several complex variables, functional analysis, measure theory and partial differential equations. In particular he pionereed some issues of geometric measure theory by means of the introduction of some sets today called Caccioppoli’s sets. • CANTOR George F.L.P. (1845, St Petersburg, Russia - 1918, Halle, Germany): Cantor founded the set theory and introduced the concept of infinite numbers with his discovery of cardinal numbers. He also advanced the study of trigonometric series. • CARATHEODORY´ Constantin (1873, Berlin - 1950, Munich): Carathéodory made significant contributions to the calculus of variations, the theory of point set measure, and the theory of functions of a real variable. • CAUCHY Augustin-Louis (1789, Paris, France - 1857, Sceaux (near Paris), France) Cauchy pioneered the study of analysis, both real and complex, and the theory of permutation groups. He also researched in convergence and divergence of infinite series, differential equations, determinants, probability and mathematical physics. • DE GIORGI Ennio (1928, Lecce, Italy - 1996, Pisa, Italy): he was an Italian mathematician, who worked on calculus of variations, geometric measure theory, partial differential equations and the foundations of mathematics, giving fundamental 154 contributions. In particular he did the last step for solving the 19th Hilbert problem on the regularity of solutions of elliptic partial differential equations (togheter with J. F. Nash, but independently) and solved the so-called Bernstein problem for minimal surfaces (in collaboration with E. Bombieri and E. Giusti). • DE LA VALLEE´ POUSSIN Charles J. (1866, Louvain, Belgio-1962, Louvain, Belgio): he was a Belgian mathematician and is best known for proving the prime number theorem. • DIRICHLET Gustav L. (1805, Düren,French Empire (now Germany)- 1859, Göttingen,Hanover (now Germany)) He made valuable contributions to number theory, analysis, and mechanics. In number theory he proved the existence of an infinite number of primes in any arithmetic series. In mechanics he investigated the equi- librium of systems and potential theory, which led him to the Dirichlet problem concerning harmonic functions with prescribed boundary values. • EGOROFF Dimitri F. (1869, Moscow, Russia - 1931, Kazan, USSR): He worked on integral equations and a theorem in the theory of functions of real variable is named after him. Luzin was Egorov’s first student and became a member of the school Egorov created in Moscow dealing with functions of real variable. • EULER Leonhard (1707, Basel, Switzerland -1783, St Petersburg, Russia) He was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function.He is also renowed for his work in mechanics, fluid dynamics, optics, and astronomy. • FATOU Pierre J.L. (1878, Lorient, France - 1929, Pornichet, France): Fatou worked in the fields of complex analytic dynamic and iterative and recursive processes. • FISHER Ernst (1875, Vienna, Austria - 1954, Cologne, Germany): Ernst Fischer is best known for the Riesz-Fischer theorem in the theory of Lebesgue integration. • FOURIER Joseph J.B. (1768, Auxerre, Francia - 1830, Parigi): Fourier studied the mathematical theory of heat conduction. He established the partial differential equation governing heat diffusion and solved it by using infinite series of trigonometric functions. • FRECH´ ET´ Maurice (1878, Maligny - 1973, Paris) Fréchétwas a French mathematician who made major contributions to the topology of point sets and defined and founded the theory of abstract spaces.In particular, in his thesis he introduced the concept of a metric space, although he did not invent the name ’metric space’ which is due to Hausdorff. • FRIEDRICHS Kurt Otto (1901, Kiel, Germany- 1982, New Rochelle, New York, USA) Friedrichs’ greatest contribution to applied mathematics was his work on partial differential equations. He also did major research and wrote many books. • FUBINI Guido, (1879, Venice, Italy -1943, New York, USA) Fubini may be considered one of the founder of modern projective-differential geometry. He has been very important as an analyst (Lebesgue integrals) and in the study of automor- phic functions and discontinuous groups. He has been also engaged in mathematical physics and in applied mathematics. • GAUSS Carl F. (1777, Brunswick, Duchy of Brunswick (now Germany) - 1855, Göttingen,Hanover (now Germany)) Gauss worked in a wide variety of fields in both 155 mathematics and physics incuding number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. His work has had an immense influence in many areas. • GREEN George (1793, Sneinton, Nottingham, England - 1841, Sneinton, Notting- ham, England) George Green was an English mathematician best-known for Green’s function and Greeen’s theorems in potential theory. • HADAMARD Jacques S. (1865, Versailles, Francia - 1963, Parigi)??? • HAHN Hans (1879, Vienna, Austria - 1934, Vienna, Austria): Hahn was an Austrian mathematician who is best remembered for the Hahn-Banach theorem. He also made important contributions to the calculus of variations, developing ideas of Weierstrass. • HAUSDORFF Felix (1868, Breslau, Germany (now Wroclaw, Poland)- 1942, Bonn, Germany ): Hausdorff worked in topology creating a theory of topological and metric spaces. In particular, he introduced the modern notion of metric space. He also worked in set theory and introduced the concept of a partially ordered set. • HILBERT David (1862, Königsberg, Prussia (now Kaliningrad, Russia)-1943, Göttingen,Germany): Hilbert’s work in geometry had the greatest influence in that area after Euclid. A systematic study of the axioms of Euclidean geometry led Hilbert to propose 21 such axioms and he analysed their significance. He made contributions in many areas of mathematics and physics. • HOLDER¨ Otto L. (1859, Stuttgart, Germany - 1937, Leipzig, Germany): Hölder worked on the convergence of Fourier series and in 1884 he discovered the inequality now named after him. He became interested in group theory through Kronecker and Klein and proved the uniqueness of the factor groups in a composition series. • KOLMOGOROV Andrey N. (1903, Tambov, Tambov province, Russia - 1987, Moscow) He was a Soviet Russian mathematician, preeminent in the 20th century, who advanced various scientific fields, among them probability theory, topology, in- tuitionistic logic, turbulence, classical mechanics and computational complexity. • JORDAN Camille M.E. (1838, La Croix-Rousse, Lyon, France - 1922, Paris, France): Jordan was highly regarded by his contemporaries for his work in algebra, group theory and Galois theory. Jordan is best remembered today among analysts and topologists for his proof that a simply closed curve divides a plane into exactly two regions, now called the Jordan curve theorem. He also originated the concept of functions of bounded variation and is known especially for his definition of the length of a curve. • LAGRANGE Joseph-Louis (1736, Turin, Sardinia-Piedmont (now Italy) - 1813 in Paris, France) Born Giuseppe Lodovico (Luigi) Lagrangia, he was a mathematician and astronomer, lived part of his life in Prussia and part in France, making great contributions to all fields of analysis, to number theory, and to classical and celestial mechanics. In particular he was one of the founders of the caculus of variations. • LEBESGUE Henry L. (1875, Beauvais, Oise, Picardie, France-1941, Paris, France): Lebesgue formulated the theory of measure in 1901 and the following year he gave the definition of the Lebesgue integral that generalises the notion of the Riemann integral. 156

• LEVI Beppo (1875, Turin, Italy - 1961, Rosarno, Argentina): He studied singu- larities on algebraic curves and surfaces. Later he proved some foundational results concerning Lebesgue integration. • LIPSCHITZ Rudolf O.S.( 1832, Könisberg, Germany (now Kaliningrad, Russia) -1903, Bonn, Germany) He was a German mathematician and professor at the Uni- versity of Bonn from 1864. Dirichlet was his teacher. While Lipschitz gave his name to the Lipschitz continuity condition, he worked in a broad range of areas. These included number theory, algebras with involution, mathematical analysis, differential geometry and classical mechanics. • LUSIN Nikolai N. (1883, Irkutsk, Russia - 1950, Moscow, USSR): Lusin’s main contributions are in the area of foundations of mathematics and measure theory. He also made significant contributions to descriptive set topology. • MINKOWSKI Hermann (1864, Alexotas, Russian Empire (now Kaunas, Lithua- nia) - 1909, Göttingen,Germany): Minkowski developed a new view of space and time and laid the mathematical foundation of the theory of relativity. • MORSE Anthony P. (1911-1984): he was an American mathematician who worked in both analysis, especially measure theory, and in the foundations of mathematics. He is best known as the co-creator, together with John L. Kelley, of Morse- Kelley set theory. This theory first appeared in print in Kelley’s General Topology. He is also known for his work on the Morse-Sard theorem and the Federer-Morse theorem. • NIKODYM Otto M. (1887, Zablotow, Galicia, Austria-Hungary (now Ukraine) - 1974, Utica, USA ): Nikodym’s name is mostly known in measure theory (e. g. the Radon-Nikodym theorem and derivative, the Nikodym convergence theorem, the Nikodym-Grothendieck boundedness theorem), in functional analysis (the Radon- Nikodym property of a Banach space, the Frechet-Nikodym metric space, a Nikodym set), projections onto convex sets with applications to Dirichlet problem, generalized solutions of differential equations, descriptive set theory and the foundations of quantum mechanics. • PEANO Giuseppe (1858, Cuneo, Italy - 1932, Turin, Italy): Peano was the founder of symbolic logic and his interests centred on the foundations of mathematics and on the development of a formal logical language. Among his important contributions, let us recall he invented ’space-filling’ curves in 1890, these are continuous surjective mappings from [0,1] onto the unit square. • PLATEAU Joseph A. F. (Tournal, Belgium, 1801- Ghent, Belgium, 1883): he was a Belgian physicist, who studied the phenomena of capillary action and surface tension. The mathematical problem of existence of a minimal surface with a given boundary is named after him Plateau’s problem. He conducted extensive studies of soap films and formulated Plateau’s laws which describe the structures formed by such films in foams. • POINCARE´ Jules Henri (1854, Nancy, Meurthe-et-Moselle - 1912, Paris) He was a French mathematician, theoretical physicist, and a philosopher of science. He is often described as a polymath, and in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime. As a mathematician and physicist, he made many original fundamental contributions to pure and 157 applied mathematics, mathematical physics, and celestial mechanics. He is considered to be one of the founders of the field of topology. • POISSON S. Denis (1781, Pithiviers, France - 1840, Sceaux near Paris) He was very well-known for his work on definite integrals, electromagnetic theory, and probability. PoissonOs˜ most important work concerned the application of mathematics to electricity and magnetism, mechanics, and other areas of physics. Poisson contributed to celestial mechanics by extending the work of Lagrange and Laplace on the stability of planetary orbits and by calculating the gravitational attraction ex- erted by spheroidal and ellipsoidal bodies. He also did important investigation of probability. In pure mathematics his most important works were a series of papers on definite integrals and his advances in Fourier analysis, which paved the way for the research of the German mathematicians Peter Dirichlet and Bernhard Riemann. • PRYM Friedrich E.F. (1841 Düren- 1915 Bonn) He was a German mathematician who introduced Prym varieties and Prym differentials. • RADEMACHER Hans (1892, Wandsbeck, now Hamburg-Wandsbek - 1969, Haver- ford, Pennsylvania, USA) Rademacher performed research in analytic number theory, mathematical genetics, the theory of functions of a real variable, and quantum theory. Most notably, he developed the theory of Dedekind sums. Rademacher’s name is also known for the result about the differentiability of Lipschitz functions. • RADON Johann (1887, Tetschen, Bohemia (now Decin, Czech Republic) - 1956, Vienna, Austria): Radon worked on the calculus of variations, differential geometry and measure theory. • RIEMANN G. F. Bernhard (1826, Breselenz, Hanover (now Germany)- 1866, Selasca, Italy): Riemann’s ideas concerning geometry of space had a profound effect on the development of modern theoretical physics. He clarified the notion of integral by defining what we now call the Riemann integral. • RIESZ Frigyes (Friedrich) (1880, Györ, Austria-Hungary (now Hungary) - 1956, Budapest, Hungary ): Riesz was a founder of functional analysis and his work has many important applications in physics. • RIESZ Marcel (1886, Györ,Austria-Hungary (now Hungary) - 1969, Lund, Swee- den) He was a Hungarian mathematician and moved to Sweden in 1908 and spent the rest of his life there. He was known for work on classical analysis, on fundamental solutions of partial differential equations, on divergent series, Clifford algebras, and number theory. He was the younger brother of the mathematician Frigyes Riesz. • SCHAUDER Juliusz P. (1899, Lemberg, Austrian Empire (now Lviv, Ukraine) - 1943, Lwów,Poland (now Ukraine)) He was a Polish mathematician of Jewish origin, known for his fundamental work in functional analysis, partial differential equation and mathematical physics. Schauder was Jewish, and after the invasion of German troops in Lwówit was impossible for him to continue his work. He was executed by the Gestapo, probably in September 1943. • SERRIN James (1926, Chicago, Illinois, USA - 2012, Minneapolis, Minnesota, USA) Serrin is a mathematician well-known for his contributions to continuum mechanics, nonlinear analysis, and partial differential equations. • SOBOLEV Sergei L. (1908, S.Petersburg - 1989, Moscow): He introduced the notions that are now fundamental for several different areas of mathematics. Sobolev spaces and their embedding theorems are an important subject in functional analysis. 158

• STEINHAUS Hugo D. (1887, Jaslo, Galicia, Austrian Empire (now Poland) - 1972, Wroclaw, Poland): He did important work on functional analysis. Some of Steinhaus’s early work was on trigonometric series. He was the first to give some examples which would lead to marked progress in the subject. • TONELLI Leonida (1885, Gallipoli, Italy - 1946, Pisa, Italy) Tonelli discovered the importance of the semicontinuity in calculus of variations in order to get the existence of minima or maxima for functionals. He also advanced the study of the integration theory. • URYSOHN Pavel S. (1898, Odessa, Ukraine- 1924, Batz-sur-Mer, France): Urysohn is best known for his contributions in the theory of dimension, and for Urysohn’s Metrization Theorem and Urysohn’s Lemma, both of which are fundamental results in topology. • VITALI Giuseppe (1875, Ravenna, Italy - 1932, Bologna, Italy): Vitali made significant mathematical discoveries including a theorem on set-covering, the notion and the characterization of an absolutely continuous functions and a criterion for the closure of a system of orthogonal functions. • VOLTERRA Vito (1860, Ancona, Italy - 1940, Roma, Italy) Volterra was an Ital- ian mathematician and physicist, known for his contributions to mathematical biology and integral equations, being one of the founders of functional analysis. Volterra is the one among few people who was a plenary speaker in the International Congress of Mathematicians four times (1900, 1908, 1920, 1928). In 1922, he joined the oppo- sition to the Fascist regime of Benito Mussolini and in 1931 he was one of only 12 out of 1,250 professors who refused to take a mandatory oath of loyalty. As a result of his refusal to sign the oath of allegiance to the fascist government, he was compelled to resign his university post and his membership of scientific academies, and, during the following years, he lived largely abroad, returning to Rome just before his death. • VON NEUMANN John (1903, Budapest, Hungary - 1957, Washington D.C., USA): Von Neumann was generally regarded as the foremost mathematician of his time and said to be ”the last representative of the great mathematicians”;a genius who was comfortable integrating both pure and applied sciences. He made major contributions to a number of fields, including mathematics (foundations of mathematics, functional analysis, ergodic theory, representation theory, operator algebras, geometry, topology, and numerical analysis), physics (quantum mechanics, hydrody- namics, and quantum statistical mechanics), economics (game theory), computing (Von Neumann architecture, linear programming, self-replicating machines, stochas- tic computing), and statistics.He was a pioneer of the application of operator theory to quantum mechanics in the development of functional analysis, and a key figure in the development of game theory and the concepts of cellular automata, the universal constructor and the digital computer. • WEIERSTRASS Karl (1815, Ostenfelde, Germania -1897, Berlino): Weierstrass is best known for his construction of the theory of complex functions by means of power series. Known as the father of modern analysis, Weierstrass devised tests for the convergence of series and contributed to the theory of periodic functions, functions of real variables, elliptic functions, Abelian functions, converging infinite products, and the calculus of variations.