Lecture Notes Geometric Measure Theory

Home , Geometric measure theory, Hausdorff measure, Locally finite measure, Vitali covering lemma

Lecture Notes

Geometric Measure Theory

Course held by Notes written by

Prof. Giovanni Alberti Francesco Paolo Maiale

Department of Mathematics Pisa University December 27, 2017 Disclaimer

These notes came out of the Geometric Measure Theory course, held by Professor Gio- vanni Alberti in the second semester of the academic year 2016/2017.

They include all the topics that were discussed in class; I added some remarks, simple proof, etc.. for my convenience.

I have used them to study for the exam; hence they have been reviewed thoroughly. Unfortunately, there may still be many mistakes and oversights; to report them, send me an email at francescopaolo (dot) maiale (at) gmail (dot) com. Contents

I Introduction to Measure Theory5

1 Introduction6 1.1 Plateau’s Problem...... 6 1.2 Geodesics problem (d = 1)...... 7 1.3 ”Surface” Problem (d > 1)...... 8

2 Measure Theory9 2.1 Definitions and Elementary Properties...... 9 2.2 Weak-∗ Topology on Measures Space...... 12 2.3 Outer Measures...... 17 2.4 Carathéodory Construction...... 21 2.5 Hausdorff d-dimensional Measure Hd ...... 23 2.6 Hausdorff Dimension of Cantor-Type Sets...... 31

II Covering Results 34

3 Covering Theorem 35 3.1 Vitali Covering Theorem...... 36 3.2 Besicovitch Covering Theorem...... 41

III Geometric Measure Theory 45

4 Density of Measures 46 4.1 Density of Doubling Locally Finite Measures...... 46 4.2 Upper Density of the Hausdorﬀ Measure...... 47 4.3 Upper d-Dimensional Density...... 52 4.4 Applications...... 60

5 Self-Similar Sets 63 IV Rectiﬁability 71

6 Other measures and dimensions 72 6.1 Geometric Integral Measure...... 72 6.2 Invariant Measures on Topological Groups...... 74

7 Lipschitz Functions 78 7.1 Deﬁnitions and Main Properties...... 78 7.2 Diﬀerentiability of Lipschitz Functions...... 79 7.3 Area Formula for Lipschitz Maps...... 83 7.4 Coarea Formula for Lipschitz Maps...... 87

8 Rectifiable Sets 89 8.1 Introduction and Elementary Properties...... 89 8.2 Tangent Bundle...... 95 8.3 Rectifiability Criteria...... 101 8.4 Area Formula for Rectifiable Sets...... 106

V Finite Perimeter Sets 108

9 Bounded Variation Functions 109 9.1 Deﬁnition and Elementary Properties...... 109 9.2 Alternative Deﬁnitions: Bounded Variation on the Real Line...... 112 9.3 Functional Properties...... 114

10 Finite Perimeter Sets 121 10.1 Main Deﬁnitions and Elementary Properties...... 121 10.2 Approximation Theorem via Coarea Formula...... 123 10.3 Existence Results for Plateau’s Problem...... 125 10.4 Capillarity Problem...... 128 10.5 Finite Perimeter Sets: Structure Theorem...... 132 10.6 Back to the Capillarity Problem...... 146

11 Appendix 149 11.1 First Variation of a Functional...... 149 11.2 Regularity in Capillarity Problems...... 154 11.3 Regularity in the F.P.S. Setting...... 155 11.4 Structure of Finite Length Continua in Rn ...... 157 11.5 Lower Semi-Continuity: Golab Theorem...... 159 11.6 Simons’ Cone...... 159 Part I

Introduction to Measure Theory

5 Chapter 1

Introduction

In this chapter, we introduce the main topics of the course and give a brief overview of what we will see and what we will be able to prove by the end of the course.

1.1 Plateau’s Problem

The primary goal and the motivating example of this course is the Plateau’s problem, that is, the problem to find the d-dimensional surface Σ of the minimal area with prescribed (d − 1)-dimensional boundary Γ. By the end, we will be able to prove that a solution indeed exists, but we will not find it explicitly since it is a NP (hard) numerical problem. As of now, the problem is not well defined. In fact, the notions of surface, area, and boundary make sense in the smooth setting but, as the examples below show, we need to work in a less regular setting. More precisely, requiring the surface to be smooth is not enough for modeling reasons (e.g., dip a wire frame into a soap solution, form a soap film, and look for the minimal surface whose boundary is the wire frame), and also for existence reasons. Example 1.1. Here we give a list of Plateau’s problems with prescribed boundary conditions, and we write down the correct solutions, without proving anything.

(a) Let us identify R4 =∼ C × C and, if d = 2, let us consider the smooth boundary given by 1 1 Γ1 := S × {0} ∪ {0} × S . Surprisingly, every minimizing sequence of smooth surfaces converges to a surface which is not smooth at all. Indeed, the solution of the problem is

2 2 Σ1 := D × {0} ∪ {0} × D .

The surface Σ1 is clearly singular at the origin, but the singularity may be removed (by factorizing it into two nonsingular surfaces). 7 1.2. GEODESICS PROBLEM (D = 1)

(b) Let us identify R4 =∼ C × C and, if d = 2, let us consider the smooth boundary given by 2 3 1 Γ2 := (z , z ): z ∈ S . The solution to the Plateau’s problem is 2 3 2 Σ2 := (z , z ): z ∈ D , which is a non-smooth surface, whose singularity cannot be removed (since the poly- 3 2 nomial z1 = z2 cannot be factorized). (c) Let us identify R8 =∼ R4 × R4 and, if d = 7, let us consider the smooth boundary given by 3 3 Γ3 := S × S .

The minimal surface of prescribed boundary Γ3 is 4 4 Σ3 := (x1, x2) ∈ R × R : |x1| = |x2| ≤ 1 .

To conclude this introductive chapter, we give a brief overview of the main approaches (studied in this course) to the Plateau’s problem, as d ranges between 1 and ∞.

1.2 Geodesics problem (d = 1)

The geodesics problem (that is, find the shortest curve connecting two points) is, surprisingly, still an open in the non-Riemannian setting. However, in the Riemannian setting, the geodesics problem is completely solved. Indeed, if we consider the curves parametrized by paths, the length is a well-defined notion, and the associated functional is lower semi-continuous and coercive; hence the compactness is easy to prove. There are many possible approaches to the geodesics problem, e.g., the Steiner approach and the set theoretical approach, which we describe briefly in the remainder of the section.

Steiner Problem. It is also called networks approach, and it is used to prove the existence of the geodesics and ﬁnd the explicit expression for it. The reader may consult [1] for a detailed dissertation on the topic.

Set Theoretical Approach. The main idea is to find a closed and connected set Σ of minimum length, containing a given finite set Γ. As we shall see later in the course, in this case the length is a well defined concept: the Hausdorff distance. In fact, if X is a suitable space (metric, endowed with Hausdorff distance, etc...), then the class defined by X := {K ⊆ X : K compact and connected} is compact and, by Gotab theorem1, H1 is lower semi-continuous on X.

1 n [4] Let C be an inﬁnite collection of non-empty compact sets all lying in a bounded portion B of R . Then there exists a sequence {Ej } of distinct sets of C convergent in the Hausdorﬀ metric to a non-empty compact set E. 8 1.3. ”SURFACE” PROBLEM (D > 1)

1.3 ”Surface” Problem (d > 1)

3. The Plateau’s problem is much harder when d = 2, but there are still many approaches possible some of which relying, in a certain sense, on the work already done in the geodesics case.

Set Theoretical Approach. This approach is highly nontrivial. For example, one may ask what does it mean that a compact set Σ spans a boundary Γ? Moreover, there is another problem one should deal with: the 2-dimensional Hausdorﬀ measure H2 is, generally, not lower semi-continuous. The reader may consult [9] for a complete treatise of the topic. Remark 1.1. Suppose that d = 2, n = 3 and that Σ is a surface with boundary Γ. If γ is another closed curve, linked to Γ (by a nonzero linking number), then γ ∩ Σ 6= ∅.

Parametric Approach. This method is essentially due to Douglas [3]. The main idea is the following: since a parametrization φ : D2 → Rn defines surfaces in Rn, the area functional is well-defined and given by the formula Z ∂ φ ∂ φ A(φ) := ∧ ds1 ds2. D2 ∂ s1 ∂ s2 On the other hand, the existence through lower semi-continuity and the compactness are a delicate matter, since coercivity is not an easy property to obtain (the integrand is similar to a determinant). There is a trick which is similar to the one we can use to find geodesics in the differential geometry setting. More precisely, we consider the functional Z 1 2 E(φ) := |∇φ| ds1 ds2. 2 D2 If we find a minimal point φ for E, then φ will be a conformal parametrized minimum for A. This trick, on the other hand, heavily depends on a nontrivial theorem: every such Σ admits a conformal re-parametrization. The lack of conformal parametrization, though, is what stop us from extending the same trick to dimension d strictly bigger than 2.

Higher Dimension. If the codimension of Σ is equal to 1 (that is, n = d + 1), then finite perimeter sets generalize the notion of open (d + 1)-dimensional sets with smooth boundary in Rn. The class of finite perimeter sets has excellent compactness properties and a notion of area lower semi-continuous. This approach is called ”weak” surfaces approach, and it is essentially due to Caccioppoli [2] and De Giorgi [7]. A different approach, working for any d and n, referred to as integral currents, was introduced by Federer and Flaming in their joint paper [6]. Chapter 2

Measure Theory

In this chapter, the chief goal is to give a brief introduction to the theory of measure and acquire the fundamental notions needed for the remainder of the course. The reader with an adequate background, may immediately jump to the next section, and use this one as a reference. In the first part, we introduce the only three classes of measures we shall be concerned about in this course, and then we study the weak-∗ topology and some of the fundamental properties of the weak-∗ convergence. In the second part, we introduce the notion of outer measure, and we show how it can be used to construct a measure that belongs to one of the classes mentioned above. In the final part, we construct the Hausdorff d-dimensional measure, denoted by Hd, and we prove some of the main properties (e.g., the relation with Lipschitz functions). The last section is devoted to the computation of the Hausdorff dimension of a Cantor-type set.

2.1 Deﬁnitions and Elementary Properties

Introduction. In this course, we will essentially be only concerned with measures that belong to one of the following classes:

(1) Positive, σ-additive measures, deﬁned on the Borel σ-algebra of a reasonable space X, e.g. we might assume X metric, locally compact and separable.1

(2) Positive, σ-subadditive measures, deﬁned on the set of all parts P (X). These are generally called outer measures. (3) Real-valued and vector-valued (bounded) σ-additive measures, deﬁned on the Borel σ-algebra.

Remark 2.1. The Borel σ-algebra B(X) is not closed under the action of continuous maps.

1We are not very speciﬁc here since the reader may try, whenever possible, to derive, as an exercise, the minimal assumption on X for an assertion to be true. 10 2.1. DEFINITIONS AND ELEMENTARY PROPERTIES

Proof. Let X = [0, 1] ⊂ R, and let us consider a space-ﬁlling curve g : [0, 1] −→ [0, 1]2 and the projection π : [0, 1]2 −→ [0, 1]; the reader may easily check that the map f := π ◦ g : [0, 1] −→ [0, 1] is a continuous function. Thus, given a Borel set S ⊂ [0, 1]2 with π (S) not Borel, it is easy to prove that the image of the Borel set g−1(S) under f is not Borel.

In this course, we will often introduce a suitable2 outer measure defined on X and, by taking the restriction to the Borel σ-algebra (see Theorem 2.14), we automatically obtain a measure which belongs to the class (1). Remark 2.2. If X is a separable Banach space3, then the class of measures (3) is the dual space of a particular subspace of C0(X). In the next section, we will also prove that, if the space of measures is endowed with the weak-∗ topology, then this class has good compactness properties (that is, the Banach-Alaoglu theorem holds true). Definition 2.1 (Vector-valued Measure). Let B(X) be a Borel σ-algebra on X, and let F be a normed space. A function µ : X −→ F is a σ-additive F -valued measure if ! X [ µ (En) = µ En , (2.1) n∈N n∈N for any countable disjoint family of Borel sets {En}n∈N ⊂ B(X). Remark 2.3. The identity (2.1) does not only imply the σ-additivity of the measure, but it also gives us a stronger property. More precisely, the sum on the left-hand side does not depend on the order of the indexes, and thus, if F is a finite-dimensional space, then it is enough to infer that the sum converges absolutely.

Representation Theorem. In this ﬁnal brief paragraph, we sketch the proof of the Riesz representation theorem, according to which the class of measures (3) is the dual of a suitable subset of the space of all continuous functions. Deﬁnition 2.2 (Total Variation). Let µ be a measure. The total variation of a set E ⊆ X is the supremum of the measure over the possible countable partitions, that is,

( +∞ ) X |µ| (E) := sup |µ(E )| {E } countable partition of E . (2.2) n n n∈N n=0

Clearly, the total variation is a positive bounded measure, deﬁned on the Borel σ-algebra B(X). The mass of µ is deﬁned by setting

kµk := |µ| (X). (2.3)

The formula (2.3) deﬁnes a norm on the space of all the F -valued σ-Borel measures, which is also complete. Moreover, it is easy to see that µ is absolutely continuous with respect

2See Section 2.4 for a more precise definition of what suitable means here. 3It is not necessary to require X to be a separable Banach space, but it is more than enough for our purposes. 11 2.1. DEFINITIONS AND ELEMENTARY PROPERTIES to its total variation |µ|, and hence by Radon-Nikodym Theorem there exists a Borel function f : X −→ F such that Z µ = f · |µ| µ(E) := f(x) d |µ| (x), E with the additional property that its norm is almost everywhere equal to one, that is, |f(x)|F = 1 for |µ|-almost every x ∈ X. Consequently, vector-valued measures may be identified with the product between a positive measure λ and a function λ-summable, that is, there exists f ∈ L1(λ) such that µ = f · λ. This identification is particularly useful when we need to integrate a function g : X −→ F with respect to a vector valued measure. Indeed, it turns out that Z Z g(x) dµ(x) = g(x) · f(x) d|µ|(x), X X where · is a suitable notion of product4 between f and g. n Notation. Let µ be an R -valued measure defined on X. We denote by Λµ the functional given by Z Z g 7−→ g dµ = g(x) · f(x) d |µ| (x), (2.4) X X where · is the scalar product in Rn.

The map (2.4) is well-defined at every |µ|-summable function g : X −→ Rn function; thus, the functional is well-defined at every continuous function h : X −→ Rn which is infinitesimal at ∞ (i.e., equal to zero in the one-point compactification of Rn.) n n In particular, Λµ is well-defined on C0 (X; R ) - or, equivalently, on C (X; R ) if X is compact -, which is a separable Banach spaces with respect to the supremum norm. The functional n Λµ : C0 (X; R ) −→ R is linear and bounded (i.e., continuous). More precisely, from (2.4) it follows that Z |Λ (g)| ≤ |g(x)| d |µ| (x) ≤ kgk n kµk, µ C0(X; R ) X which, in turn, implies that kΛµk ≤ kµk. If we set g := f, then it turns out that kgk = kfk = 1 and that the equality holds true, i.e.

kΛµk = kµk.

Theorem 2.3 (Riesz). Let M (X; Rn) be the space of all the Rn-valued measures. The map n n ∗ M (X; R ) −→ (C0 (X; R )) , µ 7−→ Λµ is an isometry. Moreover, if X is a compact space, then

n n ∗ M (X; R ) −→ (C (X; R )) , µ 7−→ Λµ is also an isometry.

4For example it could be a scalar product, or an external product. 12 2.2. WEAK-∗ TOPOLOGY ON MEASURES SPACE

Theorem 2.4 (Riesz). Let F be a ﬁnite-dimensional normed space. The map

∗ ∗ ∗ M (X; F ) −→ (C0(X; F )) , µ 7−→ Λµ is an isometry. Moreover, if F is a separable Banach space, then

∗ ∗ ∗ M (X; F ) −→ (C0(X; F )) , µ 7−→ Λµ is also an isometry, provided that F ∗ is good enough.

2.2 Weak-∗ Topology on Measures Space

In this section, we endow the space of measures M (X; Rn) with the weak-∗ topology since the associated notion of convergence is particularly pleasant.

n n Deﬁnition 2.5 (Measures convergence). A sequence (µn)n∈N ⊂ M (X; R ) of R -valued measures converges weakly-∗ to a measure µ ∈ M (X; Rn) if and only if Z Z 0 n lim g(x) dµn(x) = g(x) dµ(x), ∀g ∈ C0 (X; R ) . (2.5) n→∞ X X

n Remark 2.4. If X is a compact space, then (µn)n∈N ⊂ M (X; R ) converges weakly-∗ to a measure µ ∈ M (X; Rn) if and only if Z Z 0 n lim g(x) dµn(x) = g(x) dµ(x), ∀g ∈ C (X; R ) . (2.6) n→∞ X X

Remark 2.5. The notion of strong convergence in the space M (X; Rn) is, unfortunately, ”too strong”, and, in general, it is not interesting at all. For this reason, from now on we say that µn converges (in the sense of measures) to µ if (2.5) is satisﬁed. Remark 2.6.

(1) The Banach-Alaoglu Theorem, in the particular case of M (X; Rn), may be stated in the following, particularly simple, way.

n Theorem 2.6 (Banach-Alaoglu). Let (µn)n∈N ⊂ M (X; R ) be a sequence with uniformly bounded masses, that is, there exists C > 0 such that

kµnk ≤ C < ∞ ∀ n ∈ N. Then, up to subsequences, it converges in the sense of measures to an element µ ∈ M (X; Rn).

n (2) Let (µn)n∈N ⊂ M (X; R ) be a sequence of measures with uniformly bounded masses. n Then µn converges to µ ∈ M (X; R ) if and only if Z Z lim g(x) dµn(x) = g(x) dµ(x), ∀g ∈ D, n→∞ X X

n where D is a dense subset of C0 (X; R ), e.g. the space of compactly supported functions. 13 2.2. WEAK-∗ TOPOLOGY ON MEASURES SPACE

(3) The ”local version of the theory” works in a very similar way. For example, if we n assume that (µn)n∈N ⊂ M (X; R ) is a sequence of measures with locally bounded masses, that is, for any K ⊆ X compact there exists C(K) > 0 such that

kµnkK ≤ C(k) < ∞, ∀n ∈ N,

then we can infer that µn converges in the sense of measures.

Proposition 2.7. Let (µn)n∈N ⊂ M (X; R) be a sequence of real-valued positive measures, and assume that it converges to µ ∈ M (X; R).

(a) For any open subset A ⊆ X, it turns out that

lim inf µn(A) ≥ µ(A). n→+∞

(b) For any compact subset K ⊆ X, it turns out that

lim sup µn(K) ≤ µ(K). n→+∞

lim µn(E) = µ(E). n→+∞

Proof.

(a) The ﬁrst assertion is equivalent to the fact that the functional Z ΦA : M (X; R) −→ R, λ 7−→ χA dλ X

is (weakly-∗) lower semi-continuous for every A ∈ B(R). The characteristic function of an open set can be approximated by an increasing sequence of continuous function, e.g.,  0 if x∈ / A (A)  fn (x) := min 1, sup{n · r | B(x, r) ⊆ A} if x ∈ A.  r>0 The supremum of weakly-∗ lower semi-continuous function5 is weakly-∗ lower semi- continuous; thus from the relation

χA(x) = sup fn(x), n∈N

it follows easily that χA is lower semi-continuous. Therefore, if we apply the standard Fatou Lemma, it turns out that

lim inf µn(A) ≥ µ(A), ∀ A ∈ B( ). n→+∞ R

5 The sequence (fn)n∈N is made up of continuous function, but the supremum will lose the upper semi- continuity. 14 2.2. WEAK-∗ TOPOLOGY ON MEASURES SPACE

(b) This assertion follows immediately from the previous one by taking the complement. On the other hand, we can mimic the proof above and obtain the result directly by proving that Z ΦK : M (X; R) −→ R, λ 7−→ χK dλ X is weakly-∗ upper semi-continuous for every K ⊆ X compact. The characteristic function of a compact set can be approximated by a decreasing sequence of continuous function, e.g.  1 if x ∈ K  gn(x) := 1 − min 0, sup{n · r | B(x, r) ⊆ Kc} if Kc.  r>0

The inﬁmum of a collection of (weakly-∗) upper semi-continuous functions is (weakly- ∗) upper semi-continuous; thus from the equality

χK (x) = inf gn(x), n∈N

it follows that χK is upper semi-continuous. Therefore, if we apply the reverse inequality of the Fatou Lemma, it turns out that

lim sup µn(K) ≤ µ(K). n→+∞

(c) Let E ⊆ X be a relatively compact set such that µ (∂E) = 0. The interior part Int E has the same measure of E and it is open; thus it follows from (a) that

lim inf µn(E) = lim inf µn (Int E) =≥ µ (Int E) = µ(E). n→+∞ n→+∞

In a similar fashion, since E is relatively compact, the closure E is compact and has the same measure of E; thus it follows from (b) that lim sup µn E = lim sup µn(E) ≤ µ(E) = µ E , n→+∞ n→+∞

and this proves the sought identity, i.e., limn→+∞ µn(E) = µ(E).

Proposition 2.8. Let (µn)n∈N ⊂ M (X; R) be a sequence of real-valued positive measures, and assume that it converges to µ ∈ M (X; R).

(a) If X is compact, then lim kµnk = kµk. n→∞ (b) If X is locally compact, then

lim inf kµnk ≥ kµk. n→∞

In particular, there exists a sequence (µn)n∈N of measures, converging to some µ, such that the mass kµnk does not converge to kµk. 15 2.2. WEAK-∗ TOPOLOGY ON MEASURES SPACE

Proof.

(a) The norm is weakly-∗ lower semi-continuous; hence it suﬃces to prove that

µn → µ and X compact =⇒ lim sup kµnk ≤ kµk. n→∞

By assumption (µn)n∈N is a sequence of positive measures, which means that the mass is simply given by kµnk = µn(X), that is, Z kµnk = dµn, ∀ n ∈ N. X By compactness of X, the convergence in sense of measures is equivalent to (2.6), and thus Z Z kµnk = dµn → dµ = kµk. X X

(b) It is enough to provide a counterexample to the convergence. Let X = R and let (xn)n∈N be a sequence of points in R such that xn → +∞ as n → +∞. If we deﬁne

µn := δxn ,

where δy is the delta of Dirac at y, then it is straightforward to prove that µn → 0 and kµnk = 1 for every n ∈ N, that is,

lim inf kµnk = 1 > kµk = 0. n→+∞

Deﬁnition 2.9 (Tightness). Let (µn)n∈N ⊂ M (X; R) be a sequence of real-valued positive measures. The sequence is tight if, for every > 0, there exists a compact subset K ⊂ X such that µn (X \ K) ≤ , ∀ n ∈ N.

Lemma 2.10. Let (µn)n∈N ⊂ M (X; R) be a sequence of real-valued positive measures, converging to an element µ ∈ M (X; R). Then the sequence is tight if and only if the mass converges, that is, n→+∞ kµnk −−−−−→kµk. (2.7)

Proof. If the sequence of the masses converges to kµk, then the tightness of (µn)n∈N follows from the deﬁnition (namely, the tail of the masses sequence is as small as we want.)

Vice versa, suppose that the sequence (µn)n∈N is tight. The norm k · k is weakly-∗ lower semi-continuous; hence it suﬃces to prove the opposite inequality, that is,

lim sup kµnk ≤ kµk. n→∞

Let > 0 and let K ⊂ X be the compact subset satisfying (2.7). If we consider the c decomposition of the ambient space X = K ∪ K , then it turns out that Z Z Z

µn(X) = dµn + dµn ≤ χK (x) dµn(x) + . K X\K X 16 2.2. WEAK-∗ TOPOLOGY ON MEASURES SPACE

Finally, we take the limit as n → +∞, and we notice that from (b) of Proposition 2.7 it follows that that Z

lim sup µn(X) ≤ lim sup χK dµn + ≤ µ(K) + , n→+∞ n→+∞ X which is enough to conclude the proof since > 0 may be taken arbitrarily small.

”Strong” Convergence. In this ﬁnal paragraph, we introduce a stronger notion of convergence, which turns out to be the right replacement for the convergence in norm for the space of all measures.

Deﬁnition 2.11 (Convergence in Variation). Let (µn)n∈N ⊂ M (X; R) be a sequence of real-valued measures. The sequence converges in variation to some µ ∈ M (X; R) if and only if n→+∞ µn −→ µ and kµnk −−−−−→kµk. (2.8) Remark 2.7. The convergence in variation induces a ﬁner topology than the weak-∗ one, and it is the right replacement for the norm convergence.

Remark 2.8. Let (µn)n∈N be a sequence of positive measures converging in variation. The assertions of Proposition 2.7 hold true, provided that we modify them accordingly with the new deﬁnition:

(b1) For any closed subset C ⊆ X, it turns out that

lim sup µn(C) ≤ µ(C). n→+∞

(c1) For any E ⊆ X such that µ (∂E) = 0, it turns out that

lim µn(E) = µ(E). n→+∞ n Remark 2.9. If µn → µ is a converging sequence of R -valued measures (not necessarily positive), then there is a subsequence (nk)k∈N ⊂ (n)n∈N such that

|µnk | → λ, where λ is a positive measure satisfying λ ≥ |µ|. Moreover, given a relatively compact set E ⊆ X with λ(E) = 0, the reader may prove, as an exercise, that

µn(E) → µ(E).

n Exercise 2.1. Let (µn)n∈N be a sequence of R -valued measures converging in variation to an element µ. Prove that:

(1) The sequence of positive measures (|µn|) weakly-∗ converges to |µ|. n∈N

(2) For every subset E ⊆ X such that |µ| (E) = 0, it turns out that µn(E) → µ(E).

Exercise 2.2. Let X be a suitable ambient space, let (xn)n∈N be a sequence of points in

X such that xn → x as n → +∞, and set µn := δxn − δx. Prove that

|µn| = δxn and |µn| → |µ| = 0. 17 2.3. OUTER MEASURES

2.3 Outer Measures

In this section, we introduce the notion of outer measure, and we set the ground for the main result of this chapter: the Carathéodory construction of a measure defined on the Borel σ-algebra B(X). Definition 2.12 (Outer Measure). Let X be a set. An outer measure on X is a set function µ : P (X) −→ [0, ∞], where P(X) denotes the power set, such that

(a) µ(∅) = 0; (b) µ is monotone, i.e. A ⊆ B =⇒ µ(A) ≤ µ(B); P (c) µ is σ-additive, i.e. {An}n∈N ⊂ P (X) =⇒ µ (∪nAn) ≤ n µ (An). Example 2.1. A simple example of an outer measure, which is deﬁned on every set X, is given by (n if |E| = n, µ(E) := +∞ if |E| ≥ ℵ0. Example 2.2. Let X := R. The outer measure from which the Lebesgue measure derives is deﬁned by ( )

X \ µ(E) := inf |In| E ⊂ In , n∈N n∈N where the In are, for example, open intervals. Definition 2.13 (Carathéodory measurable). Let µ be an outer measure defined on X.A subset A ⊆ X is Carathéodory µ-measurable if and only if µ(E) = µ(E ∩ A) + µ(E ∩ Ac), ∀ E ⊆ X. (2.9) Theorem 2.14 (Carathéodory). Let M be the class of Carathéodory µ-measurable sets in X. Then M is a σ-algebra and the restriction of µ to M, denoted by µ ¬ M, is σ-additive.

Proof. To ease the notation for the reader we divide the proof into three steps.

Step 0. The relation (2.9) is symmetric with respect to the complement; hence A ∈ M immediately implies Ac ∈ M, and vice versa. In particular, the empty set ∅ belongs to M.

Step 1. Let A1,A2 ∈ M. For every E ⊆ X it turns out that c µ(E) = µ(E ∩ A1) + µ(E ∩ A1) =

c c = µ(E ∩ A1 ∩ A2) + µ(E ∩ A1 ∩ A2) + µ(E ∩ A1) ≥

c ≥ µ (E ∩ (A1 ∩ A2)) + µ (E ∩ (A1 ∩ A2) ) , where the last inequality follows from the monotonicity property of µ. The opposite inequality is always true as a consequence of the subadditivity of the measure; hence we can infer that A1,A2 ∈ M =⇒ A1 ∩ A2 ∈ M. 18 2.3. OUTER MEASURES

Step 2. Let {An}n∈N be a disjoint family of elements of M. Then c µ(E) = µ(E ∩ A1) + µ(E ∩ A1) =

c c c = µ(E ∩ A1) + µ(E ∩ A1 ∩ A2) + µ(E ∩ A1 ∩ A2) =

c c = µ(E ∩ A1) + µ(E ∩ A2) + µ(E ∩ A1 ∩ A2) =

m m !c ! X [ = [µ(E ∩ An)] + µ E ∩ An . n=1 n=1 If we take the supremum, the identity above yields to

+∞ +∞ !c ! X [ µ(E) ≥ [µ(E ∩ An)] + µ E ∩ An , n=1 n=1 and thus, using the σ-subadditivity of the outer measure µ, we obtain the nontrivial inequality +∞ !! +∞ !c ! [ [ µ(E) ≥ µ E ∩ An + µ E ∩ An , n=1 n=1 that is, the countable union of disjoint elements of M still belongs to M.

Step 3. In a similar fashion, one can prove that µ ¬ M is σ-additive. In fact, one can take S a family of disjoint elements {An}n∈ and take E := An. N n∈N

This theorem, despite the simple proof, is incredibly powerful. Indeed, it is relatively easy to find an outer measure on X and derive a σ-additive measure, while it is much harder to define it directly. Remark 2.10. The Carathéodory result, on the other hand, gives no information whatso- ever on the σ-algebra M. For example, if we consider the outer measure ( 0 if E = µ(E) := ∅ 1 if E 6= ∅, then it is easy to prove that the σ-algebra associated via Theorem 2.14) is the trivial one, that is, M = {∅,X} , and thus it is not an interesting object to deal with.

The next theorem will give us an easy-to-check criterion for an outer measure to be ”interesting,” in the sense that the σ-algebra M contains (at least) the Borel σ-algebra. Theorem 2.15. Let X be a metric space and let µ be an outer measure deﬁned over X. If

dist(A1,A2) > 0 =⇒ µ(A1 ∪ A2) = µ(A1) + µ(A2), then M contains the Borel algebra.

Proof. First, we notice that it is enough to prove that closed sets belong to M. We divide the argument into two steps: a particular, fairly straightforward, case and the general case. 19 2.3. OUTER MEASURES

Step 1. Let X be a Cantor-type set (in particular, it is totally disconnected). The identity

µ(E) = µ(E ∩ I) + µ(E ∩ Ic), ∀ E ⊆ X, holds for any interval I contained in X. By construction, the interval I and the complement Ic are distant, and this concludes the proof since the intervals generate the Borel σ-algebra of X.

Step 2. Let C ⊆ X be a closed subset, and let us consider the sequence given by 1 An := x ∈ X d(x, C) ≥ . n

The sequence (An)n∈N is increasing, and its limit is the complement of C, that is, c An ↑ (X \ C) = C .

We may always assume, without loss of generality, that µ(E) < +∞. It follows from the monotonicity of µ that

µ(E) ≥ µ(E ∩ C) + µ(E ∩ An), ∀ n ∈ N, and hence, by taking the limit as n → +∞, an application of Lemma 2.16 proves that

c µ (E ∩ An) → µ (E ∩ C ) , which is the nontrivial inequality.

Lemma 2.16. Let An ↑ A be an increasingly converging sequence of subsets, and assume that c dist(An,An+1) > 0 ∀ n ∈ N. (2.10)

If the measure of the limit is ﬁnite (µ(A) < +∞), then µ(An) ↑ µ(A).

Proof. Let a ∈ R be the real number such that µ(An) ↑ a (it exists, as a consequence of the monotonicity of the sequence). The reader can easily prove that

a ≤ µ(A).

To prove the opposite inequality, we consider the sequence of sets deﬁned by induction as ( B1 := A1,

Bn := An \ Bn−1.

By construction, for every k ∈ N it turns out that

+∞ X µ(Ak) ≥ µ(A) − µ(Bn), n=k+1 P and thus, if we can prove that the sum µ(Bn) is ﬁnite, then we can take the limit as n∈N k → +∞ and ﬁnd that a = lim inf µ(Ak) ≥ µ(A). k→+∞ 20 2.3. OUTER MEASURES

The assumption (2.10) implies that Bn and Bn+2 are distant for every n ∈ N; therefore we can decompose the sum as

+∞ +∞ X X X µ(Bn) = µ(B2k) + µ(B2k+1). n∈N k=0 k=0 The measure µ satisﬁes the assumption of Theorem 2.15; hence

+∞ ! X [ µ(B2k) = µ B2k ≤ µ(A). k∈N k=0 In conclusion, the assumption µ(A) < ∞ implies that X µ(Bn) ≤ 2µ(A) < ∞, n∈N which is exactly the estimate we needed.

Lemma 2.17. Let (µn)n∈N be a sequence of outer measures. If µn % µ or µn & µ, then also µ is an outer measure.

Proof. We may assume that (µn)n∈N is an increasing sequence of outer measures converging to some µ, that is, µn % µ. The opposite case is obtained in a similar way.

Step 1. The equality µ(∅) = 0 is trivial. If A ⊂ B, then

µn(A) ≤ µn(B) =⇒ µ(A) := sup µn(A) ≤ sup µn(B) =: µ(B). n∈N n∈N

Step 2. Let {Ak}k∈N be any countable collection of subsets of X. Then ! [ X µn Ak ≤ µn(Ak) ∀ n ∈ N, k∈N k∈N and it follows from Fatou’s lemma that ! ! [ [ µ Ak = lim sup µn Ak ≤ n→+∞ k∈N k∈N X ≤ lim sup µn(Ak) ≤ n→+∞ k∈N X ≤ lim sup µn(Ak) = n→+∞ k∈N X = µ(Ak). k∈N 21 2.4. CARATHEODORY´ CONSTRUCTION

Proposition 2.18. Let X be a compact space, and let µ be a positive ﬁnite measure, deﬁned on the Borel σ-algebra. For every E ⊆ X Borel it turns out that µ(E) = inf {µ(A) | A ⊇ E, A is open} =

= inf {µ(K) | K ⊆ E, K compact} . The ﬁrst identity is usually referred to as outer-regularity of a measure; the second one inner-regularity of a measure.

Sketch of the Proof. First, we notice that the space X is compact, and thus the inner- regularity follows from the outer-regularity by passing to the complement. If we deﬁne the measure ν(E) := inf {µ(A) | A ⊇ E, A is open} , then it is easy to prove that this is an outer measure (thus outer-regular by deﬁnition), which is additive on distant sets. It follows from Theorem 2.15 that ν is a positive measure on the Borel σ-algebra. Finally, since µ(A) = ν(A) for every open subset A, a straightforward application of the Monotone Class Theorem6 allows us to infer that µ ≡ ν on the whole σ-algebra of Borel. Remark 2.11. Let X and Y be spaces satisfying suitable assumption (e.g., metric and locally compact.) If E is a Borel subset of X, and f : E −→ Y is a continuous function, then f(E) is universally measurable7, but it is, in general, not Borel.

2.4 Carath´eodory Construction

Let X be a metric space, let F ⊆ P(X) be a family of subsets, and let ρ : F −→ [0, +∞] be the associated gauge function. We also assume that

(a) ∅ ∈ F; (b) ρ(∅) = 0.

For every positive real number δ ∈ (0, +∞] and every E ∈ P(X), we define ( ) X [ Ψ (E) := inf ρ(F ) {F } ⊂ F, diam(F ) < δ, F ⊇ E , (2.11) δ n n n∈N n n n∈N n∈N where the diameter of a set F is defined by setting diam(F ) := sup {d(x, x0) | x, x0 ∈ F } . It may not be clear at this moment, but it is crucial to take the infimum only over countable covers in the definition (2.11). We assume that

inf ∅ = +∞,

6A monotone class is a collection of sets which is closed under countable monotone union and intersection. 7Deﬁnition. The set f(E) ⊂ Y is universally measurable if and only if, for every ﬁnite positive measure µ on Y , there are two Borel sets A ⊆ f(E) ⊆ B such that µ(A) = µ(B). 22 2.4. CARATHEODORY´ CONSTRUCTION in such a way that the function

(0, +∞] 3 δ 7−→ Ψδ is (weakly) decreasing. In particular, for any E ∈ P(X) we deﬁne

Ψ(E) := lim Ψδ(E) = sup Ψδ(E). (2.12) δ→0+ δ>0 Lemma 2.19.

(a) For any δ > 0, the set function Ψδ is an outer measure and it is additive on sets which are distant more than δ in F, that is, for every A1,A2 ∈ F such that d(A1,A2) > δ, it turns out that Ψδ(A1 ∪ A2) = Ψδ(A1) + Ψδ(A2).

(b) The limit set function Ψ is an outer measure and it is additive on distant set of F, that is, for every A1,A2 ∈ F such that d(A1,A2) > 0, it turns out that

Ψ(A1 ∪ A2) = Ψ(A1) + Ψ(A2).

In particular, the outer measure Ψ satisﬁes the assumption of Theorem 2.15.

Proof.

(a) The reader may prove that Ψδ is an outer measure, as a relatively straightforward check of the characterizing properties; here we only show that it is additive on distant sets.

Let {En}n∈N be an admissible countable cover of A1 ∪ A2, with diam(En) < δ. By assumption d(A1,A2) > δ,

therefore it is immediate to check that, for any n ∈ N, either En ∩A1 6= ∅ or En ∩A2 6= ∅. For i = 1, 2, let us set

Ii := {n ∈ N :: En ∩ Ai 6= ∅} ,

and consider the countable covers E := {En}n∈I1 and F := {En}n∈I2 of A1 and A2 respectively. Then the nontrivial inequality follows easily: X X Ψδ(A1) + Ψδ(A2) ≤ ρ(En) + ρ(En) =

n∈I1 n∈I2

X = ρ(En) = Ψδ(A1 ∪ A2). n∈N

(b) This assertion is an immediate consequence of (a) and Lemma 2.17.

Corollary 2.20. The limit set function Ψ is a σ-additive measure deﬁned on the σ-algebra of Borel. 23 2.5. HAUSDORFF D-DIMENSIONAL MEASURE HD

Example 2.3 (Lebesgue Measure). Let X = Rd, and let us consider the collection of all rectangles, that is, F = {I1 × · · · × Id | Ii ⊂ R interval} . The gauge function associated to this collection is given by

d Y ρ (I1 × · · · × Id) = |Ii| , i=1 where |I| denotes the length of the interval I ⊂ R. For every δ ∈ (0, +∞], it is easy to prove that d Ψδ = Ψ = L , where Ld denotes the d-dimensional Lebesgue measure. Moreover, the limit function coincides with the gauge function on rectangles set, i.e.,

Ψ(I1 × · · · × Id) = ρ (I1 × · · · × Id) .

Exercise 2.3. Let X = Q and F be the family of 1-dimensional rectangles, that is,

F = {I | I ⊂ R interval} . Prove that ρ (I) = |I| =⇒ Ψδ ≡ 0. Hint: the set of all rational numbers Q is totally disconnected countable space, and thus we can take countable covers made up of points, whose length - in the sense of ρ - is zero.)

2.5 Hausdorﬀ d-dimensional Measure Hd

Let X be a metric space, let d ∈ [0, ∞) be any positive real number, let F = P(X) be the family of all subsets, and let ρ (E) = (diam(E))d be the gauge function. The Hausdorﬀ outer measure is deﬁned by formula (2.11), but there is also a multiplicative constant (depending on d only), that is,

d Hδ (E) = cdΨδ(E). The Hausdorﬀ measure is the limit as δ → 0+, that is, it is deﬁned by formula (2.12) up to the same multiplicative constant:

d H (E) = cdΨ(E).

The constant cd, for integer values of d, is deﬁned as follows: α c := d , d 2d where αd is the measure of the d-dimensional unitary ball, that is,

πd/2 αd = d . Γ( 2 + 1) 24 2.5. HAUSDORFF D-DIMENSIONAL MEASURE HD

Lemma 2.21 (Properties of the Hausdorﬀ Measure).

(a) If d = 0, then Hd is the counting measure.

(b) If X = Rn, then d d d n H (λE) = λ H (E) , ∀ λ > 0, ∀ E ⊆ R .

Hd (f(E)) = Hd (E) .

(d) If f : E ⊆ X −→ Y is an L-Lipschitz map, then

Hd (f(E)) ≤ LdHd (E) .

n (e) Let X := R , and let Ψδ be the outer measure associated to the rectangles gauge function. For every δ ∈ (0, +∞] and every E ⊆ Rn, it turns out that d Hδ (E) = Ψδ(E). Moreover, the Hausdorﬀ d-dimensional measure and the Lebesgue d-dimensional measure coincide on Rn, that is, Hd ≡ Ld.

Proof.

(a) Let E be a ﬁnite set with n := |E|. Since E is discrete, there exists a real number > 0 such that any countable cover of E made up by sets of diameter less than , must be of cardinality at least n. On the other hand, there is a trivial covering by n nonempty sets of diameter less than (i.e., small neighborhoods of the n points); hence H0(E) = n. Employing the monotonicity of the outer measures, this proves the claim.

(b) First, we observe that any subset A ⊂ Rn has the following property: for every positive constant λ > 0, it turns out that

diam (λA) = λ · diam(A).

n 8 Let E ⊆ R be given, and let {En}n∈N be an optimal cover for E of diameter δ > 0. It follows easily that {λEn}n∈N is a countable cover for the rescaling λE, of diameter λδ. Hence d X d Hλδ (λE) ≤ cd (diam (λEn)) = n∈N

d X d d d = cdλ (diam (En)) = λ Hδ (E), n∈N and, by taking the limit as δ approaches 0+, we infer that

Hd (λE) ≤ λdHd (E) .

8A cover realizing the inﬁmum in deﬁnition (2.11). 25 2.5. HAUSDORFF D-DIMENSIONAL MEASURE HD

In a similar fashion, let {Fn}n∈N be an optimal cover for λE of diameter δ > 0. It −1 −1 follows easily that {λ En}n∈N is a countable cover for E, of diameter λ δ. Hence d X d Hλ−1δ (E) ≤ cd (diam (En)) = n∈N

−d X d d d = cdλ (diam (λEn)) = λ Hδ (λE), n∈N and, by taking the limit as δ approaches 0+, we infer that 1 Hd (λE) ≥ Hd (E) . λd (c) First, we notice that for every isometry f : X −→ Y and every subset E ⊆ X, it turns out that d(x, x0) = d(f(x), f(x0)) =⇒ diam(E) = diam (f(E)) . The argument shown in (b) applies here without any signiﬁcant change, provided that we take into account the property above.

(d) Let {En}n∈N be a countable cover of E with diameter less than or equal to δ > 0. The map f is L-Lipschitz, and thus {f(En)}n∈N is a countable cover of f(E) with diameter less than or equal to Lδ. Therefore

d X d HLδ (f(E)) ≤ cd (diam (f(En))) ≤ n∈N

d X d d d ≤ cdL (diam (En)) = L Hδ (E), n∈N and the thesis follows by taking the limit as δ approaches 0+. (e) This assertion is rather nontrivial, and we give a sketch of the proof after we introduce the Steiner symmetrization and the isoperimetrical inequality (see Theorem 2.27.)

Lemma 2.22. Let E be a Borel set contained in a d-dimensional surface Σ ⊂ Rn of class C1. Then the d-dimensional Hausdorﬀ measure of E is the d-dimensional volume, that is, d H (E) = vold(E).

Proof. The proof presented here is not rigorous, but the reader may try to expand it, ﬁll the gaps and ﬁx the imprecision as a particularly useful exercise.

Step 1. Fix δ > 0 and cover the surface Σ with a countable collection U := {Un} of n∈N sets such that, for every n ∈ N, there is an almost-isometry ∼ d fn : Un −−→ An ⊆ R , d where An is a ﬂat subset of R . More precisely, there exists δ > 0 such that 0 0 0 (1 − δ) · d (x, x ) ≤ d (fn(x), fn(x )) ≤ (1 + δ) · d (x, x ) , where d is the distance of the metric space X. 26 2.5. HAUSDORFF D-DIMENSIONAL MEASURE HD

Step 2. The set E may be written as the disjoint union of a collection of subsets E := {En} satisfying the inclusion En ⊆ Un for every n ∈ , that is, n∈N N G E = En. n∈N The d-dimensional volume of E is given by the sum of the d-dimensional volumes of the Ens, i.e., X vold(E) = vold(En), n∈N which is equal, taking the preimages via the almost-isometries, to ! X d −1 vold(E) = L fn (An) ∩ En · (1 + O(δ)). n∈N

The function fn is (1 + δ)-Lipschitz; hence ! d X d X d H (E) = H (En) = H (fn(En)) · (1 + O(δ)), n∈N n∈N and this concludes the proof since the right-hand side coincides with vold(E). Remark 2.12. The Hausdorﬀ measure Hd does not change if we replace the family F with the following alternatives:

(1) Closed Sets. The diameter of a set A coincides with the diameter of its closure A.

(2) Open Sets. Given a set A, for every > 0 one can ﬁnd an open set B such that

A ⊆ B and diam(B) ≤ diam(A) + .

(3) Convex Sets. In the particular case X = Rn, the reader may prove that the diameter of the convex hull of A is equal to the diameter of A.

On the other hand, one cannot replace the family F with, e.g., the family of balls9.

Hausdorff Spherical Measure. Let X be a metric space, let d ∈ [0, ∞) be any positive real number, let Fs be the family of all the balls, and let ρ (E) = (diam(E))d be the gauge function. The spherical Hausdorff outer measure is defined by formula (2.11), but there is also a multiplicative constant (depending on d only), that is,

d Hδ, s(E) = cdΨδ, s(E). The spherical Hausdorﬀ measure is the limit as δ → 0+, that is, it is deﬁned by formula (2.12) up to the same multiplicative constant:

d Hs (E) = cdΨ(E).

9The reader may prove this assertion formally, but the rough idea behind it is evident: in general, a r-diameter set is not contained in a r-diameter ball (see, e.g., an equilateral triangle). 27 2.5. HAUSDORFF D-DIMENSIONAL MEASURE HD

Lemma 2.23. There is a constant C > 0 such that

d d d H (E) ≤ Hs (E) ≤ C ·H (E), ∀ E ⊆ X.

Proof. Let E ⊆ X be a subset of X, and let {En}n∈N be a countable covering of E with sets whose diameter is strictly less than δ.

Step 1. Let xn ∈ En be a sequence of points, and let us consider the family of balls deﬁned by Bn := B (xn, diam(En)) . By construction, one can easily check that

En ⊂ Bn and diam (Bn) = 2diam(En), and thus {Bn}n∈N is a covering of E, made up of balls whose diameter does not exceed 2δ.

Step 2. A straightforward application of the deﬁnitions proves the inequality

d X d d X d H2δ, s(E) ≤ cd (diam(Bn)) = 2 cd (diam(En)) . n≥0 n≥0

d If we take the inﬁmum over all such families {En}n∈N, then we get the thesis with C = 2 , that is, d d d d H (E) ≤ Hs (E) ≤ 2 ·H (E), ∀ E ⊆ X. One easily notices that 2d is not the sharp constant, and it can be improved up to

d 2n 2 C(d) := , n + 1 but we will not furnish a proof of this result in the course.

Lemma 2.24. Let E be a Borel set contained in a d-dimensional surface Σ ⊂ Rn of class C1. Then d d H (E) = vold(E) = Hs (E).

Proof. The argument is along the lines of the one used in Lemma 2.22. Lemma 2.25. Let d < d0 be two real numbers, and let E ⊆ X be any subset.

0 (1) If Hd(E) is ﬁnite, then Hd (E) = 0.

0 (2) If Hd (E) > 0, then Hd(E) = +∞.

In particular, for a ﬁxed set E, the function d 7−→ Hd(E) is decreasing, and it attains a ﬁnite nonzero value at most once.

Proof. 28 2.5. HAUSDORFF D-DIMENSIONAL MEASURE HD

(1) Let {En}n∈N be a covering of E, whose diameter does not exceed a ﬁxed δ > 0. By deﬁnition, it turns out that

d0 X d0 d0−d X d Hδ (E) ≤ (diam(En)) ≤ δ · (diam(En)) , (2.13) n≥0 n≥0

which, in turn, implies that

d0 d0−d d Hδ (E) ≤ δ ·Hδ (E).

In conclusion, we notice that d0 − d is strictly greater than 0, and hence the thesis follows by taking the limit as δ → 0+. (2) The inequality (2.13) may be rewritten as

d d−d0 d0 Hδ (E) ≥ δ ·Hδ (E),

and thus we conclude as in (1) since d − d0 is strictly less than 0.

Definition 2.26 (Hausdorff Dimension). Let E ⊆ X. The Hausdorff dimension of E, denoted by dimH E, is the unique real number such that

 d H (E) = 0 d > dimH E

d H (E) = +∞ d < dimH E.

Remark 2.13 (Basic Properties).

d (a) If d = dimH E is the Hausdorff dimension of E, then H (E) might be either zero or infinite (i.e., it is not necessarily finite). (b) The Hausdorff dimension of a countable union is equal to the supremum of the Haus- dorff dimensions, that is, [ dimH En = sup {dimH En} . n∈N n∈N

(c) Let (En)n∈N ⊂ P(R) and assume that dimH En % d. The dimension of the union is equal to d, that is, [ dimH En = d, n∈N and the Hausdorﬀ measure of the union is zero: ! d [ H En = 0. n∈N Exercise 2.4. Let E ⊆ X. The following properties are equivalent:

(a) The Hausdorﬀ measure of E is zero, i.e. Hd(E) = 0. 29 2.5. HAUSDORFF D-DIMENSIONAL MEASURE HD

(b) For every > 0 there is a countable cover {En}n∈N, whose diameters satisfy the inequality X d (diam (En)) ≤ . n≥0

d (c) The ∞-Hausdorﬀ measure of E is zero, i.e. H∞(E) = 0.

Theorem 2.27 (Hd ≡ Ld). Let X := Rn. For every δ > 0 and every E ⊆ X, it turns out that d d d H (E) = Hδ (E) = L (E).

Before we can give the proof of this result, we need to introduce two fundamental tools (the isoperimetrical property of balls and the Steiner symmetrization), which are also extremely important per se. Theorem 2.28 (Isoperimetrical Property of Balls). Let B(r) be the closed ball centered at the origin of radius r in Rd. Among all closed sets with given diameter in Rd, the ball B(r) has maximum Lebesgue measure, that is,

d d d cd (diam(Br)) = L (B(r)) ≥ L (E), ∀ E ⊆ X : diam(E) = 2r.

Sketch. The proof of the isoperimetrical property is a direct consequence of the Steiner symmetrization (the reader may consult [5, pp 195–198] for a rigorous argument.)

Symmetrization Construction. Let E ⊆ Rd be a bounded closed subset and let V be an aﬃne hyperplane in Rd. For every point x ∈ V , we may consider the 1-dimensional subspace ex, orthogonal to V , and replace the intersection ex ∩ E by a closed segment, 10 centered at x0, with the same length . From now on, we denote by Ee the symmetrization of E.

Symmetrization Properties. Let E ⊆ Rd be a bounded closed subset and let V be an aﬃne hyperplane in Rd. The reader may prove that the Lebesgue measure does not change, that is, Ld(E) = Ld Ee , while the diameter decreases, that is, diam(E) ≥ diam Ee .

Moreover, one could easily prove that the set Ee is closed.

10Hausdorﬀ measure. 30 2.5. HAUSDORFF D-DIMENSIONAL MEASURE HD

Proof (d = 2). In this particular case the proof is fairly straightforward (it suﬃces to look at Figure 2.1), but the idea for d > 2 is exactly the same.

2 Let E be any subset of R , and consider an orthogonal basis {e1, e2} of the real plane. If we denote by Ee the Steiner symmetrization of E with respect to e1 ﬁrst, and e2 after, then it is easy to prove that Ee symmetric with respect to the origin, and it is thus contained in a ball of diameter diam Ee .

Figure 2.1: Idea of the proof in dimension 2

Proof of Theorem 2.27. Recall that the normalization constant of the Hausdorﬀ measure is given by α c = d , d 2d d and thus, for every ball Bx, r := B(x, r) ⊆ R , it turns out that

d vold(Bx, r) = cd (diam(Bx, r)) .

Step 1. First, we notice that it suﬃces to prove that

d d d H∞(E) ≥ L (E), ∀ E ⊆ R , in order to obtain the ﬁrst inequality, that is,

d d d H (E) ≥ L (E), ∀ E ⊆ R .

Let {En}n∈N be a countable cover of E. It follows from the isoperimetrical property of balls (see Theorem 2.28) that d d cd (diam(En)) ≥ L (En), 31 2.6. HAUSDORFF DIMENSION OF CANTOR-TYPE SETS and thus X d X d d cd (diam(En)) = cd (diam(En)) ≥ L (E). n∈N n∈N By taking the inﬁmum of the left-hand side, we conclude that

d d H∞(E) ≥ L (E).

Step 2. The opposite inequality follows if we are able to prove that

d d d Hδ (E) ≤ L (E), ∀ E ⊆ R , ∀ δ > 0. 11 Let {En} be a countable optimal cover for computing the Lebesgue measure of E. The optimal cover cannot be made up of cubes or rectangles12, but by Vitali’s Covering Lemma 3.1 it follows that it can be chosen among all covers made up of (closed) balls. Moreover, for every ﬁxed > 0, there exists a collection of balls {Bn}n∈N such that

d X d X d d + L (E) ≥ L (Bn) = cd (diam(Bn)) ≥ Hδ (E), n∈N n∈N and this concludes the proof by arbitrariness of δ > 0.

2.6 Hausdorﬀ Dimension of Cantor-Type Sets

In this brief section, we compute the Hausdorff dimension of the standard Cantor set C, and we generalize the process to more elaborate sets. Lemma 2.29. Let C be the standard Cantor set. Its Hausdorff dimension is given by ln 2 d := dim (C) = , H ln 3 and the Hausdorff measure is finite, that is,

0 < Hd (C) < +∞.

Proof. In this proof for simplicity purposes we assume that the renormalization constant cd is equal to 1, and we prove that the Hausdorﬀ measure of C is bounded as follows: 1 ≤ Hd(C) ≤ 1. 2

Upper bound. The bound from above is, as usual, relatively easy to prove: it is enough to ﬁnd a cover for the Cantor set C satisfying the bound. By deﬁnition

∞  2k  \ [ C =  Ik, i , k=0 i=1

11In particular, the cover does not need to be optimal for computing the Hausdorﬀ measure. 12The reader may prove, as an exercise, that covers made up of cubes or rectangles yield to the sought inequality up to a constant strictly bigger than one. 32 2.6. HAUSDORFF DIMENSION OF CANTOR-TYPE SETS where the diameter of each interval is given by

−k |Ik, i| = 3 .

Fix δ > 0, and consider the minimal integer k ∈ N such that 3−k < δ. Clearly

2k [ C ⊂ Ik, i, i=1 and this implies, by σ-subadditivity, that

2k d X d 1 H (C) ≤ H (I ) ≤ 2k = 1. δ δ k, i 3dk i=1

Lower bound. First, we notice that

d d Hδ (C) ≥ H∞ (C) , and thus it is enough to prove the lower bound for the ∞ measure, that is,

? 1 Hd (C) ≥ Hd (C) ≥ . δ ∞ 2

Let {En}n∈N be a countable cover of C, and assume that each En is open and convex. Moreover, by compactness, we may assume that there are only ﬁnitely many. Fix n ∈ N and take the smallest interval Ik, i := In that contains En. The reader may prove by herself that |In| ≤ 3 |En| , 0 00 13 as a consequence of the fact that, if In splits into In and In , then En must intersect both (otherwise In would not be minimal, as required.) As a consequence of our claim, it turns out that X d −d X d (diam (En)) ≥ 3 (diam (In)) = n∈N n∈N

1 = 3−d = . 2 The equality is left as an exercise for the reader. The rough idea behind it is to throw away the repeated intervals In in the previous sum; it is not hard to prove that, in this way, we end up with ﬁnitely many. Remark 2.14. Let C be a Cantor set and let d be its Hausdorﬀ dimension. The lower bound is not sharp, and one could prove that

Hd(C) = 1.

13 0 00 N.B. There could be points of En lying between In and In , but, since they do not belong to the Cantor set, we can ignore them (with some care). 33 2.6. HAUSDORFF DIMENSION OF CANTOR-TYPE SETS

Exercise 2.5 (Cantor-Type Set). Fix 0 < λ < 1, and let us consider the following partition of the unitary interval:

λ λ λ λ I = 0, t , 1 − t 1 − , 1 . λ 2 2 2 2

Let Cλ be the generalized Cantor set, that is

∞  2k  \ [ (λ) Cλ =  Ik, i  , k=0 i=1

(λ) λ k where |Ik, i | = 2 . Prove that the Hausdorﬀ dimension of Cλ is given by the solution of the equation λdλ 2 = 1, 2 and, actually, it turns out that dλ 0 < H (Cλ) ≤ 1.

Remark 2.15. The Hausdorﬀ dimension of Cλ is explicitly given by log 2 d = , λ log 2 − log λ and thus we get any Hausdorﬀ dimension in (0, 1) as λ ranges in [0, 1]. Part II

Covering Results

34 Chapter 3

Covering Theorem

In this chapter, we investigate several covering theorems, that is, the possibility, given a family F of balls covering E, to extract a subfamily F 0 which covers E and behaves ”better.” More precisely, let X be a metric space and µ a measure deﬁne on B(X); we would like to extract a subfamily F 0 satisfying one of the following properties:

(a) The collection F 0 is a disjoint covering of E. It is, clearly, the best possibility we could be hoping for, but, unfortunately, it is false1 even in R2. (b) The collection F 0 is a covering of E made up of balls that do not overlap ”too much” (i.e., the measure of the overlapping portion is arbitrarily small.) (c) The collection F 0 is a disjoint covering of µ-almost all of E, that is, the portion of E that is not covered has measure zero.

(d) The collection F 0 is a covering of E satisfying the inequality X µ(B) ≤ µ(E) + . B∈F 0

In this chapter, we are mainly concerned with two classes of covering theorems, depending on the ambient space:

(1) Covering theorems that works on every metric space X, provided that the measure µ satisﬁes some extra assumptions.

(2) Covering theorems that works only on X := Rn, with no requirement on the measure µ.

1 2 2 Exercise. Prove that the unitary square [0, 1] in R cannot be covered by disjoint balls. 36 3.1. VITALI COVERING THEOREM

3.1 Vitali Covering Theorem

In this section, to ease the notation we denote by B(x, r) the closed ball of center x and radius r, and by Bb(x, r) the closed ball of center x and radius 5r.

Lemma 3.1 (Vitali Covering Lemma). Let X be a metric space, let F := {B(xi, ri)}i∈I be a family of closed balls with uniformly bounded radii that covers a set E ⊆ X. Then there exists a disjoint subfamily F 0 such that the rescaling

0 0 Fb := {Bb(xi, ri) | B(xi, ri) ∈ F } is a covering of E. Remark 3.1. The uniform bound on the radii of the family F is fundamental, and it is impossible to drop it. Indeed, there is a simple counterexample in the real plane R2. The family F := {B(n, n) | n ∈ N} , is an increasing covering of the upper-half plane E. Therefore, any disjoint subfamily is necessarily formed by a single ball, and hence it cannot cover E for any n ∈ N (see Figure 3.1.)

Figure 3.1: Necessity of the uniform boundedness on radii

Proof. We ﬁrst prove the result by assuming some additional properties, and then we reduce the general case to it with a simple trick.

Step 1. Assume that the family F = {B(xn, rn)}n∈N is a countable cover of E, and assume also that the sequence of the radii is weakly decreasing, i.e. r1 ≥ r2 ≥ ... . To construct the disjoint subfamily, we proceed as follows: 37 3.1. VITALI COVERING THEOREM

0 (1) The ﬁrst ball B1 automatically belongs to F .

(2) The second ball B2 either intersect B1 or it does not. If the latter holds true, then we 0 add B2 to the family F ; if the former alternative holds true, we solely throw it away. (3) Iterate the process for every n ≥ 3 - taking into account all of the intersections with the balls that were previously added to the subfamily F 0.

At this point, we only need to prove that the rescaled collection of balls Fb0 is a covering of 0 E. Clearly, we may equivalently show that every ball Bn is a subset of an element in Fb . In particular, for every n ∈ N there are only two possible outcomes:

0 (i) The ball Bn already belongs to F , and thus there is nothing to prove.

(ii) The ball Bn intersects the ball Bm for some m < n. The distance between the centers xn and xm is less than or equal to the sum of the radii, which means that

d(xm, xn) ≤ rm + rn ≤ 2rm =⇒ B(xn, rn) ⊂ B(xm, 3rm) ⊂ Bbm.

Step 2. In the general case, we cannot assume that the family is countable or that the radii give an increasing sequence, and thus everything depends on the boundedness assumption. More precisely, there exists a real number R > 0 such that

B(xi, ri) ∈ F =⇒ ri ≤ R.

Let us consider the following partition R R Fn := B ∈ F < r ≤ , ∀ n ∈ N, 2n+1 2n in such a way that we can extract a subfamily from each Fn as follows:

0 (i) Extract a maximal (with respect to the inclusion) disjoint subfamily F0 from F0. 0 (ii) Extract a maximal disjoint subfamily F1 from F1, satisfying the additional require- 0 ment that it does not intersect any element of F0. (iii) Iterate the process for n ≥ 3 - taking into account all of the intersections with the subfamilies already chosen in the previous steps.

0 0 0 Let F = ∪n∈NFn: we need to show that the rescaled family Fb is a covering of E. Clearly, 0 we may equivalently show that every ball Bn is a subset of an element in Fb . In particular, for every B(x, r) ∈ FN there are only two possible outcomes:

0 (i) The ball B(x, r) already belongs to FM for some M ∈ N, and thus there is nothing to prove.

0 (ii) The ball B(x, r) intersects the ball B(y, s), with B(y, s) ∈ FM for some M ≤ N. It follows that r ≤ 2s, and therefore the distance between the centers is less than or equal to the sum of the radii, which means that

d(x, y) ≤ r + s ≤ 3s =⇒ B(x, r) ⊆ Bb(y, s). 38 3.1. VITALI COVERING THEOREM

Theorem 3.2 (Vitali Covering Theorem). Let X be a metric space, let µ be a doubling2 locally ﬁnite measure3, and let E ⊆ X be a Borel set. Then, for every > 0 and every F ﬁne cover4. of E made up of closed balls, there exists a disjoint subfamily F 0 ⊂ F which covers µ-almost all of E, that is, ! [ µ E \ B = 0, B∈F 0 and the mass does not exceed µ(E) ”too much”, that is, X µ(B) ≤ µ(E) + . (3.1) B∈F 0 Remark 3.2. The statement of the Vitali covering theorem makes sense, but there is a topological issue which may not be apparent: The disjoint subfamily needs to be countable, and thus we need some additional assumptions on X (locally compact, separable, etc.)

Figure 3.2: Idea of the proof

Proof. Assume, without loss of generality, that µ(E) < ∞, and ﬁx an open neighborhood A0 of E satisfying the additional property µ(E) µ (A \ E) ≤ ∧ , (3.2) 0 2C3

2Deﬁnition. A measure µ is doubling if and only if there exists a positive constant C > 0 such that µ (B(x, 2r)) ≤ C µ (B(x, r)) .

3Definition. A measure µ is locally finite if and only if for every x ∈ X there exists a neighborhood Ux 3 x such that µ(Ux) < +∞. 4Definition. Let F be a family of closed balls that covers E. We say that F is fine if and only if each x ∈ E is the center of a closed ball in F with arbitrarily small radius. 39 3.1. VITALI COVERING THEOREM where C is the doubling constant.

Step 1. Let us consider the collection of balls

F0 := {B(x, r) ∈ F | B(x, r) ⊆ A0 } . By construction, this is still a cover of E, and thus, by Lemma 3.1, it follows that there 0 0 exists a disjoint subfamily of balls F0 such that the rescaling Fb0 covers E.

Step 2. We apply the doubling property of µ three times (since 22 < 5 < 23), and it is 0 easy to see that we can ﬁnd a lower bound for the measure of F0:

X 3 X X µ(E) µ(E) ≤ µ Bb ≤ C µ(B) =⇒ µ(B) ≥ 3 . 0 0 0 C B∈F0 B∈F0 B∈F0 In a similar fashion5, we obtain the following estimate6:

3 X 3 [ 0 [ 0 C µ(B) ≤ C µ F0 ∩ E + µ F0 \ E ≤ 0 B∈F0

[ µ(E) ≤ C3µ F 0 ∩ E + , 0 2 from which it follows that µ(E) [ ≤ µ F 0 ∩ E . 2 C3 0 As an immediate consequence, the measure of the portion of E that has not been covered 0 yet by F0 may be estimated as follows: [ 1 µ E \ F 0 ≤ 1 − µ(E). (3.3) 0 2C3

00 S 00 Step 3. Let us take a suitable ﬁnite subfamily F0 , and let E1 := E\ F0 . The inequality (3.3) proves that 1 µ (E ) ≤ 1 − µ(E). 1 3C3

The process can now be iterated in the following way: Let A1 be an open neighborhood 7 S 00 c of E1 satisfying the additional requirement that E1 ⊂ A1 ⊂ ( F0 ) , and notice that everything works out smoothly as in the previous steps. In particular, it turns out that

1 k µ (E ) ≤ 1 − µ(E), k 3C3

5Hint. Decompose E as the disjoint union of the interior part (∩) and the exterior part (\). Then, apply the subadditivity of the measure and the assumption (3.2) on the measure of the open setA0. 6Notation. The union of all the elements in a collection is denoted by [ [ F := B. B∈F

7 00 Here we use the ﬁniteness of the subfamily F0 to ensure that the complement is an open set. 40 3.1. VITALI COVERING THEOREM and thus the inevitable candidate is given by

0 [ 00 F := Fn . n∈N The reader may check, as an exercise, that the following properties hold (and conclude the proof):

(1) The family F 0 is disjoint. (2) The family F 0 is a covering of µ-almost all of E. Hint: prove that

[ 00 \ E \ Fn = En. n∈N n∈N (3) The measure is arbitrarily near to the measure of E, that is, X µ(B) ≤ µ(E) + . B∈F 0

Lemma 3.3. Let X be a metric space, and let µ be a doubling locally finite measure. Let E ⊆ X be a Borel subset, and let F be a fine cover of E. Then there exist a finite disjoint subfamily F 00 and a positive real number δ > 0 such that [ µ E \ F 00 ≤ δ · µ(E).

Proof. The statement follows easily from the proof of the Vitali’s Covering Theorem 3.2. Lemma 3.4. Let X be a metric space, and let µ be a doubling locally ﬁnite measure. Let E0 ⊂ X be a Borel null-set, let > 0 be a positive real number, and let F be a ﬁne cover of 00 E. Then there exist a subfamily F , which covers E0, such that X µ(B) ≤ . B∈F 00

Proof. Let A0 be an open neighborhood of E0 satisfying the inequality µ(A ) ≤ . 0 C3 Let us consider the family of balls n o

G := Bi := B(xi, ri) Bi ⊂ A0 and Bbi ∈ F .

The reader may check, as an exercise, that G is a ﬁne cover of E0. By Vitali’s Covering 0 0 Lemma 3.1 we can ﬁnd a disjoint subfamily G such that Gb covers E0; hence, we set F 00 := Gb0. 00 By construction F ⊂ F, and it is also a covering for E0 such that

X 3 X 3 µ Bb ≤ C µ(B) ≤ C µ(A0) ≤ , B∈G0 B∈G0 which is exactly what we wanted to prove. 41 3.2. BESICOVITCH COVERING THEOREM

Remark 3.3. The cover F need not be fine in the statement of Lemma 3.4, but, since this is the only result in this section whose assumptions can be weakened, we will just ignore it. Theorem 3.5. Let X be a metric space, and let µ be a doubling locally finite measure. Let E ⊆ X be a Borel subset, let > 0 be a positive real number, and let F be a fine cover of E. Then there exists a subfamily F 0, which covers E, such that X µ(B) ≤ µ(E) + , B∈F 0 that is we can cover E completely with balls, whose measure does not exceed too much µ(E), by dropping the disjoint requirement.

Proof. First, we notice that the Vitali’s Covering Theorem 3.2 allows us to restrict out attention to null-set (i.e., µ(E) = 0.) Then, we apply Lemma 3.4 to conclude.

3.2 Besicovitch Covering Theorem

In this section, we investigate a diﬀerent type of covering theorem, which is, actually, very similar to the Vitali’s result. The originality of the Besicovitch covering theorem is that it works without further assumption on the measure µ, provided that X is the Euclidean space Rn. Lemma 3.6 (Besicovitch Covering Lemma). Let X := Rn, and let

F := {B(xi, ri)}i∈I be a family of closed balls with uniformly bounded radii, which is also a ﬁne cover of a Borel set E ⊂ Rn. Then there exist a natural number N := N(n), which depends only on the dimension n, and F1,..., FN+1 ⊂ F disjoint subfamilies such that

N+1 [ Fi ⊇ E. i=1

Before we work out the details of the proof of this result, we need to state a technical lemma that allows us to infer easily that the natural number N(n) depends solely on the dimension of the ambient space.

Lemma 3.7. There exists a constant N := N(n) ∈ N, depending only on the dimension n of the ambient space, such that for any ﬁnite collection of closed balls B0,...,Bp ⊂ R satisfying

(a) B0 ∩ Bi 6= ∅ for every i = 1, . . . , p;

(b) the 0th ball has the smaller radius, that is r0 ≤ ri, for any i = 1, . . . , p;

(c) the center xi of the ball Bi does not belong to the ball Bj for every i 6= j ∈ {1, . . . , p}; it turns out that p ≤ N(n). Moreover, the conclusion does not change if we replace the assumption (b) with a slightly diﬀerent one, that is 42 3.2. BESICOVITCH COVERING THEOREM

0 (b) r0 ≤ 2 ri, for any i = 1, . . . , p.

Proof. See [10, pp. 99–101].

Proof of Lemma 3.6. We ﬁrst prove the result by assuming some additional properties, and then we reduce the general case to it with a simple trick.

Step 1. Assume that the family F = {B(xn, rn)}n∈N is a countable ﬁne cover of E, and assume also that the sequence of the radii is weakly decreasing, i.e. r1 ≥ r2 ≥ ... . To construct the disjoint subfamilies, we proceed as follows:

(1) Select any index j ∈ {1,...,N + 1} and add the ball B(x0, r0) to the subfamily Fj.

(2) Assume that n − 1 balls have already been placed in F1,..., FN+1, or thrown away. Let {i1, . . . , ik} ⊂ {1, . . . , n − 1} be the subset of the indices of the kept balls. (3) We consider the n-th ball, and we notice that there are only two possible outcomes: Sk (i) If xn belongs to j=1 Bij , then we throw the ball Bn away. Sk (ii) If xn does not belong to j=1 Bij , then there is an index i ∈ {1,...,N +1} such that Bn can be added to Fi, and that subfamily is still disjoint. We argue by contradiction. If such an index i does not exist, then, for any j ∈ {1,...,N + 1}, there exists a ball Bj ∈ Fj such that Bj ∩ Bn 6= ∅. By Lemma 3.7 we immediately derive the contradiction N + 1 ≤ N, which is exactly what we needed to conclude the ﬁrst part of the proof.

B(xi, ri) ∈ F =⇒ ri ≤ R. (3.4)

Let us put a well order on F := {Bα}α<ω for some ordinal ω satisfying the additional property 0 α < α =⇒ rα < 2 rα0 , which is always possible as a result of (3.4). The construction is no diﬀerent than the previous one, but we do need to use the transﬁnite induction. In particular, we proceed as follows:

(i) Select any index j ∈ {1,...,N + 1} and add the ball B(x0, r0) to the subfamily Fj.

0 (ii) Assume that α balls have already been placed in F1,..., FN+1, or thrown away. Let τ ⊂ α0 be the subset of the indices of the kept balls. (iii) Let α > α0. There are two possible outcomes: S (i) If xα belongs to j∈τ Bj, then we throw the ball Bα away. S (ii) If xα does not belong to j∈τ Bj, then there is an index i ∈ {1,...,N + 1} such that Bα can be added to Fi, and that subfamily is still disjoint. 43 3.2. BESICOVITCH COVERING THEOREM

Lemma 3.8. Let X = Rn, and let µ be a locally finite measure. Let E ⊆ X be a Borel subset, and let F be a fine cover of E. Then there exist a finite disjoint subfamily F 00 and a positive real number δ > 0 such that [ µ E \ F 00 ≤ δ · µ(E).

Proof. By Besicovitch Covering Lemma 3.6 there are N := N(n) disjoint subfamilies F1,..., FN ⊂ F, whose union covers E, that is N [ E ⊂ Fi. i=1 For every i ∈ {1,...,N} we set [ Ei := E \ Fi, and the reader may readily check that there exists an index i ∈ {1,...,N} such that µ(E) µ(E ) ≤ . i N

The thesis follows immediately by taking to a proper finite subfamily of Fi. Remark 3.4. It is not necessary for F to be a fine cover, but it is enough to have each point of E be the center of such a ball. n Lemma 3.9. Let X := R , and let µ be a locally finite measure. Let E0 ⊂ X be a Borel null-set, let > 0 be any positive number, and let F be a fine cover of E. Then there exist 00 a subfamily F , which covers E0, such that X µ(B) ≤ . B∈F 00

Proof. Let A0 be an open neighborhood of E0 satisfying the inequality µ(A ) ≤ . 0 N Let us consider the family of balls

G := {Bi := B(xi, ri) | Bi ⊆ A0 } .

The reader may check, as an exercise, that G is a ﬁne cover of E0. By Besicovitch Covering Lemma 3.6 we can ﬁnd disjoint subfamilies G1,..., GN covering E0. Set

N 00 [ F := Gi, i=1 and notice that it is also a covering of E0 satisfying the additional property

N " # X X X µ(B) = µ(B) ≤ N · µ(A0) ≤ . 00 B∈F i=1 B∈Gi

Indeed, the sum of the measures of the balls inside any subfamily Gi (as a consequence of the disjointness) needs to be equal or less than the measure of A0. 44 3.2. BESICOVITCH COVERING THEOREM

Theorem 3.10 (Besicovitch Covering Theorem). Let X := Rn and let µ be a locally ﬁnite measure. Let E ⊆ X be a Borel subset, let > 0 be any positive real number and let F be a ﬁne cover of E. Then there exists a disjoint subfamily F 0, which covers µ-almost all of E, satisfying the inequality X µ(B) ≤ µ(E) + . B∈F 0

Proof. This is a direct consequence8 of the Besicovitch Covering Lemma 3.6 and Lemma 3.8.

8The argument is very similar to the one used in Vitali’s lemma. The reader may try to ﬁll-in the details as an exercise. Part III

Geometric Measure Theory

45 Chapter 4

Density of Measures

In this brief chapter, we investigate the notion of density associated with a measure µ. In particular, we ﬁrst study the upper density related to the d-dimensional Hausdorﬀ measure, and then we show that some of its essential properties may be generalized to other measures satisfying certain assumptions.

4.1 Density of Doubling Locally Finite Measures

In this section, we prove that the density of a doubling1 locally finite measure µ is a well- defined quantity. Definition 4.1 (Density). Let E ⊆ X be a Borel subset of a metric space. The density of a measure µ of the set E is given, at the point x ∈ X, by the following limit: µ (E ∩ B(x, r)) Θ(µ, E, x) := lim . r→0 µ (B(x, r)) Theorem 4.2. Let µ be a (doubling) locally finite measure defined on a metric space X, and let E ⊆ X be a Borel subset. Then the density is either 1 or 0, that is, ( µ (E ∩ B(x, r)) 1 for µ-almost every x ∈ E, Θ(µ, E, x) := lim = r→0 µ (B(x, r)) 0 for µ-almost every x∈ / E. Remark 4.1. The theorem holds true even if µ is not a doubling measure. Indeed, one can require µ to be asymptotically doubling, that is µ (B(x, 2r)) lim < +∞ r→0 µ (B(x, r)) for µ-almost every x ∈ X.

Proof. Here we only prove that the density is 1 for µ-almost every x ∈ E. The reader may either prove the other case with a similar argument or obtain it automatically by taking the complement.

1 n This assumption is necessary if X is a general metric space, but we can drop it if X = R . 47 4.2. UPPER DENSITY OF THE HAUSDORFF MEASURE

Step 1. Fix λ ∈ (0, 1), and let us consider the set µ (E ∩ B(x, r)) Eλ := x ∈ E lim inf < λ . r→0 µ (B(x, r))

2 It is easy to see that we may equivalently prove that measure of Eλ is zero for any λ, that is µ(Eλ) = 0, ∀ λ ∈ (0, 1).

Step 2. Let F be the family of all the closed balls B(x, r) with center x ∈ Eλ and radius satisfying the inequality µ (E ∩ B(x, r)) ≤ λ, µ (B(x, r)) which is possible because the inferior limit of the ratio is estimated by λ in Eλ. Then F is a ﬁne cover of Eλ, and thus by Vitali’s Covering Theorem 3.2 there exists a disjoint subfamily 0 F , covering µ-almost all of Eλ, such that X µ(B) ≤ µ(Eλ) + . B∈F 0

An easy estimate for the measure of Eλ follows: X X µ(Eλ) ≤ µ (E ∩ B) ≤ λ µ (B) ≤ λ(µ(Eλ) + ). B∈F 0 B∈F 0

In conclusion, if we take the limit as → 0+, then we obtain the estimate

µ(Eλ) ≤ λµ(Eλ), and this the sought contradiction since we assumed λ to be strictly less than 1.

4.2 Upper Density of the Hausdorﬀ Measure

In this section, we focus our efforts on the upper density of the d-dimensional Hausdorff measure since we can find a lower bound and an upper bound explicitly.

Deﬁnition 4.3 (Upper Density). Let E be a Borel subset of a metric space X. The upper d-dimensional density (with respect to Hd) of E at the point x is deﬁned by setting

d ∗ H (E ∩ B(x, r)) Θd(E, x) := lim sup d , (4.1) r→0+ αdr

−d where αd2 is the renormalization constant introduced the deﬁnition of the Hausdorﬀ measure, and B(x, r) denotes the closed ball of center x and radius r.

2 Here, we would need to check carefully that the set Eλ is actually Borel! The intuitive idea behind this goes as follows: The density ratio is increasing with respect to r, and thus it is a Borel function. The inferior limit for r → 0, on the other hand, can be computed using rational sequences for the radii; thus it is Borel, and so is the set Eλ. The reader may try to ﬁll in the details here as an exercise. 48 4.2. UPPER DENSITY OF THE HAUSDORFF MEASURE

Theorem 4.4. Let E be a Borel subset of a metric space X, and assume that E is locally Hd-ﬁnite. Then the following properties hold true:

(a) The upper density of external points (outer density) is almost everywhere zero, that is, ∗ d Θd(E, x) = 0, for H -almost every x∈ / E. (b) The upper density is bounded from below, that is, 1 ≤ Θ∗(E, x), for Hd-almost every x ∈ E. 2d d

(c) If X =∼ Rn, then the upper density is bounded from above by 1, that is, ∗ d Θd(E, x) ≤ 1, for H -almost every x ∈ E. (d) If X is a generic metric space, then the upper density is bounded from above by a slightly different constant, that is, ∗ d d Θd(E, x) ≤ 5 , for H -almost every x ∈ E. Remark 4.2. The reason we are only concerned with upper density is the following: There exists E ⊆ R with finite and positive Hausdorff measure such that the lower density is 0 for Hd-almost every x ∈ E. On the other hand, there is a dual concept of the Hausdorff dimension, called the packing dimension. Similar statements hold for this kind of measure, but the lower density replaces the upper one (see, e.g., [8, pp. 81–86].)

Proof.

(a) Fix m > 0 and let us consider the set ∗ Em := {x∈ / E | Θd(E, x) > m} . The thesis follows easily if we can prove that the d-dimensional Hausdorff measure of Em is equal to zero for any fixed m. Let µ := Hd ¬ E the restriction to E of the d-dimensional Hausdorff measure, and notice that µ is locally finite by construction. Let A be an open neighborhood of Em, and let us consider the family of balls n o d d F = B(x, r) B(x, r) ⊂ A, x ∈ Em and H E ∩ B(x, r) > m · αdr .

The reader may easily prove that F is a ﬁne cover of Em. Thus, by Vitali’s Covering Lemma 3.1 it follows that there exists a disjoint subfamily F 0 ⊂ F such that Fb0 is also a covering of Em. A straightforward computation proves that

X X d µ(A) ≥ µ(B) ≥ m · αd r(B) = B∈F 0 B∈F 0

d m αd X = diam(Bb) ≥ 5d 2d B∈F 0

m ≥ Hd (E ), 5d ∞ m 49 4.2. UPPER DENSITY OF THE HAUSDORFF MEASURE

since Em ⊆ ∪B∈F 0 Bb. By deﬁnition, the set Em is disjoint from E, and thus µ(Em) = 0 (as µ is the restriction to E), and, by regularity (of the measure µ), we can always choose A in such a way that µ(A) is as small as we want. It turns out that m ≥ Hd (E ), ∀ > 0 =⇒ Hd(E ) = 0 5d ∞ m m d d since H (E) = 0 is equivalent to H∞(E) = 0. (b) We divide the proof into three steps: we prove it in a very particular case, and then we reduce it to the general case using a simple trick. Fix m < 2−d and let us consider the set

∗ Em := {x ∈ E | Θd(E, x) < m} .

Let µ := Hd ¬ E the restriction to E of the d-dimensional Hausdorﬀ measure, and notice that µ is locally ﬁnite by construction. By assumption, for every x ∈ Em there is a real number r0(x) := r0 > 0 such that

d d H E ∩ B(x, r0) < m · αdr0 . (4.2)

Step 1. Assume that3 the inequality above holds for every r > 0, uniformly with respect to x. Let F := {Fi | i ∈ I } be a covering of Em made up of balls, and consider 0 the family of balls F := {Bi | i ∈ I } deﬁned in the following way:

1) The radius of Bi is equal to the radius of Fi for any i ∈ I.

2) The center of Bi is a point of Em for any i ∈ I. 0 3) The collection of balls F covers Em.

We can easily estimate the Hausdorﬀ measure of Em as follows:

X X d 1 X 2dHd(E ) ≥ α µ(F ) ≥ α (diam(B )) ≥ Hd (E ∩ B ) , m d i d i m i Fi∈F Bi∈I i∈I from which it follows that 1 2dHd(E ) ≥ Hd(E ) =⇒ Hd(E ) = 0 m m m m since m < 2−d by assumption.

Step 2. Assume now that the inequality (4.2) does not hold uniformly for every r > 0, but there is a uniform constant r0 > 0 such that (4.2) holds true for every x ∈ E and every r ≤ r0. d In a similar fashion, we can estimate the outer measure Hδ for δ < r0/2 - up to an arbitrarily small error > 0 -, that is, for every > 0 we obtain the inequality 1 2dHd(E ) + ≥ Hd(E ), m m δ m which immediately implies d H (Em) = 0, since m < 2−d by assumption.

3We will get rid of this assumption in the next step using a simple trick. 50 4.2. UPPER DENSITY OF THE HAUSDORFF MEASURE

Step 3. Finally, assume that the inequality (4.2) is not uniform with respect to x, that is, for any x ∈ Em, it holds for r0 := r0(x) > 0. We consider the set n o E := x ∈ E Hd E ∩ B(x, r) < m · α rd, ∀ r ≤ r m, r0 d 0 and we prove, using the previous step, that this set has measure zero for any choice −d of r0 > 0 and m < 2 . (c) We ﬁrst give a rough idea of the proof of this assertion and then we formalize it correctly.

Step 0. Fix m > 1 and let us consider the set

∗ Em := {x ∈ E | Θd(E, x) > m} .

Let µ := Hd ¬ E the restriction to E of the d-dimensional Hausdorff measure, and notice that µ is locally finite by construction. By assumption, there are infinitely many balls such that

d d µ B(x, r) = H B(x, r) ∩ E > m · αdr ,

and thus X αd X d µ(E ) ' µ(B ) ≥ m · (2r ) ≥ m ·Hd(E ). m i 2d i m i i d In conclusion, since µ is the restriction of H to E, it suﬃces to notice that Em ⊂ E to infer that d d H (Em) = µ(Em) ≥ m ·H (Em), which is absurd since m > 1.

Step 1. Fix δ > 0 and let us consider the collection of balls δ d d F := B(x, r) x ∈ Em, r ≤ and H B(x, r) ∩ E > m · αdr . 2

It is easy to prove that F is a ﬁne cover of Em. By Besicovitch Covering Lemma 3.6, 0 0 for every > 0 we can extract a subfamily F ⊂ F such that F is a covering of Em and X µ(Em) + ≥ µ(B). B∈F 0 A straightforward computation proves that X µ(Em) + ≥ µ(B) ≥ B∈F 0

X d ≥ m αdr(B) = B∈F 0

αd X = m · (2r(B))d ≥ 2d B∈F 0

d ≥ mHδ (Em). 51 4.2. UPPER DENSITY OF THE HAUSDORFF MEASURE

Therefore, if we take the limit as δ and goes to 0+, then it turns out that

d d d d H (Em) + = µ(Em) + ≥ m Hδ (Em) =⇒ H (Em) ≥ m H (Em),

which is absurd since m > 1.

(d) Let µ := Hd ¬ E the restriction to E of the d-dimensional Hausdorff measure, and notice that µ is locally finite by construction. Unfortunately, the measure µ does not have the doubling property (or the asymptotic one); therefore we need to rely on a different covering theorem. d Fix m > 5 , fix δ > 0, let A be an open neighborhood of Em, and let us consider the family of closed balls n o d d F := B(x, r) ⊂ A x ∈ Em, r ≤ δ and H B(x, r) ∩ E > m · αdr .

The cover F is ﬁne, and thus there exists a disjoint subfamily F 0 ⊂ F such that Fb0 covers Em. Then X µ(A) ≥ µ(B) ≥ B∈F 0

X d ≥ m αd r(B) = B∈F 0

d m αd X = 2r Bb ≥ 5d 2d B∈F 0

m d ≥ H 0 (E ), 5d δ m where δ0 = 10δ. We conclude the proof by noticing that, since µ is a regular measure (both inner and outer regular), we have

d H (Em) = µ(Em) = inf µ(A) A⊃Em

and hence, by taking the limit as δ, → 0+, it turns out that

d m d d H (E ) ≥ H 0 (E ) =⇒ H (E ) = 0. m 5d δ m m

Remark 4.3. In the previous results, we used the covering theorems assuming either that X = Rn or µ doubling, but for the applications we need them to be true for a larger class of measures. More precisely, the two statements

A1) If E ⊂ X is a Borel subset and F is a ﬁne cover of E, then for any > 0 there exists a disjoint subfamily F 0 ⊂ F, covering E µ-almost everywhere, such that X µ(B) ≤ µ(E) + . B∈F 0 52 4.3. UPPER D-DIMENSIONAL DENSITY

A2) If E ⊂ X is a Borel subset and F is a ﬁne cover of E, then for any > 0 there exists a subfamily F 0 ⊂ F, covering E, such that X µ(B) ≤ µ(E) + . B∈F 0 hold true even for a measure µ = µ0 ¬ F , where µ0 is doubling and F is arbitrary (actually, it is enough to ask µ µ0 ¬ F .)

4.3 Upper d-Dimensional Density

Support. Let µ be a measure deﬁned on a metric space X. The support of µ is the smallest closed subset F ⊂ X such that the complement F c is a null set, that is,

spt(µ) = inf {F ⊂ X | F is closed and µ(F c) = 0} .

The measure µ is supported on a Borel set E ⊂ X if the complement of E is a null set for µ. In particular, the set E contains the support of µ, that is,

spt(µ) ⊆ E.

Definition 4.5. Let µ and λ be two measures defined on the same metric space X. We say that µ is orthogonal to λ, and we denote it by µ ⊥ λ, if and only if they are supported on disjoint Borel sets. Remark 4.4. The support of a measure µ is a well-defined notion, but it is not the optimal one concerning the orthogonality. Indeed, the reader may prove, as an exercise, that there exists a measure µ orthogonal to a measure λ such that

spt(µ) ∩ spt(λ) 6= ∅. Hint. Consider the Lebesgue measure and the Dirac measure on the real line.

Lemma 4.6. Let λ and µ be locally finite measures defined on a metric space X. Assume that λ is orthogonal to µ and λ satisfies the assumption A2). Then the Radon-Nikodym derivative is zero, that is, dλ λ B(x, r) (x) := lim = 0, dµ r→0 µ B(x, r) for µ-almost every x ∈ X.

Proof. Fix m > 0, let F ⊂ X be a Borel set satisfying

λ (F c) = 0 and µ(F ) = 0, and let dλ Em := x∈ / F lim sup (x) > m . r→0+ dµ 53 4.3. UPPER D-DIMENSIONAL DENSITY

Since F is a µ-null set, we may equivalently prove that µ(Em) = 0 for every ﬁxed m > 0. Consider the family of closed balls n o

F := B(x, r) x ∈ Em, λ B(x, r) ≥ mµ B(x, r) .

The reader may check that F is a ﬁne cover of Em; hence, by assumption A2) it follows that for every > 0 there exists a subfamily F 0 ⊂ F such that [ X Em ⊆ B and λ(B) ≤ λ(Em) + . B∈F 0 B∈F 0 The complement of F is a λ-null set, and thus we ﬁnally infer that X λ(Em) + ≥ m µ(B) ≥ mµ(Em) =⇒ µ(Em) = 0. | {z } 0 =0 B∈F

Corollary 4.7. Let λ and µ be locally finite measures defined on a metric space X. Assume that λ is orthogonal to µ and λ satisfies the assumption A2). Then the Radon-Nikodym derivative is infinite, that is, dµ (x) = +∞, dλ for µ-almost every x ∈ X. Theorem 4.8. Let µ be locally finite measures defined on a metric space X satisfying the p assumption A2), and let f ∈ Lloc(X, µ) be a function. Then Z − |f(y) − f(x)|p dµ(y) −−−→r→0 0, B(x, r) for µ-almost every x ∈ X.

Proof. Fix > 0. By Lusin’s Theorem4 we can always ﬁnd a continuous function fe : X −→ R and a subset E ⊂ X such that

f E = fe E and µ (X \ E) < . The complement of E can be taken arbitrarily small; thus it suﬃces to prove the theorem for µ-almost every x ∈ E. In particular, given x ∈ E it turns out that Z Z Z − |f(y) − f(x)|p dµ(y) = − |f(y) − f(x)|p dµ(y) + − |f(y) − f(x)|p dµ(y) = B(x, r) B(x, r)∩E B(x, r)\E

" # 1 Z p Z (∗) p = fe(y) − fe(x) dµ(y) + |f(y) − f(x)| dµ(y) ≤ µ Br Br ∩E Br \E

(∗) p p−1 Z h i 2 p p ≤ ω fe + [|f(x)| + |f(y)| ] dµ(y), µ Br B(x, r)\E

4Lusin’s Theorem: Let (X, Σ, µ) be a Radon measure space and let Y be a second-countable topological space. If f : X → Y is a measurable function and > 0, then for any A ∈ Σ there is a closed set E with µ(A \ E) < such that f E is continuous. 54 4.3. UPPER D-DIMENSIONAL DENSITY

where ω fe denotes the oscillation of fe, and the inequality (∗) follows from the following trivial fact: If p ≥ 1, then for any real numbers a and b it turns out that

|a − b|p ≤ 2p−1(|a|p + |b|p).

Let us consider the measures

p λ1 := |f| 1X\E · µ and λ2 := 1X\E · µ, and let µe be the restriction of µ to the set E. Then Z p λ1 B(x, r) λ2 B(x, r) p h i p−1 p−1 p − |f(y) − f(x)| dµ(y) ≤ ω fe + 2 + 2 |f(x)| , B(x, r) µe B(x, r) µe B(x, r) from which it easily follows that:

1) The oscillation ω fe goes to 0 as r → 0+ for every x ∈ E, as a consequence of the continuity of fe.

2) The measures λ1 and λ2 are both orthogonal to µe; hence both the ratios go to 0 as + r → 0 , for µe-almost every x ∈ X, which means for µ-almost every x in E.

Deﬁnition 4.9. The function f : X −→ R is Lp-approximately continuous if and only if

Z + − |f(y) − f(x)|p dµ(y) −−−−→r→0 0 B(x, r) for µ-almost every x ∈ X. Corollary 4.10. Let µ be locally ﬁnite measures deﬁned on a metric space X satisfying the p assumption A2), and let f ∈ Lloc(X, µ) be a function. Then

Z + − f(y) dµ(y) −−−−→r→0 f(x), B(x, r) for µ-almost every x ∈ X.

Proof. By Jensen’s inequality5, it turns out that

q p Lloc(Ω) ⊂ Lloc(Ω), ∀ q < p, and thus we can apply Theorem 4.8 with p = 1.

5 Let (Ω, G, µ) be a probability space. If g :Ω −→ R is a µ-summable function, and if ϕ : R −→ R is a convex function, then Z Z ϕ g(x) dµ(x) ≤ ϕ ◦ g(x) dµ(x). Ω Ω 55 4.3. UPPER D-DIMENSIONAL DENSITY

Theorem 4.11. Let µ and λ be locally ﬁnite measures satisfying the assumption A2). Assume that there is a decomposition

λ = f · µ + λs, where λs denotes the singular part of the measure λ. Then the density of λ with respect to µ can be computed pointwise, and it is equal to dλ (x) = f(x), dµ for µ-almost every x ∈ X.

Proof. By Lemma 4.6 it turns out that dλ s (x) = 0 dµ for µ-almost every x ∈ X. On the other hand, by Theorem 4.8 it follows that dλ ac (x) = f(x), dµ where λac denotes the part of λ which is absolutely continuous with respect to µ, that is,

λac = f · µ.

Upper Density. Let µ be a locally finite measure defined on a metric space X, and let d > 0 be a positive real number. The d-dimensional upper density at the point x with respect to µ is defined by setting µ B(x, r) ∗ Θd(µ, x) := lim sup d , (4.3) r→0+ αdr

αd where 2d is the renormalization constant used in the definition of the Hausdorff measure. Theorem 4.12. Let µ be a locally finite measure defined on a metric space X, and let d > 0 be a positive real number. The following properties are equivalent:

(1) The upper density is ﬁnite and nonzero, that is,

∗ 0 < Θd(µ, x) < +∞, for µ-almost every x ∈ X.

1 d (2) There is a locally summable function f ∈ Lloc X, H such that

µ = f ·Hd.

The proof of this theorem is rather involved. Hence, we split it into diﬀerent propositions and lemmas and, at the end of the section, we get much more than the statement above. 56 4.3. UPPER D-DIMENSIONAL DENSITY

Proposition 4.13. Let µ be a locally ﬁnite measure on X and let d > 0. Suppose that there 1 d d is f ∈ Lloc H such that µ = f ·H . Then

−d ∗ d d 2 f(x) ≤ Θd(µ, x) ≤ 5 f(x) for H -almost every x ∈ X.

∗ In particular the upper-density of µ is ﬁnite and nonzero (0 < Θd(µ, x) < +∞) for µ-almost any x ∈ X.

Proof. We ﬁrst prove the result in some particular cases, and then we generalize it using a simple trick.

Step 1. Assume that X = Rn and assume that there are constants m < M ∈ R such that 0 < m < f(x) ≤ M < +∞ for Hd-almost every x ∈ X s.t. f(x) 6= 0.

d Set E := {x ∈ X | f(x) 6= 0} and consider the measure λ := 1E ·H . Clearly λ is locally ﬁnite, and thus we can easily compute the density ratio, that is, µ B(x, r) Z 1 d d = d f(y) dH (y) = αdr αdr B(x, r)

1 Z = d f(y) dλ(y) = αdr B(x, r)

λ B(x, r) Z − = d f(y) dλ(y). αd r B(x, r) Since λ is equal to the restriction to E of the d-dimensional Hausdorﬀ measure, it follows from Theorem 4.4 that λ B(x, r) 1 d lim sup d ∈ d , 5 r→0 αdr 2 for Hd-almost every x ∈ E. On the other hand, the Lebesgue result (see Corollary 4.10) implies that Z − f(y) dλ(y) −−−→r→0 f(x), B(x, r) for Hd-almost every x ∈ E, which is exactly what we wanted to prove.

Step 2. Let X be any reasonable6 space and assume that there are constants m < M ∈ R such that 0 < m < f(x) ≤ M < +∞ d for H -almost every x ∈ X such that f(x) 6= 0. Let f1 and f2 be ﬁnite sums of step functions in such a way that f1(x) ≤ f(x) ≤ f2(x).

6 n Here we do not specify what kind of assumptions are needed on X, but any space that ”looks like” R will do. The reader may try, as an exercise, to ﬁnd the minimal assumptions that can make the argument of the second step work. 57 4.3. UPPER D-DIMENSIONAL DENSITY

We claim that −d ∗ d 2 f1(x) ≤ Θd(µ, x) ≤ 5 f2(x) for Hd-almost every x ∈ X such that f(x) 6= 0.

Step 2.1. Let α1, . . . , αm ∈ R and A1,...,Am ⊂ X, and assume that m X 1 f1(x) = αi Ei , i=1 where Ei = E ∩ Ai for every i = 1, . . . , m. The following inequality of measures is tirvial m X 1 d µ ≥ αi Ei ·H , i=1 and hence ∗ −d (x∈Ei) −d Θd(Ei, x) ≥ αi2 = f1(x) 2 d for H -almost every x ∈ Ei. The union of the Eis is almost all of E, and thus it turns out that −d ∗ 2 f1(x) ≤ Θd(µ, x) for Hd-almost every x ∈ E.

Step 2.2. Let us consider the function

m X 1 f2(x) = βi Ei , i=1 F where the family {Ei := Ai ∩ E}i=1, ..., m is disjoint. Since E = i=1, ..., m Ei, it turns out ∗ d that Θd(µ, x) = 0 for H -almost every x∈ / E. If x ∈ Ej, then m ∗ X ∗ ∗ d Θd(µ, x) ≤ βiΘd (Ei, x) = βjΘd(Ej, x) ≤ 5 f2(x) i=1

d for H -almost every x ∈ Ej, as a consequence of Theorem 4.4.

1 n d Step 3. Assume f ∈ Lloc R , H . Fix m > 1, set 1 Em := x ∈ E ≤ f(x) ≤ m , m and let µ = µm + µfm be the Radon-Nikodym decomposition of µ associated to Em. More precisely, we consider the decomposition µ = f 1 ·Hd and µ = f 1 ·Hd, m Em fm E\Em in such a way that µ B(x, r) r→0 −−−→ 1 for µm-almost every x ∈ E. µm B(x, r) 58 4.3. UPPER D-DIMENSIONAL DENSITY

Since µfm is singular, it turns out that ∗ ∗ d Θd(µ, x) = Θd(µm, x) ≤ 5 f(x)

d for µm-almost every x ∈ E, i.e., for H -almost every x ∈ Em The union of the Ems covers almost all of E; thus ∗ d Θd(µ, x) ≤ 5 f(x) for µ-almost every x ∈ E. If, on the other hand, x∈ / E, then we can prove7 that the d-dimensional upper density is equal to 0 for Hd-almost every x∈ / E.

1 d Step 4. Let X be a reasonable metric space and let f be a function in Lloc X, H . This step follows from the previous ones since for a locally summable function, for every > 0, there are constants m, M > 0 and Em, M ⊆ X such that

0 < m < f(x) ≤ M < +∞

d d for H -almost every x ∈ E such that f(x) 6= 0 and H (Em, M ) < . Lemma 4.14. Let µ be a locally finite measure with finite d-dimensional upper density, that is, ∗ Θd(µ, x) < +∞ for µ-almost every x ∈ X. Then µ is absolutely continuous with respect to the Hausdorff measure Hd, that is, Hd(E) = 0 =⇒ µ(E) = 0.

Sketch of the Proof. By assumption, if r > 0 is small enough, it turns out that µ B(x, r) d ≤ m. αdr

Let E be a Hd-null subset of X, that is Hd(E) = 0. For any m, ρ > 0 let us consider the collection of sets   µ B(x, r)   Em, ρ = x ∈ E ≤ m for every r < ρ . α rd  d  The countable union over the rational numbers covers almost every point of E, that is, [ Em, ρ = µ-almost all of E, 2 (m, ρ)∈Q since there may be A ⊂ X null set such that

∗ x ∈ A =⇒ Θd(µ, x) = +∞.

It is clearly enough to prove that the measure of Em, ρ is equal to 0 for every ﬁxed couple (m, ρ) ∈ Q2.

7The reader may prove this assertion in a similar fashion to Theorem 4.4. 59 4.3. UPPER D-DIMENSIONAL DENSITY

d d By assumption H (Em, ρ) = 0, and thus Hδ (Em, ρ) = 0. In particular, for every > 0 there exists a covering {Ei}i∈Im, ρ for Em, ρ satisfying the additional property

X d diam(Ei) < δ and (diam(Ei)) ≤ . i

Let us consider the closed ball Bi := B(xi, ri), centered at xi ∈ Ei ∩ E with radius ri = diam(Ei), for every i ∈ Im, ρ. The collection of balls {Bi}i∈Im, ρ covers Em, ρ, and thus

X X d X d µ(Em, ρ) ≤ µ(Bi) ≤ mαd ri ≤ mαd (diam(Ei)) ≤ mαd · ,

i∈Im, ρ i∈Im, ρ i∈Im, ρ which is exactly what we wanted to prove. Lemma 4.15. Let µ be a locally ﬁnite measure deﬁned on a metric space X. Fix m > 0 and let ∗ Em := {x ∈ E | Θd(µ, x) > m} .

The d-dimensional Hausdorﬀ measure of Em is bounded from above, that is,

5d Hd(E ) ≤ µ(E ). m m m

Proof. Let A be an open neighborhood of Em, and let us consider the collection of closed balls   µ B(x, r)   F := B(x, r) B(x, r) ⊆ A and > m . α rd  d 

The reader may prove that F is a ﬁne cover of Em; thus by Vitali’s covering Lemma 3.1 0 0 there exists a disjoint subfamily F ⊂ F such that Fc covers Em. It follows that X µ(A) ≥ µ(B) ≥ B∈F 0

X d ≥ m αdr = B∈F 0

m αd X = (10r)d ≥ 5d 2d B∈F 0

d mαd X m ≥ diam(Bˆ) ≥ Hd (E ). 10d 5d ∞ m B∈F 0

d In particular, to estimate Hδ it is enough to consider a cover made up of balls whose diameter is less than δ/10. If we take the inﬁmum with respect to A and the supremum with respect to δ > 0, then it turns out that

5d Hd(E ) ≤ µ(E ). m m m 60 4.4. APPLICATIONS

Corollary 4.16. Let µ be a locally ﬁnite measure deﬁned on a metric space X and assume that ∗ Θd(µ, x) > 0 for µ-almost every x ∈ X. Then µ is supported on the set

∗ E := {x ∈ X | Θd(µ, x) > 0} , which is σ-ﬁnite with respect to Hd.

Proof. It follows immediately from the previous theorem since [ E = E 1 n n∈N and we have, for every n ∈ N, the estimate d d 5 H (E 1 ) ≤ µ(E 1 ) < +∞. n m n

Proposition 4.17. Let µ be a locally ﬁnite measure deﬁned on a metric space X and assume that ∗ 0 ≤ Θd(µ, x) < +∞ for µ-almost every x ∈ X. Then µ = f ·Hd 1 d for some locally summable function f ∈ Lloc(H ).

Proof. By Lemma 4.14, the measure µ is absolutely continuous with respect to the d- dimensional Hausdorﬀ measure Hd, while the previous corollary implies that µ is supported on the set E deﬁned above. Therefore

d µ λ := 1E ·H , and λ is σ-finite; thus the Radon-Nikodym theorem implies that µ = f · λ. Remark 4.5. Notice that the Radon-Nikodym theorem cannot be applied directly to Hd since the Hausdorff measure is not σ-finite (which is a necessary assumption for the result to hold true.)

4.4 Applications

In this final brief section, we investigate some of the main applications of the theory devel- oped so far (especially to Cantor sets.) Corollary 4.18. Let E be a Borel set contained in a metric space X, and let d ≥ 0 be a positive real number. Assume that there exists a finite measure µ, defined on X, such that:

∗ (a) The d-dimensional upper density is ﬁnite, that is, Θd(µ, x) < +∞ for µ-almost every x ∈ E. 61 4.4. APPLICATIONS

(b) The set E is not null with respect to µ, that is, µ(E) > 0.

Then the d-dimensional Hausdorﬀ measure of E is strictly bigger than zero.

Proof. By Theorem 4.12, the measure λ := 1E · µ is absolutely continuous with respect to Hd; therefore the Hausdorﬀ measure of the set E cannot be zero (otherwise also µ(E) would be equal to zero).

Cantor Set. We have proved in the previous chapter that log 2 d := , log 3 is the Hausdorﬀ dimension of the Cantor set C, and recall that Hd(C) > 0. Remark 4.6. The Cantor set is deﬁned by setting

+∞ \ C := Ci, i=0 k where C0 = [0, 1], C1 = [0, 1/3] ∪ [2/3, 1] and Ck is the disjoint union of 2 intervals, whose length is equal to 3−k. Proposition 4.19. Let

Φ(Ii, j) = Φ(Ii+1, k) + Φ(Ii+1, k+1) be a set function (that preserves the mass). Then Φ can be uniquely extended to a ﬁnite measure ν on R, which is supported on the Cantor set C.

Proof. Let F = {Ii, j}i, j∈N∪∅ and let µ be the outer measure on C given by the Carath´eodory construction. We claim that the following properties are satisﬁed:

(a) For every i, j ∈ N it turns out that µ(Ii, j) = Φ(Ii, j). (b) The outer measure µ is additive on distant sets (and thus the restriction to the Borel σ-algebra is a σ-additive measure.)

As a consequence of the second claim, the restriction of µ to the Borel σ-algebra, denoted by ν, is exactly the sought measure. The reader may prove, as an exercise, that the measure deﬁned in this way is unique.

(a) Let us consider the outer measure

µ (Ii, j) := inf {Φ(Ek) | Ek ∈ F and Ii, j ⊆ Ek } .

Notice that (Ek) is an open covering of C and, for every i, j ∈ , the interval Ii, j k∈N N is compact. It follows that there exists a ﬁnite subfamily Ei1 ,...,Eik such that

µ (Ii, j) = inf Φ(Ei ). `=1, ..., k `

8 This proves that µ(Ii, j) = Φ(Ii, j) for every i, j since Φ is additive on distant sets .

8It follows easily from the deﬁnition of Φ and of C. 62 4.4. APPLICATIONS

(b) We consider the outer measure

µδ (Ii, j) := inf {Φ(Ek) | Ek ∈ F, diam(Ek) ≤ δ and Ii, j ⊆ Ek } .

Then µδ coincides with µ on a suitable class of sets, and it is larger on where the distance is bigger than δ. This concludes the proof.

Canonical Measure on C. Let µ be the measure given by Proposition 4.19 associated to the set function −i Φ(Ii, j) = 2 . ∗ It is enough to prove that Θd(µ, x) < +∞ for µ-almost every x ∈ C. First, we notice that for every x ∈ C, it turns out that

−i ∗ µ(Ii, j) 2 Θfd(µ, x) := lim d = −id = 1. diam(Ii, j )→0, x∈Ii, j (diam(Ii, j)) 3 Then ∗ h ∗ ∗ i Θd(µ, x) ∈ c1Θfd(µ, x), c2Θfd(µ, x) , and this is exactly what we wanted to prove. Indeed, the closed ball B(x, r) satisﬁes the −i−1 −i inequality 3 < r ≤ 3 , and thus B(x, r) ∩ C ⊆ Ii, j for some j ∈ N. Chapter 5

Self-Similar Sets

In this brief chapter, we investigate the notion of Hutchinson’s fractals (or self-similar sets), and we prove that, under suitable assumptions, given a certain number of similarities, there exists a unique compact self-similar set with a precise Hausdorff dimension d. Definition 5.1 (Self-Similar). A subset E ⊆ Rn is self-similar if there exist a finite collection φ1, . . . , φN of similarities

φi(x) := xi + λi · Rix, where Ri ∈ O(n) and λi ∈ (0, 1), such that N [ E = φi(E). i=1 1 Remark 5.1. If the φi(E) are (essentially) disjoint , then we expect the Hausdorﬀ dimension of E to be equal to the unique solution of the equation

N X d λi = 1. (5.1) i=1

It is important to stress that the Hausdorff dimension should be equal to d, but, a priori, there is no guarantee that the d-dimensional Hausdorff measure of E is finite. We now state two lemmas that explain what we mean by should be equal to.

Lemma 5.2. The equation (5.1) admits a unique solution d ≥ 0, provided that the λis are positive and strictly less than one.

Lemma 5.3. If the φi(E) are (essentially) disjoint and if there exists d ≥ 0 such that 0 < Hd(E) < +∞, then d is the unique solution of (5.1).

Proof. This is a straightforward application of the properties of the Hausdorﬀ measure:

N N N ! G d X d X d d E = φi(E) =⇒ H (E) = H (φi(E)) = λi H (E), i=1 i=1 i=1

1 n More precisely, we say that φi(E) and φj (E) are essentially disjoint if and only if L (φi(E) ∩ φj (E)) = 0. 64

PN d that is, the factor i=1 λi needs to be equal to one for the identity to hold.

Hutchinson Construction of Self-Similar Sets. Let be given φ1, . . . , φN similarities with scaling factors λi ∈ (0, 1). For the next theorem to be true, we need to introduce the open set condition.

The open set condition (OSC). The ﬁnite family of similarities φ1, . . . , φN satisﬁes the open set condition if and only if there exists a nonempty open set V ⊂ Rn such that

N [ φi(V ) ⊆ V and φi(V ) ∩ φj(V ) = ∅ for i 6= j. i=1

Theorem 5.4 (Hutchinson). Let be given a ﬁnite collection φ1, . . . , φN of similarities with scaling factors λi ∈ (0, 1) satisfying the (OSC) condition. Then there exists a unique self- similar compact set C ⊆ Rn, that is,

N [ C = φi(C). i=1

The Hausdorff dimension of C is the unique solution d of the equation (5.1), and the d- dimensional Hausdorff measure of C is finite and nonzero, that is,

0 < Hd(C) < +∞.

The proof of this theorem is hard, and we need extra care to deal properly with the open set condition. For this reason, we replace it with a stronger condition: φi V ∩ φj V = ∅, ∀ i 6= j. More precisely, we assume that the images of V via the similarities are distant.

Proof. Let X := V , and let F := {F ⊆ X | F is closed.} be the family of all the closed (and thus compact) subsets of X.

Step 1. The Hausdorﬀ distance induces a structure of metric space on F. Indeed, if we denote by Ur(A) an open neighborhood of A with diameter r, then one can prove that

dH(C1,C2) := inf {r > 0 | C1 ⊆ Ur(C2) and C2 ⊆ Ur(C1)} , is a metric. Moreover, the following implications hold2:

(a) If X is complete, then (F, dH) is complete.

(b) If X is compact, then (F, dH) is compact.

2These two statements are not related to the content of this course; hence both are left as an exercise for the reader. 65

Step 2. Let us consider the following operator

N [ Φ: F −→ F,F 7−→ φi(F ). i=1 between two Banach spaces, and let λmax := maxi=1, ..., N λi ∈ (0, 1). By definition, it turns out that dH (Φ(F1), Φ(F2)) ≤ λmax · dH(F1,F2), which means that the operator Φ is a contraction. The Banach fixed point theorem3 proves that there exists a unique fixed point C which is given by lim Φj(F ), j→+∞ for every F ∈ F. In particular, one can choose X as a starting point for the sequence.

Step 3. First, we observe that \ X ⊃ Φ(X) ⊃ Φ2(X) ⊃ · · · =⇒ lim Φj(X) = Φj(X) = C. j→+∞ j∈N j For any j ∈ N and any j-tuple of indices (i1, . . . , ij) ∈ {1,...,N} , we denote by Xi1, ..., ij the iterated image of X, that is,

Xi1, ..., ij := φij ◦ · · · ◦ φi1 (X). This notation is particularly useful since we can now express C as an inﬁnite intersection of ﬁnite unions (a fairly straightforward generalization of the construction of the Cantor set), that is,   \ [  Xi1, ..., ij  = C. (5.2) j∈N 1≤i1, ..., ij ≤N

Step 4. In this brief step, we want to ﬁnd an upper bound on the d-dimensional Haus- dorﬀ measure of C and, more precisely, we prove that α Hd(C) ≤ d (diam(X))d . 2d j Fix δ > 0, choose j so that λmax · diam(X) < δ, and observe that j diam Xi1, ..., ij ≤ λmax · diam(X) and diam Xi1, ..., ij = λi1 . . . λij · diam(X). As a consequence of the identity (5.2), it follows that the family Xi1, ..., ij is a covering of C with diameter less than δ as (i1, . . . , ij) ranges in the set of all the j-tuples; hence

αd X d Hd(C) ≤ diam X ≤ δ 2d i1, ..., ij 1≤i1, ..., ij ≤N

αd d X d ≤ (diam(X)) λ . . . λ . 2d i1 ij 1≤i1, ..., ij ≤N

3Theorem. Let B be a Banach space, and let T : B −→ B be a contraction. Then there exists a unique b ∈ B such that T (b) = b, and for every x ∈ B it turns out that b is the limit point of the sequence n {xn := T x}n∈N. 66

The right-hand side is, up to a constant, equal to (diam(X))d since

N !j X d X d λi1 . . . λij = λi = 1.

1≤i1, ..., ij ≤N i=1

In particular, notice that the existence of the ﬁxed point C and the upper bound on the d- dimensional Hausdorﬀ measure rely on the completeness of X only: all the other assumptions are necessary for the lower bound!

Step 5. The lower bound is rather delicate, and the rough idea behind it is to construct a probability measure µ on C such that

∗ Θd(µ, x) < +∞, ∀ x ∈ C. More precisely, we construct µ as the weak limit of a sequence of probability measures

(µk)k∈N as follows. Let x ∈ C be a point, and let µ0 := δx be the Dirac measure centered at that point. We deﬁne the measure

d d µ1 := λ1δφ1(x) + ··· + λN δφN (x), where the renormalization constants need to be chosen in this way since the problem is asymmetrical4 as the weight is not uniformly distributed.

Step 5.1. Let us denote by xi1, ..., ij the image via the collection φ1, . . . , φN of x, that is,

xi1, ..., ij := φij ◦ · · · ◦ φi1 (x).

The jth element of the sequence is thus given by

X d µ := λ . . . λ δ . j i1 ij xi1, ..., ij 1≤i1, ..., ij ≤N

The set function µj is a probability measure deﬁned on C for every j ∈ N; hence there exists 5 a subsequence µjk that converges to a probability measure µ. We now claim that for every j j ∈ N and for every j-tuple of indices (i1, . . . , ij) ∈ {1,...,N} , it turns out that µ X = λd . . . λd . i1, ..., ij i1 ij It is easy to see that for any k ≥ j, we have

µ X = λd . . . λd , k i1, ..., ij i1 ij and hence our claim follows immediately if we can pass this identity to the limit as k → +∞. Here we use the strong replacement for the open set condition: For every j-tuple, the set Xi1, ..., ij is both open and closed in C, and thus we can take the limit for k → +∞.

4The reader may check that a diﬀerent choice of renormalization constants yields to a diﬀerent result (e.g., with 1/N). 5The whole sequence converges to µ, but we do not use this fact in the proof. On the other hand, we will use this property in the next chapters; thus the reader may try to prove it by herself. 67

Step 5.2. Fix x ∈ C. Then x is uniquely identiﬁed by a sequence of indices (ij)j∈N in such a way that x ∈ Xi1, ..., ij for every j ∈ N. Moreover, we have the identity

µ X 1 i1, ..., ij = , (5.3) d diam(X)d diam(Xi1, ..., ij )

since diam(Xi1, ..., ij ) = λi1 . . . λij · diam(X) by construction. It remains to prove that (5.3) is enough to infer that the d-dimensional upper density of µ is ﬁnite. Fix x ∈ C and ﬁx 0 < r < dmin, where

dmin := inf d (φi(x), φj(x)) . i, j=1, ..., N

There exists a natural number j ∈ N such that

dmin · λi1 . . . λij+1 < r ≤ dmin · λi1 . . . λij , and therefore

B(x, r) ∩ C ⊆ Xi1, ..., ij . It follows that µ B(x, r) ≤ µ Xi1, ..., ij , and this is enough to estimate the density ratio, that is, µ B(x, r) d µ Xi1, ..., ij 1 d ≤ d = < +∞. r dmin · diam(X) dmin · λi1 . . . λij

Step 5.3 It remains to prove that

∗ Θd(µ, x) < +∞, ∀ x ∈ C is enough to infer that Hd(C) > 0 strictly. Suppose that

Br ⊆ X ⊆ BR, and let ρ > 0 be a ﬁxed real number. Denote by S the space of all ﬁnite sequences (i1, . . . , ij) such that the following inequality holds:

λmin · ρ ≤ λi1 . . . λij ≤ ρ. (5.4) It follows that the collection X := Xi1, ..., ij :(i1, . . . , ij) ∈ S

is disjoint, and thus every Xi1, ..., ij ∈ X contains a ball of radius r · λi1 . . . λij and it is contained in a ball of radius R · λi1 . . . λij . By (5.4) we infer that

Br·λmin·ρ ⊆ Xi1, ..., ij ⊆ BR·ρ.

In particular, every ball of radius ρ intersects q sets of the collection X , where 1 + 2Rn q := . λminr 68

Moreover, by deﬁnition the support of µi1, ..., ij is contained in Xi1, ..., ij for every j ≥ 1, and therefore X d µ = λi1 . . . λij µi1, ..., ij .

(i1, ..., ij )∈S

For every ball Bρ of radius ρ such that Bρ ∩ Xi1, ..., ij 6= ∅, it turns out that

X d n µ(Bρ) ≤ λi1 . . . λij µi1, ..., ij (R ),

(i1, ..., ij )∈S which means that q|B |d µ(B ) ≤ qρd = ρ ρ 2d whenever |Bρ| < |X|. Let {Ui}i∈I be a countable cover of C, and notice that [ C ⊆ Bi where |Bi| ≤ 2|Ui|, i∈I from which it follows that

X X d d 1 = µ(X) ≤ µ(Bi) ≤ q |Ui| ' qH (E), i∈I i∈I and thus Hd(E) ≥ q−1 > 0, which is what we wanted to prove.

Examples. In the following ﬁgures one can see many applications of the Hutchinson The- orem.

Figure 5.1: Fractal 1 69

Figure 5.2: Fractal 2

Figure 5.3: Fractal 3 70

Figure 5.4: Fractal 4 Part IV

Rectiﬁability

71 Chapter 6

Other measures and dimensions

In this chapter, we first introduce the integralgeometric measure, and then we investigate the Haar invariant k-dimensional measure. In the second half, we show how the Haar measure may be used to define an invariant measure on the Grassmannian manifold G(n, m), which will be extremely useful to study rectifiable sets.

6.1 Geometric Integral Measure

In this ﬁrst section, we propose an alternative k-dimensional measure to the Hausdorﬀ one and, at the same time, we exhibit a motivational example for introducing the notion of invariant measures.

Definition (1-dimensional). Let E ⊆ Rn be a subset, and fix a projection n PL : R −→ L onto a 1-dimensional linear subspace (i.e., a line). We may easily define (see Figure 6.1) a measure, which is invariant under translation but not under rotations, as follows: Z −1 1 # PL (x) ∩ E dH (x), (6.1) x∈L where #(·) denotes the cardinality function. In order to define a rotation-invariant measure, the simplest idea that comes in mind is to consider the average value of (6.1), as L ranges in the set of all the 1-dimensional subspaces of Rn. More precisely, we ”define” the 1-dimensional integralgeometric measure as follows: Z Z 1 −1 1 I (E) := − # PL (x) ∩ E dH (x) . (6.2) L∈G(n, 1) x∈L The measure (6.2) is not well-defined, since we do not know which measure we need to use to integrate over all L ∈ G(n, 1); we will come back to this issue later. 73 6.1. GEOMETRIC INTEGRAL MEASURE

Figure 6.1: One-dimensional translation-invariant measure.

Deﬁnition (k-dimensional). The same construction can be easily generalized to a k- dimensional measure. Indeed, if we denote by G(n, k) the Grassmannian manifold (i.e, the set of all k-dimensional subspaces of Rn) then, it turns out that Z Z k −1 k I (E) := ck, n − # PV (x) ∩ E dH (x) (6.3) V ∈G(n, k) x∈V is a measure, which is invariant under translations and rotations. The reader may prove that, if the renormalization constant ck, n is chosen properly, then Ik(E) = Lk(E) = Hk(E), provided that E is contained in a k-hyperplane of Rn.

Deﬁnition Issue. The k-dimensional integralgeometric measure (6.3) is not well-deﬁned (as we have already mentioned in the 1-dimensional case.) Indeed, we are taking the average integral over all the elements of the Grassmannian manifold (n, k), but we have not introduced yet a measure on that space that is also invariant. This issue is the main reason why we are so interested in developing (at least partially) the theory of invariant measures. At the end of the chapter, we will be able to prove the existence of an invariant measure γn, k on the Grassmannian manifold (n, k).

Basic Properties. To conclude this introduction we give a list of relevant and compelling facts about the k-dimensional geometric integral measure and leave them as exercises for the reader. 74 6.2. INVARIANT MEASURES ON TOPOLOGICAL GROUPS

Lemma 6.1. Let E be a subset of a h-dimensional surface Σ. Assume that h > k strictly, and that the h-dimensional volume of Σ is positive. Then Ik(E) = +∞. Remark 6.1. The k-dimensional Hausdorﬀ measure is, in general, diﬀerent from the k- dimensional geometric integral measure.

More precisely, it may happen that for some subset E ⊂ Rn, the Hausdorﬀ dimension is k dimH(E) > k, but I (E) = 0. 2 Example 6.1 (Cantor Set). Let us consider the set C2 ⊂ R given by the product of two Cantor-type set with scaling factor equal to 1/4 (see Figure 6.1.)

In particular, we have the similarities Φ1,..., Φ4, all with scaling factor equal to 1/4, in such a way that C2 is a self-similar fractal set in the sense of Hutchinson, that is,

4 [ C2 = Φi(C2). i=1

By Theorem 5.4, the Hausdorﬀ dimension of C2 is the unique solution to the equation 1d 4 = 1 =⇒ d = 1, 4

1 but the reader may prove, as an exercise, that I (C2) = 0.

Figure 6.2: Cantor Square

6.2 Invariant Measures on Topological Groups

In this section, we examine the assumptions needed for the existence and uniqueness of an invariant measure defined on a topological group G. Definition 6.2 (Push-Forward). Let µ be a positive measure on X, and let f : X −→ Y be a Borel function. The push-forward measure of µ via f is defined by setting

−1 f#µ(E) := µ f (E) , ∀ E ∈ B(Y ). 75 6.2. INVARIANT MEASURES ON TOPOLOGICAL GROUPS

Lemma 6.3. Let (X, B(X)) and (Y, B(Y )) be measurable spaces, and let µ be a positive measure on X. Then the push-forward f#µ is a well-deﬁned measure on the the Borel σ-algebra of Y .

Topological Groups. Let G be a topological group. For any y ∈ G, we denote by τy the ∗ left-multiplication (x 7→ y · x) and by τy the right-multiplication (x 7→ x · y). Deﬁnition 6.4 (Invariant Measure). Let µ be a measure deﬁned on a topological group G. The measure µ is left-invariant on G if and only if

(τy)# µ = µ ∀ y ∈ G. In a similar fashion, the measure µ is right-invariant if and only if

τ ∗ µ = µ ∀ y ∈ G, y # and, clearly, µ is invariant if and only if µ is both left-invariant and right-invariant.

We are now ready to state the existence and uniqueness results. We will not prove the second theorem in this course, but we will give, at the end of the section, a complete sketch of the proof of the ﬁrst result (which is the only one we shall be using in this course). Theorem 6.5. Let G be a compact group. Then there exists a unique invariant probability measure on G, called Haar measure. Theorem 6.6. Let G be a locally compact and separable group. Then there exists a locally ﬁnite invariant measure on G, which is unique up to a multiplicative constant.

Grassmannian Manifold. The Grassmannian G(n, k) is, in general, not a group. There- fore, we cannot apply to it the theorem stated above, but there is another way around since, G as we shall see soon, we can identify G(n, k) with a quotient Γ, where Γ satisﬁes certain properties. Let G be a topological group acting on a set X, and let τ : G × X −→ X be the left action map, that is, τ(g, x) = g · x.

In order to be coherent with the previous paragraph, we denote by τg(x) the element τ(g, x) so that

τg1·g2 = τg1 τg2 . A measure µ deﬁned on X is invariant under the action of G if and only if

(τg)# µ = µ for every g ∈ G. However, in this general setting, the existence (let alone the uniqueness) of an invariant measure is not guaranteed and, actually, we can exhibit an easy counterexample assuming both G and X compact.

Example 6.2. Let X = P1(R) =∼ R ∪ {0} and let G be the group of all the projective transformation of X. Since the translation group is contained in G, the only possible invariant measure is the 1-dimensional Lebesgue measure. On the other hand, the Lebesgue measure is not invariant under homothety, and hence there are no invariant measures. 76 6.2. INVARIANT MEASURES ON TOPOLOGICAL GROUPS

Theorem 6.7. Let G be a topological group acting on a set X. An invariant measure (not necessarily unique) exists if one of the following conditions is satisﬁed:

(1) The group G is abelian (or, more generally, it satisﬁes the Weyl condition1.)

G 2 (2) The set X is isomorphic to the quotient H, where H is a closed subspace of G. G Remark 6.2. If µ is a left-invariant measure on G and π : G −→ H is a projection onto a closed subset, then the push-forward π#µ is also an invariant measure.

The non-oriented Grassmannian manifold G(n, k) is diffeomorphic to a certain quotient, so that, by Theorem 6.7, an invariant measure γn, k exists. In particular, we will finally be able to show that the integralgeometric k-dimensional measure (6.3) is well-defined. Lemma 6.8. Let O(m) denote the group of the orthogonal m × m matrices. Then there is a diffeomorphism ∼ O(n) G(n, k) = (O(k) × O(n − k)), where O(k) × O(n − k) is the space of orthogonal matrices made up of a k × k orthogonal block and a (n − k) × (n − k) orthogonal block.

The reader may work out the details of the proof by themselves, but the intuitive idea behind it is simple: Given W ∈ G(n, k) element of the Grassmannian manifold, consider an orthonormal basis w1, . . . , wk for it. Complete it to an orthonormal basis w1, . . . , wn of Rn and, at this point, deﬁne an equivalence class naturally (that is, up to change of orthonormal basis for W and the complement of W separately.) In conclusion, as promised, we sketch the proof of Theorem 6.5 in the special case of G Lie group, and we give a complete proof (of the existence, at least) in the case of G commutative).

Proof 1. Let G be a k-dimensional Lie group. The idea is to deﬁne a left-invariant k-form ω, that is, a k-form such that the pull-back according to τy is ω itself. Then it suﬃces to check that Z µ(E) := ω E is the sought invariant measure, and also that it is unique.

Proof. Let G be a commutative group, and let P be the space of probability measures deﬁned on G. For any g ∈ G, set n o P := µ ∈ P (τ ) µ = µ g g # be the subset of P containing all the g-invariant probability measures deﬁned on X.

1We do not need to deal with this delicate condition, but the interested reader may ﬁnd more information here. 2Notice that H does not need to be a normal subgroup of G. 77 6.2. INVARIANT MEASURES ON TOPOLOGICAL GROUPS

Step 1. We want to prove that, for every g ∈ G, the subset Pg is nonempty. Fix µ0 ∈ P and let us consider, for every n ∈ N, the probability measure deﬁned by setting

µ0 + (τg) µ0 + ··· + (τgn ) µ0 µ := # # ∈ P, n n + 1 where gn denotes the product of n copies of g.

By compactness there exists a subsequence µnk weakly-∗ converging to a measure µ∞. We now claim that µ∞ is a τg-invariant probability measure. Indeed, by deﬁnition of µn, it follows that

(τg)# µnk → µ∞ =⇒ (τg)# µ∞ = µ∞.

Step 2. We want to prove that the intersection of all the Pg is nonempty, which is clearly enough to infer the existence of an invariant measure.

Let g, h ∈ G be two elements, let µ0 ∈ Pg be an invariant measure, and let µ∞ be the weakly-∗ limit of the sequence

µ0 + (τh) µ0 + ··· + (τhn ) µ0 µ := # # ∈ P. n n + 1

The set Pg is weakly-∗ closed; therefore µ∞ ∈ Pg ∩ Ph. By induction we can prove that the family {Pg}g∈G has the ﬁnite intersection property, and thus, by compactness of G, it immediately follows that \ Pg 6= ∅. g∈G

Step 3. We want to prove that the intersection above contains only one element. In order to do that, we deﬁne the convolution product of two measures by setting

µ1 ∗ µ2(E) := (µ1 × µ2)({(x1, x2) | x1 + x2 ∈ E }) .

The reader may prove that the convolution is commutative, and also that

µ1 ∗ µ2 = µ1, if µ1 is an invariant measure.

This is enough to infer that the invariant measure is unique. Indeed, if λ, µ ∈ ∩g∈GPg are two invariant measures, then the properties above of the convolution implies that

µ = µ ∗ λ = λ ∗ µ = λ =⇒ µ = λ. Chapter 7

Lipschitz Functions

In the ﬁrst half of the chapter, we present some of the basic properties of Lipschitz maps and their relations to Hausdorﬀ measures. In the second half, we derive the so-called area formula for Lipschitz maps and we show the close relations with the coarea formula.

7.1 Deﬁnitions and Main Properties

In this section, we introduce Lipschitz function and Lipschitz maps (between metric spaces), and we state some of the main properties (which explains why they are such a good replacement in geometric measure theory for C1 functions.) Deﬁnition 7.1 (Lipschitz). Let X and Y be metric spaces. A map f : X −→ Y is Lipschitz if there exists a positive constant L > 0 such that

dY (f(x), f(y)) ≤ L · dX (x, y) for every x, y ∈ X. The Lipschitz constant of f is the optimal one, that is,

Lip(f) := inf {L > 0 | dY (f(x), f(y)) ≤ L · dX (x, y) for every x, y ∈ X } .

Compactness. We now present an Ascoli-Arzel`aresult, that is, a compactness criteria for Lipschitz maps.

Theorem 7.2. Let (fn)n∈N be a sequence of maps fn : X −→ Y between compact metric spaces, and assume that it is uniformly Lipschitz, that is, each fn has the same Lipschitz constant.

Then there exists a subsequence (fnk )k∈N converging uniformly to a Lipschitz map f∞ : X −→ Y with the same Lipschitz constant.

This result follows immediately from an application of the typical Ascoli-Arzel`atheorem for metric spaces, but the reader may notice that the assumptions are not optimal. Indeed, one may only assume that X is locally compact and Y is a pre-compact space. 79 7.2. DIFFERENTIABILITY OF LIPSCHITZ FUNCTIONS

Extension Property. Lipschitz functions f : X −→ R have an excellent extension property, which preserves the Lipschitz constant so that, even if it is deﬁned on a subset E ⊂ X, we can always talk about f as a function deﬁned on X.

Lemma 7.3 (Mc Shame). Let f : E ⊂ X −→ R be a Lipschitz function. There exists F : X −→ R such that

F E = f and Lip(F ) = Lip(f).

Proof. Let L := Lip(f), and set

F (x) := inf {f(y) + L · dX (x, y) | y ∈ E } . (7.1)

The reader may prove by themselves that F is an extension of f, and also that F is a L-Lipschitz map (since it is a lower envelope.)

The extension is, in general, not unique. Indeed, we can easily define a different extension by setting Fe(x) := sup {f(y) − L · dX (x, y) | y ∈ E } . (7.2) The reader may check that (7.1) and (7.2) define, in general, two different extension of f. The extension property is also true for Y := Rm-valued functions, but the argument above does not preserve the Lipschitz constant in general.

Lemma 7.4. Let f : E ⊂ X −→ Rm be a Lipschitz function. There exists F : X −→ Rm such that √ F E = f and Lip(F ) ≤ m · Lip(f).

Proof. Let L := Lip(f). We may extend each component by setting

Fi(x) := inf {fi(y) + L · dX (x, y) | y ∈ E } . (7.3)

The reader may prove by themselves that F = (F1,...,Fm) is an extension of f satisfying the properties mentioned above.

If X and Y are metric spaces, then there is no guarantee that such an extension exists. On the other hand, if they are Hilbert spaces, then there is a highly nontrivial theorem that proves the existence of an extension that preserves the Lipschitz constant. Theorem 7.5 (Kirszbraun). Let f : E ⊆ X −→ Y be a Lipschitz map between Hilbert spaces. Then there exists F : X −→ Y such that

F E = f and Lip(F ) = Lip(f).

7.2 Diﬀerentiability of Lipschitz Functions

In this section, we investigate the Lusin property of Lipschitz maps (from Rn to Rm) with C1 maps and, in particular, we prove that Lipschitz maps are Ln-almost everywhere diﬀer- entiable. 80 7.2. DIFFERENTIABILITY OF LIPSCHITZ FUNCTIONS

Theorem 7.6 (Lusin Property). Let f : Rn −→ R be a Lipschitz map. Then, for every 1 n > 0 there exists a function g ∈ C (R ; R) satisfying the following properties:

(1) The two maps coincide up to a set of measure at most , that is,

n n L ({x ∈ R | f(x) 6= g(x)}) ≤ .

(2) The Lipschitz constant is bounded from above, that is,

Lip(g) ≤ Lip(f).

We will not prove this theorem entirely as we will only sketch the proof of a ”local” statement. On the other hand, at the end of the section we brieﬂy show how to obtain the global statement via a partition of unity (but this is a highly nontrivial result!)

Regularity. First, we prove that Lipschitz maps from Rn to R (and thus to Rm) are almost everywhere diﬀerentiable with respect to the Lebesgue measure.

Theorem 7.7 (Rademacher). Let f : Rn −→ R be a Lipschitz map. Then f is Ln-almost everywhere differentiable. Remark 7.1. The extension property (i.e., Mc Shame Lemma 7.3) automatically gives us the Rademacher theorem for Lipschitz maps defined on a subset E ⊂ Rn. Remark 7.2. The statement of the Rademacher theorem presented here is not the most general possible. Indeed, the proof we are about to give works for Lipschitz maps with values in any finite-dimensional normed space. On the other hand, for infinite-dimensional Banach spaces, the statement is usually false. There is a particular class of Banach spaces for which the theorem holds, usually referred in the literature as Banach spaces with the Lusin property).

n 1, ∞ n Lemma 7.8. Let f : R −→ R be a Lipschitz map. Then f belongs to Wloc (R ), that is, the distributional gradient Df is essentially bounded.

1 n n Sketch of the Proof. Given a locally summable f ∈ Lloc(R ), and given a direction v ∈ R , it is a well-known fact that the distributional directional derivative is given by ∂f f − τ f = lim hv . ∂v h→0 h Since f is a Lipschitz map, we easily infer that

f(x) − τhvf(x) sup ≤ Lip(f)|v|, n x∈R h that is, the directional distributional derivative belongs to L∞(Rn), and so also the distributional gradient. Remark 7.3. For every p ∈ [1, +∞) it turns out that

1, ∞ n 1, p n Wloc (R ) ⊂ Wloc (R ). 81 7.2. DIFFERENTIABILITY OF LIPSCHITZ FUNCTIONS

n 1, p n Theorem 7.9. Let f : R −→ R be a continuous Sobolev function, that is f ∈ Wloc (R ) for some p ∈ (n, +∞). Then f is Ln-almost everywhere diﬀerentiable, and the pointwise gradient agrees with the distributional gradient for Ln-almost every x ∈ Rn.

Proof. Fix a ball B := B (x, r).

Step 1. We claim that

Z 1/p ω (f, B) ≤ C(n) · r − |∇f(x)|p dx , (7.4) B where C(n) is a universal constant1 and ω is oscillation function, that is,

ω (f, B) := sup {|f(x) − f(x)| | |x − x| < r } .

Step 2. In order to prove (7.4), we notice that we can always reduce to the case B = B (0, 1) by composing with the transformation x − x x 7−→ . r The Poincar´einequality gives the inequality2

Z Z − p f(x) dx − f(x) dx ≤ c(n)k∇fkL (B), B(0, 1) B(0, 1) from which we infer that

Z p − Φ(f) = k∇fkL (B(0, 1)) + f(x) dx B(0, 1) is equivalent to the usual W 1, p(B(0, 1)) norm. Therefore, we apply Sobolev embedding theorem and obtain the inequality " # Z p − kfk∞,B(0, 1) ≤ c1(n) k∇fkL (B(0, 1)) + f(x) dx . B(0, 1) In conclusion, it turns out that " # Z p − ω (f, B) ≤ 2 kfk∞ ≤ 2 c1(n) k∇ fkL (B(0, 1)) + f(x) dx , B(0, 1) and, by replacing f with f − av(f), we infer that (7.4) holds true since the left-hand side does not change if we add a constant to the function.

1We say that a constant is universal if it depends only on the dimension of the ambient space 2The constant depends only on the dimension n since the domain is the unitary ball! 82 7.2. DIFFERENTIABILITY OF LIPSCHITZ FUNCTIONS

Step 3. For every h ∈ Rn it turns out that Z 1/p |f(x + h) − f(x)| ≤ C(n)|h| − |∇f(x)|p dx . B Therefore, if L is a linear application, it turns out that

Z 1/p |f(x + h) − f(x) − Lh| ≤ C(n)|h| − |∇f(x) − L|p dx , B and this concludes the proof. Indeed, if x is a point of Lp-approximate continuity of ∇f ∈ p n L loc(R ), and if we take L := ∇f(x), then it turns out that |f(x + h) − f(x) − ∇f(x)h| ≤ C(n)|h|O(1) = C(n) O(khk).

The Rademacher theorem for Lipschitz maps follows easily combining the lemma and the theorem above. We are ﬁnally ready to state the local version of the Lusin property theorem and to give, at least, the main ideas behind it.

Theorem 7.10. Let f : Rn −→ R be a Ln-almost everywhere diﬀerentiable map, and let Ω n be an open bounded subset of R . For every > 0 there exist a compact subset K ⊂ Ω and 1 n a function g ∈ C (R ; R) satisfying the following properties:

(1) The two maps coincide up to a set of measure at most , that is,

n f = g and L (Ω \ K) ≤ . K K

(2) The Lipschitz constant is bounded from above, that is,

Lip(g) ≤ Lip(f) +

provided that f is Lipschitz.

Proof. Let n D := {x ∈ R | f is diﬀerentiable at x} be the set of points where f is diﬀerentiable. Then, for any x ∈ D, it turns out that

f(x + h) − ∇f(x)h ω(x, r) := sup → 0 |h|

+ decreasingly as r → 0 . For every > 0 we can ﬁnd a compact set K ⊂ Ω such that

n (a) L (Ω \ K) ≤ ;

(b) ∇f is continuous at each point of K (Lusin);

+ (c) ωr(x) & 0 as r → 0 , uniformly with respect to x ∈ K. 83 7.3. AREA FORMULA FOR LIPSCHITZ MAPS

3 n The reader may check these assertions as an exercise . Let A = R \ K be the complement of K in Ω; for any integer i ∈ Z we consider the set 1 1 Ai := x ∈ A < d(x, K) < . 2i+1 2i−1

Clearly {Ai}i∈Z is a covering of A; hence there exists a smooth partition of unity {σi}i∈Z subordinated to it, with the additional property

i+3 |∇σi| ≤ 2 . Let ρ be a regularizing kernel with support contained in the unitary ball, and assume also R R that ρ = 1 and xρ = 0. If we let ρi := ρ 1 be the rescaling, then we can set 2i+1  f(x) if x ∈ K ,  g(x) := X (f(x)σi(x)) ∗ ρi(x) if x∈ / K.  i∈Z To conclude the proof the reader may check that

(a) f ≡ g on K; (b) g is smooth on A;

7.3 Area Formula for Lipschitz Maps

The main result of this section is the following theorem. Theorem 7.11. Let Σ be a d-dimensional surface of class C1, and let f :Ω −→ Σ be a Lipschitz function deﬁned on an open subset Ω ⊆ Rd. Then for every E ⊆ Ω Borel it turns out that Z Z d d mf, E(y) dH (y) = Jf(x) dL (x), (7.5) Σ E −1 4 where mf, E(y) := # f (y) ∩ E and Jf(x) denotes the Jacobian of f at x. Remark 7.4. If f is an injective Lipschitz function, the area formula (7.5) reduces to an expression we are more familiar with (the substitution of variables in integral calculus): Z Hd (f(E)) = Jf(x) dHd(x). (7.6) E Remark 7.5. The codomain Σ does not need to be a d-dimensional surface, but it is enough to require Σ Riemannian manifold since a scalar product is all we need to deﬁne orthonormal basis.

3Hint. The second assertion follows from the Lusin theorem, while the third one follows from the Egorov theorem. 4The Jacobian, in this abstract setting, will be deﬁned below in Deﬁnition 7.14. 84 7.3. AREA FORMULA FOR LIPSCHITZ MAPS

Remark 7.6 (Jacobian, I). If Σ is a d-dimensional surface embedded in Rn for some n and f is diﬀerentiable at x, then the Jacobian of f at x can be easily computed using the standard formula q T Jf(x) = det (∇tf(x)) (∇tf(x)), (7.7) where ∇t denotes tangent gradient. More precisely, if we look at f as a function from Ω to Rn, then the diﬀerential at x is a linear map ∼ d n dfx : TxΩ = R −→ R . By choosing orthonormal bases of both Rd and Rn, we obtain a matrix that represents the linear map dfx, which is also the matrix representing ∇tf(x).

∗ ∼ n d Proof. We notice that the adjoint map (dfx) sends Tf(x)Σ = R to R since the scalar product canonically identiﬁes a vector space with its dual. The linear map ∗ (dfx) ◦ dfx : TxΣ −→ TxΣ

T is represented by the square matrix (∇tf(x)) (∇tf(x)) by construction, and thus we infer that the formula (7.7) holds. Remark 7.7. The main result of this section (7.5) is right as it is stated, but the reader may notice that the proof we present completely ignores the measurability issues5.

A weaker statement, which can be proved as an exercise, with no measurability issue to check, is given by the following theorem.

Theorem 7.12. Let f :Ω ⊆ Rd −→ Rn be a Lipschitz function deﬁned on an open subset Ω ⊆ Rd. Then for every F ⊂ Rn Borel set it turns out that Z Z # f −1(y) dHd(y) = Jf(x) dx. (7.8) F f −1(F )

A stronger (more general) statement, which requires a lot of work to be proved rigorously, is given by the following theorem. Theorem 7.13. Let Σ be a d-dimensional surface of class C1, and let f :Ω −→ Σ be a Lipschitz function deﬁned on an open subset Ω ⊆ Rd.For every positive Borel function h :Ω −→ [0, +∞], it turns out that Z X Z [ h(s)] dHd(x) = h(x)Jf(x) dx. (7.9) Σ s∈f(x) Ω

Area Formula. In this ﬁnal paragraph, the main goal is to sketch the proof of (7.5). The ﬁrst step is, as promised, to give meaning to the Jacobian of a Lipschitz function.

Definition 7.14 (Jacobian). Let f :Ω ⊆ Rd −→ Rn be a function differentiable at x, and d ≤ n. The d-dimensional Jacobian of f at x, denoted by Jdf(x), is defined by setting

( d P is a d-dimensional ) H (∇tf(x)(P )) Jdf(x) = sup . Hd[P ] d parallelepiped contained in R 5A nice exercise would be to clean up the proof and make it rigorous. 85 7.3. AREA FORMULA FOR LIPSCHITZ MAPS

We are now ready to prove the area formula (7.5). We shall see soon that it is an immediate consequence of two technical lemmas.

Lemma 7.15. Let A ⊆ Ω be an open set such that the restriction f A is a diﬀeomorphism, that is, injective and with maximal rank:

rank (df(s)) = d.

Then the push-forward measure

d µ := f# 1A(x)Jf(x) ·L is equal to the measure d 1f(A) ·H . In particular, given the subset F = f(A) ⊆ Σ as in (7.8), it turns out that Z Hd (F ) = Jf(s) ds, A that is, the area formula (7.5) holds if F = f(E) for every E Borel. Lemma 7.16. Let E ⊆ Ω be a Borel set such that Jf(s) = 0 for every s ∈ E (i.e., the rank of the diﬀerential map is not maximal). Then

Hd (f(E)) = 0 and the area formula (7.5) holds for every such E.

Suppose, for now, that both lemmas hold true. Then, for every Borel set E ⊂ Ω, it is enough to split it into countably many pieces Ei satisfying the following properties:

i) The restriction f is a diﬀeomorphism for every i ∈ I. Ei ii) The complement set [ E \ Ei i∈I is the set of all points s ∈ E such that the diﬀerential of f at s has rank strictly less than d.

Proof of Lemma 7.15. First, we notice that the thesis follows easily if we can prove that the Radon-Nikodym density dµ (s) = 1 (7.10) d d 1f(A) ·H for every s ∈ f(A).

Step 0. Suppose that (7.10) holds. The measure µ is supported, by deﬁnition, in the set f(A) and it has no singular part6; hence

d d f# 1A Jf ·L = 1f(A) ·H .

6Indeed, if µ has a singular part, then there exists a point s ∈ f(A) such that (7.10) is +∞, against our assumption. 86 7.3. AREA FORMULA FOR LIPSCHITZ MAPS

Step 1. Recall that

dµ µ (B(s, r)) + (s) = 1 ⇐⇒ −−−−→r→0 1, d d d 1f(A) ·H H ((B(s, r)) ∩ Σ) and thus it is enough to ﬁnd an asymptotic estimate of both the numerator and the denominator.

n Step 2. If πs : R −→ TsΣ is the orthogonal projection onto TsΣ, then it turns out that (1 − o(1)) · |x − y| ≤ |π(x) − π(y)| ≤ |x − y| that is, the projection is an almost-isometry. It follows that

d d d H (B(s, r) ∩ Σ) ∼ H (π (B(s, r) ∩ Σ)) ∼ αdr .

Step 3. In a similar fashion, given a ball Br := B(0, r) ⊆ TsΣ, one can consider the ellipsoid7 −1 Er := df (Br) .

The image of Er according to f has measure

µ (f(s + Er)) ∼ µ (B(s, r)) , and, on the other hand, it is easy to show that Z |Br| µ (f(s + Er)) = Jf(x) dx = Jf(s) |s + Er| = Jf(s) = |Br|. s+Er Jf(s) In particular, it turns out that

d µ (B(s, r)) ∼ µ (f(s + Er)) ∼ |Br| = αdr , which is exactly what we wanted to prove.

Proof of Lemma 7.16. By assumption, for every point s ∈ E and positive real number r > 0 small enough, it turns out that

f (B(s, r)) ⊆ Bk

Hd (f(B(s, r))) 'O(r)k · o(r)d−k ∼ o(rd), and this is enough to conclude since we can cover the ellipsoid with a countable union of balls satisfying the estimate above, that is, k X d O(r) diam ((B )) ∼ · o(r)d ∼ o(rd), i o(r) i which is exactly what we wanted to prove.

7The preimage of a ball via the diﬀerential is an ellipsoid because the rank is maximal. 87 7.4. COAREA FORMULA FOR LIPSCHITZ MAPS

7.4 Coarea Formula for Lipschitz Maps

Coarea. Let f :Ω ⊆ Rn −→ Rm be a map of class C1, and assume that m < n. If h :Ω −→ [0, +∞] is a positive function, then the coarea formula is given by

Z "Z # Z h(x) dHn−m(x) dy = h(x)J(f)(x) dx, (7.11) n −1 R f (y)∩Ω Ω where r t J(f)(x) = det (∇f(x)) (∇f(x)) ∈ M(n × n, R) is a suitable notion of Jacobian. The proof of formula (7.11) is simple, and it is left to the reader to ﬁll in the details in what follows.

Sketch of the Proof.

Particular Case. If f : R2 −→ R is the projection over the ﬁrst coordinate, then the formula reduces to the Fubini-Tonelli theorem.

General Case. In the general case, the main idea is to split the set Ω as

Ω = Ωs t Ωn, where Ωs is the set of all singular points (i.e., all the x ∈ Ω such that the Jacobian at x has non-maximal rank), and Ωn is the set of all points such that the Jacobian is a matrix of maximal rank. The formula (7.11) clearly holds at every point of the ﬁrst kind since the Jacobian is not invertible (i.e., J(f)(x) = 0), while the set f −1(x) ∩ Ω is Hn−m-null as expected. The formula for regular points, on the other hand, can be easily deduced from the particular case discussed above (f : R2 −→ R projection) via a simple change of variables. 88 7.4. COAREA FORMULA FOR LIPSCHITZ MAPS

Figure 7.1: Idea of the coarea formula: integrate ﬁrst along the blue lines, and then integrate the result along the violet lines Chapter 8

Rectiﬁable Sets

In this chapter, we shall be mainly concerned with the notions of d-dimensional rectifiable and purely unrectifiable sets in metric spaces. In the first section, we furnish the definitions of rectifiable (and purely unrectifiable) set through Lipschitz maps, and we state some of the main criteria in Rn. In the second section, we work towards a (suitably weak) definition of the tangent bundle for rectifiable sets, and, in the next section, we end up proving that any Borel set E with an approximate tangent cone and d-dimensional lower density bounded from below is d- rectifiable. In the last section, we prove the area formula for rectifiable sets using the characterization as the union of the graphs of Lipschitz maps.

8.1 Introduction and Elementary Properties

Definition 8.1 (Rectifiable Set). Let X be a metric space. A Borel set E ⊆ X is a d- dimensional rectifiable set if and only if there exists a collection of Borel sets {Ei}i∈N such that +∞ [ E = Ei, i=0 satisfying the following properties:

d 1) E0 is a H -null set, and

d d 2) Ei ⊂ fi(R ), where fi : R −→ X is a Lipschitz function for every i ≥ 1. Remark 8.1. Let X be a metric space, and let E ⊂ X be a d-dimensional rectiﬁable set. Then the Hausdorﬀ dimension of E is less than or equal to d, i.e.,

dimHE ≤ d 90 8.1. INTRODUCTION AND ELEMENTARY PROPERTIES

Proof. Let {Ei}i∈N be the collection given by the deﬁnition of d-rectiﬁable set. The assump- d tion that E0 is a H -null set immediately implies that

dimHE0 ≤ d.

On the other hand, by Lemma 2.21 it follows that a Lipschitz map does not increase the Hausdorﬀ dimension, that is,

d d dimHR = d =⇒ dimHfi(R ) ≤ d, which is enough to infer (as a consequence of Remark 2.13) that [ E = Ei =⇒ dimHE = sup dimHEi ≤ d. i∈N i∈N

Exercise 8.1. Prove that Hd(E) = 0 does not imply that E is contained in a countable union of Lipschitz images of Rd, that is,

[ d E 6⊆ fn(R ). n∈N

2 d Exercise 8.2. Prove that there exists a Borel set E0 ⊆ R such that H (E0) = 0 which cannot be covered by a rectiﬁable curve (hint: self-similar sets).

Proposition 8.2 (Criteria of Rectiﬁability). Let X := Rn, and let E ⊂ X be a Borel set. The following assertions are equivalent:

(a) The set E is d-rectiﬁable.

(b) There exists a collection of Borel sets {Ei}i∈N such that

+∞ [ E = Ei, i=0 satisfying the following properties:

d 1) E0 is a H -null set. d n 2) Ei ⊂ fi(Ai), where Ai ⊆ R is an open subset and fi : Ai −→ R is a diﬀeren- tiable function.

+∞ [ E = Ei, i=0 satisfying the following properties:

d 1) E0 is a H -null set. d n 2) Ei ⊂ fi(Ai), where Ai ⊆ R is an open subset and fi : Ai −→ R is a diﬀeomor- phism. 91 8.1. INTRODUCTION AND ELEMENTARY PROPERTIES

(d) There exists a collection of Borel sets {Ei}i∈N such that

+∞ [ E = Ei, i=0 satisfying the following properties:

d 1) E0 is a H -null set. 1 n 2) Ei ⊂ Σi, where Σi is a d-dimensional surface of class C contained in R .

Proof. By deﬁnition (diﬀerentiable maps are Lipschitz) (b) =⇒ (a), while the opposite implications is a straightforward consequence of the following result.

d n Lemma 8.3. Let f : R −→ R be a Lipschitz map. Then there is a collection {fi}i∈N of functions 1 d of class C and a H -null Borel set A0 such that

d [ d f(R ) ⊆ fi(R ) ∪ A0. i∈N Proof. By Theorem 7.6, Lipschitz maps have the Lusin property in the class of diﬀerentiable maps 1 d n C ; hence, for every i ∈ N, there exists a diﬀerentiable map fi : R −→ R such that

d 1 |Ei| := x ∈ | fi(x) 6= f(x) ≤ . R i The reader may easily prove that, by construction, it turns out that ! !! d [ d \ f(R ) ⊂ fi(R ) ∪ f Ei . i∈N i∈N In particular, the image of the intersection is a Hd-null set since ! ! d \ d \ L Ei = 0 =⇒ H Ei = 0, i∈N i∈N and f is a Lipschitz map (see Lemma 2.21).

Clearly (c) =⇒ (b) (since diﬀeomorphisms are also diﬀerentiable maps); thus we only need to prove the opposite implication.

Suppose (b) holds, and let {Ei}i∈N be the collection of Borel sets satisfying the deﬁnition, i.e,, +∞ [ E = Ei, i=0 92 8.1. INTRODUCTION AND ELEMENTARY PROPERTIES

d n where E0 is a H -null set and Ei ⊂ fi(Ai) for differentiable maps fi : Ai −→ R . For every i ≥ 1 we consider a partition of Ai max min Ai = Ai t Ai , where max Ai := {x ∈ Ai | dfx has maximal rank} and min Ai := {x ∈ Ai | dfx has NOT maximal rank} . min d Recall that Ai is a H -null set for every differentiable map fi. Hence, we can consider the collection of Borel sets given by   [ min max Ef0 := E0 ∪  f Ai  and Efi := fi (Ai ) i≥1 together with the diffeomorphisms fi max . Ai In conclusion, it remains to prove the equivalence (c) =⇒ (d). The ”only if” part is immediate from the definitions, while the ”if” part is left to the reader as a simple exercise (in the same spirit of the implications we have already proved here).

Remark 8.2. If X := Rn, we just proved that Lipschitz maps may be replaced by C1 maps in the definition of d-rectifiable set. On the other hand, it is not possible to replace C1 by a more regular space Ck≥2 since C1 does not have the Lusin property in C2. Definition 8.4 (Purely Unrectifiable). Let X be a metric space. A Borel set E ⊆ X is a d-dimensional unrectifiable set if and only if d d d H E ∩ f R = 0 for every Lipschitz map f : R −→ X. Proposition 8.5 (Criteria of Unrectifiability). Let X := Rn, and let E ⊂ X be a Borel set. The following assertions are equivalent:

(a) E is purely d-unrectiﬁable.

(b) For every C1 map f : Rd −→ Rn it turns out that d d H E ∩ f R = 0.

(d) For every d-dimensional surface Σ ⊆ Rn of class C1 it turns out that Hd (E ∩ Σ) = 0. Notation. From now on, we will use purely unrectifiable and 1-purely unrectifiable as synonyms. 2 2 Proposition 8.6. Let X = R . For every d ∈ (0, 2] there exists a compact set Kd ⊂ R such that dimH K = d and K purely unrectifiable.

Proof. Here we present the proof of the assertion for d ∈ (0, 2). The reader may extend the construction to dimension 2 as an exercise. 93 8.1. INTRODUCTION AND ELEMENTARY PROPERTIES

Step 1. First, we claim that if K = K1 × K2 is a product of compact subsets of R, then

1 1 L (K1) = L (K2) = 0 =⇒ K purely unrectiﬁable.

Indeed, let C be a curve of class C1 (a submanifold of R2). Since C is locally the graph of a function, we may assume, without loss of generality, that

1 C = (x1, g(x1)) g ∈ C (R; R) . The 1-dimensional Hausdorﬀ measure of the intersection K ∩ C may be computed by parts, and thus we can use the assumption on the Lebesgue measure to obtain Z 1 1 p 2 H (K ∩ C) ≤ H ((K1 × R) ∩ C) = 1 +g ˙ dt = 0. K1 Therefore, we conclude that K is purely 1-unrectiﬁable as a consequence of Proposition 8.5.

0 Step 2. If K1 = K2 are self-similar fractals (in Hutchinson sense) of dimension d , then 0 the product K = K1 × K2 has Hausdorﬀ dimension 2d . Indeed, the reader may prove that for any two separable metric spaces X and Y with Y totally bounded that

dimH X + dimH Y ≤ dimH (X × Y ) ≤ dimH X + dimB Y, where dimB Y denotes the upper box counting dimension of Y (see, e.g. [8]). In particular, if Y has equal Hausdorﬀ and upper box counting dimension (which holds if Y is a compact interval) then dimH X + dimH Y = dimH (X × Y ) .

Step 3. In conclusion, in Section 2.6 we have proved that there is a Cantor-type set of every dimension d ∈ (0, 1). Hence, for every d ∈ (0, 1), the product Cd × Cd is a purely unrectiﬁable set of dimension 2d ∈ (0, 2), which is exactly what we wanted to exhibit.

Proposition 8.7. Let E ⊂ X be a Borel set with finite Hausdorff measure Hd(E) < +∞. Then E is the union of a d-rectifiable part and a purely d-unrectifiable part, that is,

E = Er ∪ Epu.

Proof. Let us consider the family

F = {F ⊂ E | F is a d-rectiﬁable subset} .

d Let m := sup H (F ) | F ∈ F , and let (Fn)n∈N ⊂ F be a maximizing sequence, that is,

d H (Fn) % m.

If we set [ Er := Fn, n∈N 94 8.1. INTRODUCTION AND ELEMENTARY PROPERTIES

then it turns out that Er ∈ F (being rectiﬁable is closed under countable unions), and the Hausdorﬀ dimension of Er is exactly m. In conclusion, we set

Epu := E \ Er,

1 and we prove , by contradiction, that Epu is purely d-unrectiﬁable.

0 Indeed, suppose that E := E \ Er is not purely d-unrectiﬁable. Let us consider the family F 0 = {F ⊂ E0 | F is a d-rectiﬁable subset} , 0 0 which is not empty by assumption. Let m := sup {dimH F | F ∈ F }, and let (Fn)n∈N ⊂ F be a maximizing sequence, that is,

d 0 H (Fn) % m .

If we set 0 [ Er := Fn, n∈N 0 0 0 then it turns out that Er ∈ F , and the Hausdorff dimension of Er is exactly m . In particular, it turns out that 0 Efr := Er ∪ Er is a d-rectifiable set with Hausdorff dimension ≥ m; thus we have derived a contradiction with the maximality of Er.

Characterization of d-rectifiable sets in Rn. In this final paragraph, we state two fundamental criteria for rectifiable sets (but we do not give the proof of them).

Theorem 8.8 (Marstrand-Preiss). Let E ⊆ Rn be a Borel set. If Hd(E) < +∞ and Hd (E ∩ B(x, r)) lim d 6= 0, ∞ r→0 αdr exists for Hd-almost every x ∈ E, then d is an integer, and E is a d-rectiﬁable set.

Remark 8.3. If E ⊆ Rn is a d-rectiﬁable set with positive Hausdorﬀ measure Hd(E) > 0, then for almost every V ∈ G(n, d) it turns out that

d H (PV (E)) > 0.

Theorem 8.9 (Besicovitch-Federer). Let E ⊆ Rn be a Borel set. If E is purely d- unrectiﬁable, then d H (PV (E)) = 0 for γn, d-almost every V ∈ G(n, d).

Proof. This result is the main topic of my seminary. I will try to upload some written notes about it after the exam.

1The reader may prove this assertion as an exercise. 95 8.2. TANGENT BUNDLE

8.2 Tangent Bundle

In this section, we want to introduce a weaker notion of the tangent bundle for rectiﬁable sets. From now on, we will denote by E a d-rectiﬁable set in Rn unless stated otherwise. Proposition 8.10. Let Σ and Σ0 be d-dimensional surfaces of class C1. Then the tangent planes are equal at almost every point in the intersection x ∈ Σ ∩ Σ0, that is,

0 Tx Σ = Tx Σ .

The proof of this assertion is a mere consequence of the following Lemma since the set of all points x ∈ Σ ∩ Σ0 such that Σ is transversal to Σ0 (i.e., where the tangent planes do not coincide) is negligible for the Hausdorﬀ/Lebesgue measure.

Lemma 8.11. Let f, fe : Rd −→ R be functions of class C1(Rd). Then

∇f(x) = ∇fe(x) for Ld-almost every point x ∈ Rd such that f(x) = fe(x).

Proof. We may always assume without loss of generality that fe ≡ 0. Let K denote the set of the zeros of f, and let x ∈ K be a density point, that is,

Hd (K ∩ B(x, r)) lim d = 1. r→0 αd r

Suppose that ∇f(x) ≥ > 0. By continuity, there exist a positive constant δi > 0 and a n point yi ∈ R such that f(yi) 6= 0 and yi ∈ B(x, δi). In particular, by choosing a suitable sequence δi & 0, it turns out that δ B y , i ∩ K = . i 2 ∅

We are now ready to derive a contradiction. The Hausdorﬀ density of K at x cannot be equal to one because there is a sequence of balls shrinking down to x that does not intersect K. More precisely, it turns out that

Hd (Kc ∩ B(y , r )) Hd (K ∩ B(x, r)) lim i i > 0 =⇒ lim < 1. i→+∞ d r→0 d αd ri αd r

Deﬁnition 8.12 (Tangent Bundle). Let E be a Borel d-rectiﬁable set. A map T from E to the Grassmannian manifold G(n, d) that sends x to T (x) is a weak tangent bundle for the set E if and only if for every Σ d-dimensional surface of class C1 it turns out that

Tx Σ = T (x) for Hd-almost every x ∈ Σ ∩ E.

Proposition 8.13. A d-rectiﬁable Borel set E ⊂ Rn admits a ”unique” - up to Hd negligible sets - weak tangent bundle. 96 8.2. TANGENT BUNDLE

Proof. By Proposition 8.2 there exists a collection of Borel subsets {Ei}i∈N such that +∞ [ E = Ei, i=0 satisfying the following properties:

d (1) The set E0 is H -null. n 1 (2) For every i ≥ 1 there is a d-dimensional surface Σi ⊂ R of class C such that Ei ⊂ Σi.

Therefore, we set  T Σ x ∈ E ,  x 1 1 T (x) := Tx Σ2 x ∈ E2 \ E1, ..., and, for every point [ x∈ / Ei, i≥1 we set T (x) to be equal to any d-dimensional vector space. The reader may easily check S c that T is the sought map since it is unique up to the negligible set i≥1 Ei .

Approximate Tangent. In this brief paragraph, we introduce an even weaker notion called approximate tangent plane. From now on, we shall assume that E is a d-rectifiable Borel set, which is locally Hd-finite, that is for every p ∈ E there is an open neighborhood Up 3 p such that d H (Up) < +∞. Definition 8.14 (Cone). Let α be a fixed angle, let x ∈ Rn be a point and let V be a d-dimensional plane in Rn. The cone of angle α around V centered at x is defined by setting 0 n 0 0 C (x, V, α) := {x ∈ R : |x − x| · sin(α) ≥ d(x − x ,V )} . Definition 8.15 (Strong Tangent Plane). Let V ∈ G(n, d) be a d-dimensional plane. If E is a Borel set and x ∈ E a point, then V is a strong tangent plane to E at x if and only if for every α > 0 there exists a positive radius r0 > 0 such that

E ∩ B(x, r) ⊆ C (x, V, α) for every r ≤ r0. Definition 8.16 (Approximate Tangent Plane). Let V ∈ G(n, d) be a d-dimensional plane. If E is a Borel set and x ∈ E a point, then V is an approximate tangent plane to E at x if and only if for every α > 0 it turns out that Hd ((E ∩ B(x, r)) \C (x, V, α)) = o(rd), (8.1) and d d H ((E ∩ B(x, r)) ∩ C (x, V, α)) ∼ αdr . (8.2) Theorem 8.17. Let E ⊆ Rn be a Borel set. If E is a d-rectifiable Hd-locally finite set, then the weak tangent bundle T (x) is the approximate tangent plane to E at x for Hd-almost every x ∈ E.

Proof. We divide the argument into diﬀerent steps to highlight what we are doing here. 97 8.2. TANGENT BUNDLE

Step 1. First, observe that the thesis is equivalent to the following assertion: For every d i ≥ 1 the vector space Tx Σi is the approximate tangent plane to E at x for H -almost every x ∈ E ∩ Σi.

Step 2. Let us deﬁne the measures

1 d λ := Σi ·H ,

0 1 d µ := Σi\E ·H ,

00 1 d µ := E\Σi ·H .

Fix α > 0 and x ∈ Σi, and let V := Tx Σi be the d-dimensional tangent plane. By construction, it turns out that

Hd ((E ∩ B(x, r)) ∩ C (x, V, α)) = λ (B(x, r) ∩ C (x, V, α)) − µ0 (B(x, r) ∩ C (x, V, α)) +

··· + µ00 (B(x, r) ∩ C (x, V, α)) (8.3) since the restriction Hd ¬ E is equal to λ − µ0 + µ00.

Step 3. By deﬁnition, as r goes to 0

λ (B(x, r) ∩ C (x, V, α)) ∼ Hd (E ∩ B(x, r)) , and thus d λ (B(x, r) ∩ C (x, V, α)) ∼ αdr for r suﬃciently small.

Step 4. We now claim that

µ0 (B(x, r) ∩ C (x, V, α)) = o(rd)

d for H -almost every x ∈ E ∩ Σi. Indeed, we observe that

0 d µ (B(x, r) ∩ C (x, V, α)) ≤ H (B(x, r) ∩ (Σi \ E)) ,

2 and thus it suﬃces to show that the density of Σi \ E is 0 at almost every point of E with respect to the measure λ (which is, by the way, rather intuitive). Indeed, if

d H (B(x, r) ∩ (Σi \ E)) Θ (Σi \ E, λ, x) = d = 0, H (Σi ∩ B(x, r))

d d then the numerator must be an element of the class o(r ) since the denominator is ∼ αdr .

2This assertion is left as a simple exercise for the reader. 98 8.2. TANGENT BUNDLE

Step 5. In a similar fashion, we claim that µ00 (B(x, r) ∩ C (x, V, α)) = o(rd) d for H -almost every x ∈ E ∩ Σi. This is an easy consequence of the inequality µ00 (B(x, r) ∩ C (x, V, α)) ≤ µ00 (B(x, r)) . Indeed, the right-hand side is an element of the class o(rd) since µ00 is orthogonal to λ by construction, and thus the density with respect to λ is zero for λ-almost every x. More precisely, dµ00 Hd (B(x, r)) (x) = lim = 0, dλ r→0 λ (B(x, r)) d and the denominator is ∼ αdr as in the previous step.

Step 6. In the same spirit, the reader may prove that Hd ((E ∩ B(x, r)) \C (x, V, α)) = o(rd) using the same decomposition introduced in (8.3).

Exercise 8.3. Let Σ be a line in R2, and let ! [ E := Σ ∪ ∂B(xn, rn) , n∈N 2 where {xn}n∈N is a dense countable collection of points in R and (rn)n∈N is a summable family of positive real numbers. Prove that:

1) The set E is d-rectifiable. 2) The set E is locally H1-finite. 3) For almost every x ∈ Σ it turns out that H1 ((E \ Σ) ∩ B(x, r)) = o(r). Remark 8.4. According to the statement of Theorem 8.17, it might happen that the mass is distributed only on one side of the tangent plane (see Figure 8.1). To prove that the statement can be improved (i.e., the situation described above cannot happen), we need to introduce a different definition of approximate tangent plane, which is easier to deal with, and then prove that it is stronger than Definition 8.16.

Deﬁnition 8.18 (Approximate Tangent Plane). Let ψx, r : B(x, r) −→ B(0, 1) be the magnifying glass map, that is, x0 − x ψ (x0) := , x, r r and let Ex, r be the image of E via ψx, r. An element V of the Grassmannian manifold G(n, d) is an approximate tangent plane to the set E at the point x if and only if 1 d 1 d Ex, r ·H * V ·H locally weakly-∗, that is, Z Z d r→0 d 0 d ϕ(t) dH (t) −−−→ ϕ(t) dH (t), ∀ ϕ ∈ Cc (R ). Ex, r V 99 8.2. TANGENT BUNDLE

Figure 8.1: The mass is distributed only on one side of the approximate tangent plane.

Remark 8.5. If V is an approximate tangent plane to E at x in the sense of Deﬁnition 8.18, then V is an approximate tangent plane to E at x in the sense of Deﬁnition 8.16.

Proof. Let us consider the following measures:

1 d 1 d µx, r := Ex, r ·H and µx := V ·H .

The d-dimensional Hausdorﬀ measure of ∂B(0, 1) ∩ V is equal to zero since B(0, 1) ∩ V is a d-dimensional set (which means that the boundary has dimension d − 1); hence

r→0 µx, r * µx =⇒ µx, r (B(0, 1)) −−−→ µx (B(0, 1)) , as a consequence of Proposition 2.7. By deﬁnition, we have µx (B(0, 1)) = αd and 1 µ (B(0, 1)) = Hd (E ∩ B(0, 1)) = Hd (E ∩ B(x, r)) , x, r x, r rd which implies a condition weaker than (b), that is

d d + H (E ∩ B(x, r)) ∼ αdr as r → 0 .

In a similar fashion, it turns out that

r→0 µx, r (B(0, 1) ∩ C(0, V, α)) −−−→ µ (B(0, 1) ∩ C(0, V, α)) = αd, 100 8.2. TANGENT BUNDLE from which it follows that

d d + H (E ∩ B(x, r) ∩ C(0, V, α)) ∼ αdr as r → 0 , that is, the condition (b) holds. The reader may prove, in a similar way, that the condition (a) also holds true.

Theorem 8.19. Let E ⊆ Rn be a Borel set. If E is d-rectifiable and locally Hd-finite, then T (x) is the approximate tangent plane (Definition 8.18) to E at x for Hd-almost every x ∈ E.

Sketch of the Proof. We divide the argument into diﬀerent steps to ease the notation.

Step 2. Fix i ≥ 1 and let Σx, r be the image of Σi via ψx, r. We consider the measures

1 d λx, r := Σx, r ·H ,

0 1 d µx, r := Σx, r \Ex, r ·H ,

00 1 d µx, r := Ex, r \Σx, r ·H , in such a way that the following decomposition holds:

0 00 µx, r = λx, r − µx, r + µx, r.

Step 3. The reader may prove that

d λx, r * 1V ·H , locally weakly-∗, using the fact that the projection πx, r : σx, r −→ V is an almost-isometry.

Step 4. In a similar way to what we have done in the proof of Theorem 8.17, one can check that

0 0 r→0 µx, r * 0 ⇐⇒ µx, r (BR) −−−→ 0 for any ﬁxed radius R > 0 as a consequence of Proposition 2.7. We now observe that 1 µ0 (B(0, 1)) = Hd (B(0, 1) ∩ (Σ \ E )) = Hd (B(x, r) ∩ (Σ \ E)) −−−→r→0 0 x, r x, r x, r rd i

d d since H (B(x, r) ∩ (Σi \ E)) is an element of the class o(r ), as proved in Theorem 8.17. 101 8.3. RECTIFIABILITY CRITERIA

Step 5. Similarly, by Proposition 2.7 it follows that

00 00 r→0 µx, r * 0 ⇐⇒ µx, r (BR) −−−→ 0 for any ﬁxed radius R > 0.

We now observe that 1 µ00 (B(0, 1)) = Hd (B(0, 1) ∩ (E \ Σ )) = Hd (B(x, r) ∩ (E \ Σ )) , x, r x, r x, r rd i thus it suﬃces to prove that the right-hand side goes to zero as r → 0+. Indeed, the limit of the ratio is the Radon-Nikodym density, which is zero as a consequence of the fact that the two measures are orthogonal:

1 d 1 d r→0 d E\Σ ·H d H (B(x, r) ∩ (E \ Σi)) −−−→ d = 0. r d (1Σ ·H )

8.3 Rectiﬁability Criteria

The primary goal of this section is to relate d-rectifiable set with properties of the approximate tangent plane. More precisely, in this section we denote by V ∈ G(n, d) a d-dimensional plane, by E ⊆ Rn a Borel set, and we fix a point x ∈ Rn. Definition 8.20 (Tangent Cone). Fix α ∈ (0, π/2). The cone C(x, V, α) is tangent to a set E at x if there exists a positive constant r0 > 0 such that

B(x, r) ∩ E ⊆ C(x, V, α), ∀ r ≤ r0.

Theorem 8.21. Assume that E admits a tangent cone at every point x ∈ E. Then there d n are (at most) countably many Lipschitz functions fi : R −→ R such that

[ d E ⊆ fi(R ). i≥1

In particular, the set E is d-rectiﬁable.

Proof. We ﬁrst prove a particular case, and then we reduce the general case to it using a standard argument.

Step 1. Let us assume3 that:

a) The d-dimensional plane V , the angle α and the radius r0 do not depend on the point x ∈ E.

b) The linear space V is a straight d-dimensional plane.

3We will get rid of these extra assumptions later. It is important to understand that the argument presented in the ﬁrst step only proves a particular case. 102 8.3. RECTIFIABILITY CRITERIA

We divide the ambient space in strips of thickness r0 sin α, that is,

n [ R = Si, with |Si| = r0 sin α. i∈Z The set E is thus given by the union [ E = (E ∩ Si) , i∈Z which means that it is enough to prove that the intersection of E with each strip Si is contained in the graph of a Lipschitz map, that is,

E ∩ Si ⊂ Γ(gi),

⊥ 4 where gi : V −→ V is Lipschitz . By assumption, it turns out that (see Figure 8.2) the intersection of E with each strip is contained in the cone centered at any x ∈ E ∩ Si, that is, E ∩ Si ⊂ C(x, V, α) ∀ x ∈ E ∩ Si. Therefore, if we denote by (x, y) the coordinates associated with the decomposition V ⊕ V ⊥ = Rn, then one can prove that |y0 − y| ≤ tan(α) · |x0 − x|, which means that the y coordinate furnishes a Lipschitz map (see Figure 8.3).

Step 2. We are now ready to get rid of the extra assumptions, one by one.

0 (i) Assume that α and r0 do not depend on the point x ∈ E, and fix an angle α > α. The cones of the form C(0, V, α), as V ranges in the set of all admissible d-dimensional 0 planes, are contained in a finite family C(0,Vi, α ) of slightly ampler cones. We can reduce to the first step by splitting E in the finite union associated to the family above, that is, N [ E = Ei, i=1

with Vi ﬁxed d-dimensional straight plane for each i = 1,...,N.

(ii) Assume that the radius r0 does not depend on the point x ∈ E. For every α ∈ Q there exists α0(α) > α such that the cones of the form C(0, V, α), as V ranges in the set of all admissible d-dimensional planes, are contained in a given 0 ﬁnite family C(0,Vi, α (α)) of slightly ampler cones. We can reduce to the ﬁrst step by splitting E in the (at most) countable union

 Nα0(α)  [ [ E =  Ei . α∈Q i=1

4 n ⊥ n The graph of the function gi is a subset of R since gi sends V to its orthogonal, and V ⊕ V = R . 103 8.3. RECTIFIABILITY CRITERIA

(iii) Finally, if no extra assumption holds, then it suﬃces to consider the collection

Er := {x ∈ E | r0(x) < r }

and split E as follows  Nr, α0(α)  [ [ E =  Er, i . r, α∈Q i=1

Figure 8.2: Sketch of the ﬁrst step.

Deﬁnition 8.22 (Approximate Tangent Cone). Fix α ∈ (0, π/2). The cone C(x, V, α) is a approximately tangent to a set E at x if and only if

Hd ((B(x, r) ∩ E) \C(x, V, α)) = o(rd).

Theorem 8.23. Assume that E admits an approximate tangent cone at every point x ∈ E, and assume also that the lower density is bounded from below, that is,

Θ∗(E, x) > 0 ∀ x ∈ E.

d n Then there are (at most) countably many Lipschitz functions fi : R −→ R such that

[ d E ⊆ fi(R ). i≥1 In particular, the set E is d-rectiﬁable. 104 8.3. RECTIFIABILITY CRITERIA

Figure 8.3: The coordinates give a Lipschitz map.

Corollary 8.24. If E admits an approximate tangent cone at Hd-almost every point x ∈ E, and the lower density is bounded from below, i.e.,

d Θ∗(E, x) > 0 for H -almost every point x ∈ E, then the set E is d-rectiﬁable.

Proof of Theorem 8.23. We ﬁrst prove a particular case, and then we reduce the general case to it using a standard argument.

Step 1.1. Let us assume that:

a) The d-dimensional plane V and the angle α are the same for every points x ∈ E.

b) There exist r0 > 0 and δ > 0 such that for every x ∈ E it turns out that

Hd (E ∩ B(x, r)) ≥ δ · rd, (8.4)

for every r ≤ r0.

First, we notice that the existence of an approximate tangent plane implies that for every r ≤ r0 it turns out that

Hd ((E ∩ B(x, r)) \C (x, V, α)) ≤ δ0 · rd, (8.5) 105 8.3. RECTIFIABILITY CRITERIA where δ0 is a positive constant that can be chosen arbitrarily. To prove the particular case, we ﬁrst need to show that there exists a radius r > 0 such that

E ∩ (V × B (x0, r)) ⊂ Γ(g),

⊥ ⊥ where x0 ∈ V and Γ(g) is the graph of a Lipschitz function g : V −→ V .

Step 1.2. Fix an angle α0 ∈ (α, π/2), and ﬁx a point x ∈ E. We claim that r E ∩ B x , 0 ⊆ C (x, V, α0) . 0 2 We argue by contradiction. Suppose that there exists a point r x0 ∈ E ∩ B x , 0 \C (x, V, α0) , 0 2 and let r, r0 > 0 be positive real numbers such that

0 0 0 0 B(x , r ) ⊂ B(x, r) and B(x , r ) ∩ C(x, V, α) = ∅. More precisely, let us consider the following radii:

r := 2 |x − x0| and r0 := |x − x0| sin(α0 − α).

The assumptions (8.4) and (8.5) proves that

δ0 · rd ≥ Hd ((E ∩ B(x, r)) \C (x, V, α)) ≥ Hd (E ∩ B(x0, r0)) ≥ δ · (r0)d, from which we can derive a contradiction by ﬁxing δ0 such that

sin(α0 − α)d δ0 < δ · . 2

Step 1.3. In this ﬁnal step, the goal is to clean the proof of the particular case by ﬁxing the values of δ0 and α0 in such a way that the contradiction above holds. In particular, let us consider 2 α + π α0 = , 4 and δ 1 π − 2 αd δ0 = · sin . 2 2 4 In the previous step, we proved that r E ∩ B x , 0 ⊆ C (x, V, α0) , 0 2 and this is enough to infer that

E ∩ (V × B (x0, r)) ⊆ Γ(g), where g : V −→ V ⊥ is a Lipschitz function of constant tan(α0), provided that the thickness 0 of the cylinder strips 2r is less or equal than (r0/2) · sin(α ), that is, r π + 2α r ≤ 0 sin . 4 4 106 8.4. AREA FORMULA FOR RECTIFIABLE SETS

Step 2. In the general case, it is enough to consider the countable covering of E deﬁned by setting 1 En := x ∈ E (8.4) and (8.5) holds for r = δ = . n

Corollary 8.25. If E admits a tangent plane at every point x ∈ E, then the set E is d-rectiﬁable.

8.4 Area Formula for Rectiﬁable Sets

Summary. Let Σ be a d-dimensional surface of class C1, and let f :Ω −→ Σ be a Lipschitz function deﬁned on an open subset Ω ⊆ Rd. In Section 7.3 we proved that for every Borel set E ⊆ Ω it turns out that Z Z d d mf, E(y) dH (y) = Jf(x) dL (x), (8.6) y∈Σ E

−1 where mf, E(y) := # f (y) ∩ E and Jf(x) is the Jacobian of f at x. The formula presented here may be extend with little eﬀorts to a more general setting. More precisely, let Σ0 be a d-dimensional surface of class C1, and let f :Σ0 −→ Σ be a Lipschitz function. The (area) formula   Z Z X d d  h(s) dH (x) = h(x)Jf(x) dL (8.7) 0 Σ s∈f −1(x) Σ holds, and the proof is very similar5 to the one presented in Section 7.3.

Rectiﬁable Sets. Let E ⊂ Rn be a d-rectiﬁable set, and let f : E −→ f(E) be a Lipschitz function6. In this paragraph, we want to prove that for every Borel set F ⊆ f(E) it turns out that Z Z d d mf, E(y) dH (y) = Jf(x) dL (x), (8.8) y∈F f −1(F ) where Jf(x) denotes the Jacobian of f at x, that is q T Jf(x) := det (∇tf(x)) (∇tf(x)). (8.9)

Definition 8.26 (Tangent Differential). The function f : E −→ X ⊆ Rm is tangentially d differentiable at H -almost every x ∈ E if and only if there exists a linear map L : TxE −→ Rm such that f(x0) = f(x) + L (π(x − x0)) + o(|x − x0|) 0 for every x ∈ E, where π is the projection onto the tangent space TxE.

5The reader may try to ﬁll in the details as an exercise. Indeed, it is enough to take a compatible atlas ∼ for the surface Σ0 and use the previous result (8.6) on each chart ϕ : U −−→ V . To conclude, the reader may use a suitable partition of unity and prove that the formula glues as expected. 6The codomain is not important since the image of E via f is always a d-rectiﬁable set. 107 8.4. AREA FORMULA FOR RECTIFIABLE SETS

Remark 8.6. If we consider a Lipschitz extension fe : Rn −→ Rm of f, then f is tangentially diﬀerentiable at x ∈ E if and only if there exists a linear map L : Rn −→ Rm such that

fe(x + h) = f(x) + Lh + o(|h|) ∀ h ∈ TxE.

Proof of Area Formula. Let {Ei}i≥0 be given by the equivalent definition of rectifiable set (see Proposition 8.2), that is +∞ [ E = Ei, i=0 and the collection satisfies the following properties:

(a) The d-dimensional Hausdorﬀ measure of E0 is zero.

1 (b) The set Ei is contained in a surface of class C , denoted by Σi, for every i ≥ 1.

m (c) The restriction f may be extended to a diﬀerentiable function Fi :Σi −→ . Ei R 0 00 0 (d) The set Ei is equal to the disjoint union of Ei and Ei , where Ei is the set of points 00 x ∈ Ei where the rank of dfx is not maximal, and Ei is the set of points where it is maximal.

0 For every i ≥ 1, the function Fi :Σi −→ Σi is diﬀerentiable, and thus the area formula −1 0 00 holds for every x ∈ Fi (Σi) ∩ Ei :   Z Z X d d  h(s) dH (x) = h(x)Jf(x) dL (x). (8.10) 0 00 Σi −1 00 Si s∈Fi (x)∩Si

−1 0 0 In a similar fashion, the area formula also holds for every x ∈ Fi (Σi) ∩ Ei since the Jacobian is zero at every point, and the image of a Hd-null set via a diﬀerentiable function is also Hd-null. Remark 8.7. Recall that, if two Lipschitz functions agree on a subset, then they have the same tangential gradient at almost every x in the intersection (see Lemma 8.11).

The area formula introduced in this ﬁnal section can be slightly improved. Indeed, we do not need to consider a Lipschitz function f, but it is enough to require the following properties:

(i) The function f is diﬀerentiable almost everywhere. (ii) The function f sends Hd-null sets into Hd-null sets. Exercise 8.4. Prove that for every d ∈ (0, +∞) there exists a set E such that Hd(E) is ﬁnite and nonzero, but the lower density is zero:

d Θ∗(E, x) = 0 for H -almost every x ∈ E. Part V

Finite Perimeter Sets

108 Chapter 9

Bounded Variation Functions

In this brief chapter, we introduce the notion of bounded variation functions, and we investigate some of the main properties (extension, approximation, compactness, trace, etc.) Notation. In this chapter, we denote by | · | the Lebesgue measure Ln(·), and we denote by dx the diﬀerential dLn(x), unless it is necessary to indicate the dimension.

9.1 Deﬁnition and Elementary Properties

Deﬁnition 9.1 (Bounded Variation). Let Ω ⊆ Rn be an open set. A function u :Ω −→ R is of bounded variation on Ω, and we denote it by u ∈ BV(Ω), if and only if the following properties hold:

(i) The function u belongs to L1(Ω).

n (ii) There exists a vector-valued measure µ = (µ1, . . . , µn) ∈ M (Ω; R ) such that Z Z ∂ϕ ∞ (x)u(x) dx = − ϕ(x) dµi(x), ∀ ϕ ∈ Cc (Ω) (9.1) Ω ∂xi Ω for every i = 1, . . . , n. Remark 9.1. Let u ∈ L1(Ω) be a summable function, and let us denote by Du the weak derivative (in the sense of distributions). Then the condition (9.1) is equivalent to requiring that u is weakly diﬀerentiable and Du is Rn-valued measure. Remark 9.2. In a similar fashion, the condition (9.1) is equivalent to the existence of a positive real-valued measure λ and a vector-valued function τ :Ω −→ Rn such that Z Z ∂ϕ ∞ (x)u(x) dx = − ϕ(x)τi(x) dλ(x), ∀ ϕ ∈ Cc (Ω) (9.2) Ω ∂xi Ω for every i = 1, . . . , n.

There is an equivalent deﬁnition of the set BV(Ω) that relies more on the vector-valued structure of Rn (that is, it is not equivalent in a generic metric space.) 110 9.1. DEFINITION AND ELEMENTARY PROPERTIES

Deﬁnition 9.2 (Bounded Variation). Let Ω ⊆ Rn be an open set. A function u :Ω −→ R is of bounded variation on Ω, and we denote it by u ∈ BV(Ω), if and only if the following properties hold:

(i) The function u belongs to L1(Ω).

(ii) There exists a vector-valued measure µ ∈ M (Ω; Rn) such that Z Z ∞ div (ϕ)(x)u(x) dx = − ϕ(x) dµ(x), ∀ ϕ ∈ Cc (Ω) (9.3) Ω Ω for every i = 1, . . . , n.

Remark 9.3. In a similar fashion, the condition (9.3) is equivalent to the existence of a positive real-valued measure λ and a vector-valued function τ :Ω −→ Rn such that Z Z ∞ div (ϕ)(x)u(x) dx = − ϕ(x) · τ(x) dλ(x), ∀ ϕ ∈ Cc (Ω) (9.4) Ω Ω for every i = 1, . . . , n, where · denotes the scalar product between two Rn vectors. Exercise 9.1. Let Ω ⊆ Rn. A function u satisfies (9.1) if and only if it satisfies (9.3), i.e., the given definitions are actually equivalent.

Remark 9.4. The space BV(Ω), endowed with the norm

kukBV(Ω) = kukL1(Ω) + |Du| (Ω), (9.5) is a non-separable Banach space.

Proof. The space BV(Ω), k · kBV(Ω) is clearly a Banach space, as it can be checked quickly by the reader; thus it is enough to prove that it is not separable. We prove that BV ([0, 1]) is not separable since the general case follows from a similar argument. For every α ∈ (0, 1) let us consider the characteristic function

1α := 1[α, 1].

For every choice α 6= β ∈ [0, 1], it turns out that

k1α − 1βkBV([0, 1]) = 2 + |α − β|, and hence we can consider the family of balls Bα := ψ ∈ BV ([0, 1]) kψ − 1αkBV([0, 1]) ≤ 1 .

This collection of balls is indexed by the interval [0, 1], which means that it is a family with the cardinality of the continuum. Therefore we can infer that BV ([0, 1]) is not separable since each dense subset must intersect every ball Bα in at least one point, which means that any dense subset has a cardinality bigger or equal than the continuum. Remark 9.5. The measure µ given by (9.1) is unique (since the weak derivative in the sense of distribution is unique, up to set of measure zero). We shall denote it by Du through the entire chapter since the symbol ∇u is reserved for the classical/pointwise gradient. 111 9.1. DEFINITION AND ELEMENTARY PROPERTIES

Lemma 9.3. Let u ∈ L1(Ω) be a summable function. Then u belongs to BV(Ω) if and only if the induced linear functional, deﬁned by Z ∞ Λu : Cc (Ω) −→ R, ϕ 7−→ div (ϕ)(x)u(x) dx, Ω is bounded with respect to the uniform norm k · k∞.

Proof.

” =⇒ ” Assume that u ∈ BV(Ω). The absolute value of the functional can be easily estimated using the identity (9.3) as follows: Z Z

div (ϕ)(x) u(x) dx = − ϕ(x) dµ(x) ≤ kϕk∞|µ|(Ω), Ω Ω where |µ|(Ω) denotes the total variation. In particular, the linear functional Λu is bounded with respect to the uniform norm, and thus it can be extended up to the closure of the domain, that is, ∞ 0 Λu : Cc (Ω) = C0 (Ω) −→ R, 0 where C0 (Ω) denotes the set of all continuous functions on Ω that are inﬁnitesimal on the 1 boundary ∂Ω. Notice that the identity (9.3) holds for every function ϕ ∈ C0 (Ω) since the divergence operator is not deﬁned on the set of all continuous functions.

1 ” ⇐= ” Vice versa, assume that u ∈ L (Ω) and Λu is a bounded functional with respect to the uniform norm. From the Riesz Representation Theorem 2.3 one can find a measure µ ∈ M (Ω; Rn) such that Z Z div(ϕ)(x) u(x) dx =: Λu(ϕ) = ϕ(x) dµ(x). Ω Ω In particular, the function u satisfies the property (9.3), that is u is weakly differentiable (in the sense of distributions) and its weak derivative is equal to −dµ, accordingly with the definition of bounded variation functions. Example 9.1.

1) The Sobolev space W 1, 1(Ω) is contained in BV(Ω) for every bounded set Ω ⊂ Rn. Indeed, let f ∈ W 1, 1(Ω) be a Sobolev function, and denote by g ∈ L1(Ω) its weak derivative. It suffices to show that the distribution associated with g is a vector-valued measure in M (Ω; Rn) satisfying (9.3) holds. By definition of weak derivative Z Z n n ∞ div (ϕ)(x)u(x) dL (x) = − ϕ(x)g(x) dL (x), ∀ ϕ ∈ Cc (Ω), Ω Ω thus it is enough to set µ := g ·Ln, which is well defined because g is a summable function. 1129.2. ALTERNATIVE DEFINITIONS: BOUNDED VARIATION ON THE REAL LINE

2) If E ⊆ Ω is an open bounded set with diﬀerentiable (C1 is enough) boundary, then 1 n the characteristic function E belongs to BV (R ). The distributional derivative is zero on both Int(E) and Ω \ E; hence it is a measure supported on the boundary ∂E. More precisely, one can check that

n−1 D1E = νi 1∂E ·H ,

∞ n n where νi denotes the inner normal vector. Indeed, for any ϕ ∈ Cc (R ; R ) it turns out that Z Z Z n−1 div(ϕ)(x)1E(x) dx = div(ϕ)(x) dx = − ϕ(x)νi(x) dH (x), n R E ∂E where the red identity follows from the divergence theorem.

Remark 9.6. The characteristic 1E introduced above is the ﬁrst example of a function of bounded variation on Rn which is not also Sobolev function. Moreover, it turns out that n−1 sup |Λ1E (ϕ)| ≤ H (∂E). kϕk≤1 We will see (in the next chapter) that the inequality above is, for every smooth set E, an equality: n−1 kΛ1E (ϕ)k∗ = H (∂E).

9.2 Alternative Deﬁnitions: Bounded Variation on the Real Line

In this brief section, we want to discuss a different definition of the space BV ([a, b]; R), which is still used in some fields of mathematics (as partial differential equations evolution theory).

Deﬁnition 9.4 (Bounded Variation, II). A function u is of bounded variation on [a, b] ⊆ R if and only if the total variation is ﬁnite, i.e.

n−1 X Tv(u) := sup |u(xi+1) − u(xi)| < +∞, (9.6) P ∈P i=0 where P is the set of all ﬁnite partitions of the interval [a, b]. Deﬁnition 9.5 (Essential Variation). A function u is of bounded essential variation on [a, b], and we denote it by u ∈ BVess ([a, b]), if and only if

n−1 X Tvess (u) := sup |u(xi+1) − u(xi)| < +∞, (9.7) P ∈Pess i=0 where Pess is the set of all ﬁnite partitions of the interval [a, b] with extremal points xi such 1 that u is L -approximate continuous at xi for every i. Remark 9.7. The essential total variation (9.7) is closely related to the total variation introduced in (9.6) since, as one can easily check, it turns out that

Tvess (u) = inf Tv(w). w(x) = u(x) a.e. 1139.2. ALTERNATIVE DEFINITIONS: BOUNDED VARIATION ON THE REAL LINE

The next result shows that the definition of bounded variation introduced in this section is related to the one we will be dealing with in this course through the finiteness of the essential variation. Theorem 9.6. A function u belongs to BV ((a, b)) (in the sense of definition (9.1)) if and 1 only if u ∈ L ((a, b)) and Tvess (u) < +∞.

Lemma 9.7. Let u ∈ BV ((a, b)). There exists a constant1 c ∈ R such that Z x u(x) = c + dµ(x) = c + µ ((a, x)) (9.8) a for L1-almost every x ∈ (a, b).

Proof. It is enough to notice that the identity (9.8) is equivalent to the following identity:

Z x 0 u(x) − dµ(x) = 0. (9.9) a

Proof of Theorem 9.6.

” =⇒ ” Assume that u ∈ BV ((a, b)). By (9.8) there exists a constant c ∈ R such that u(x) = c + µ ((a, x)) .

The function u is summable by deﬁnition; thus it is enough to prove that the total essential variation is ﬁnite. The explicit expression for u immediately yields to

n−1 n−1 X X Tvess (u) := sup |µ ((a, xi+1)) − µ ((a, xi))| = sup |µ ((xi, xi+1))| . P ∈Pess i=0 P ∈Pess i=0 In particular, we notice that the total essential variation is bounded by the mass of µ, and thus it is ﬁnite by assumption:

n−1 X Tvess (u) = sup |µ ((xi, xi+1))| ≤ |Du| ((a, b)) < +∞. P ∈Pess i=0

” ⇐= ” This assertion is left as an exercise for the reader.

1In the proof of Theorem 9.6 is not necessary to know the exact value of c, but the reader may try to prove that it is equal to the trace. 114 9.3. FUNCTIONAL PROPERTIES

9.3 Functional Properties

In this ﬁnal section, we investigate some of the main functional properties of bounded variation functions (extension operator, trace operator, approximation results, etc.) which will be extensively used when we introduce the notion of ﬁnite perimeter set.

Deﬁnition 9.8 (Lipschitz Boundary). A domain Ω ⊂ Rn with Lipschitz boundary is an open subset, which is locally the graph of a Lipschitz map with respect to some choice of orthogonal bases.

Theorem 9.9 (Extension). Let Ω ⊂ Rn be a bounded set with Lipschitz boundary ∂Ω. Then there exists2 an extension operator

n E : BV(Ω) −→ BV(R ). Theorem 9.10 (Approximation, I). The immersion

∞ n n C (R ) ,→ BV (R ) n ∞ n is dense, that is, for every u ∈ BV (R ) there exists a sequence (un)n∈N ⊂ C (R ) such that  u −→ u in L1( n),  n R  n ∇ un ·L * Du weakly (in the sense of distributions),    1 n k∇ unkL (R ) −→ kDuk .

Proof.

∞ n Step 1. Let ρ ∈ Cc (R ) be a molliﬁer and, for any > 0, consider the scaling 1 x ρ (x) := ρ . n ∞ n The function u := ρ ∗ u belongs to Cc (R ) and a standard approximation argument is enough to infer that 1 n u → u in L (R ).

Step 2. The key point of the proof is the following inequality (from which the last two properties will follow immediately):

? kρ ∗ Duk 1 n = k∇uk 1 n ≤ kDuk . (9.10) L (R ) L (R ) If we give this inequality for granted, then we can easily conclude that the family of functions 1 n d (∇u)>0 is uniformly bounded in L (R ). In particular, ∇u ·L >0 is a uniformly bounded sequence of measures, and therefore (by an Ascoli-Arzel`acompactness theorem) it converges in sense of measures to a vector-valued measure µ ∈ M(Rn; Rn). For every ∞ n ϕ ∈ Cc (R ) it turns out that Z Z ∇u(x)ϕ(x) dx = − u(x)div(ϕ)(x) dx, n n R R

2The operator E is, in general, not unique in any sense. 115 9.3. FUNCTIONAL PROPERTIES and the left-hand side converges to Z ϕ(x) dµ(x), n R while the right-hand side converges to Z − u(x)div(ϕ)(x) dx n R as → 0+. In particular, we obtain the identity Z Z Z ∞ n ϕ(x) dµ(x) = − u(x)div(ϕ)(x) dx = ϕ(x) d(Du)(x), ∀ ϕ ∈ Cc (R ), n n n R R R which implies that µ = Du (e.g., using the fundamental lemma in Calculus of Variations).

1 n Step 3. The third property follows easily from the fact that k · kL (R ) is a lower semi- continuous function. Indeed, we have that

lim inf k∇ukL1( n) ≥ kDuk , →0+ R which gives the sought convergence of the masses, as the opposite inequality holds by (9.10).

Theorem 9.11 (Approximation, II). Let Ω ⊂ Rn be a bounded set with Lipschitz boundary ∂Ω. The immersion C∞ Ω ,→ BV (Ω) ∞ is dense, that is, for every u ∈ BV (Ω) there exists a sequence (un)n∈N ⊂ C (Ω) such that  u −→ u in L1(Ω),  n  n ∇ un ·L * Du in the weak sense of distributions,    1 n k∇unkL (R ) −→ |Du| (Ω).

Proof. The key idea is to extend u to Rn via an extension operator n E : BV(Ω) −→ BV(R ), which exists as a consequence of Theorem 9.9, and apply the ﬁrst approximation result to the function E(u) (see Theorem 9.10). We need to be careful though: It is in general false that every extension operator E can be used here since, e.g., the sequence ue, which approximates E(u), does not satisfy the third property (see Figure 9.1). More precisely, the norm is lower semi-continuous and hence

lim inf k∇ukL1( n) ≥ kDuk , →0+ R but the opposite inequality needs not to hold. Indeed, we need to ask E to be an extension operator satisfying the following key property: |DE(u)| (∂Ω) = 0. (9.11) Such an extension operator exists, but it needs to be chosen carefully (we will not discuss this further, but the reader may try to do it as an exercise). 116 9.3. FUNCTIONAL PROPERTIES

Remark 9.8. The immersion C∞ Ω ,→ BV (Ω) is not dense with respect to the norm (9.5).

Exercise 9.2. Let Ω be a bounded set with Lipschitz boundary ∂Ω, and let u ∈ BV(Ω). Prove that the following is an extension operator  u(x) x ∈ Ω n  E : BV(Ω) −→ BV(R ), u 7−→ ue(x) := 0 x∈ / Ω, and prove also that it is not the right one for the proof of Theorem 9.11.

Figure 9.1: What could go wrong in the proof?

Remark 9.9. Recall that the Sobolev critical exponent in Rn is deﬁned by setting n 1∗ := . n − 1

Theorem 9.12 (Sobolev Embedding). Let Ω ⊂ Rn be a bounded set with Lipschitz boundary ∂Ω. The immersion BV (Ω) ,→ Lp(Ω) is continuous for every 1 ≤ p ≤ 1∗ and compact for every 1 ≤ p < 1∗.

Characterization of BV(Rn). In this brief paragraph, we want to state and prove a useful characterization of BV(Rn), which is exactly the one that does not work for L1(Rn). Proposition 9.13. Let u ∈ L1(Rn). The following assertions are equivalent: 117 9.3. FUNCTIONAL PROPERTIES

(1) The function u belongs to BV(Rn). (2) There is a constant C > 0 such that

kτhu − uk 1 n ≤ C|h|. L (R )

Proof.

” =⇒ ” Assume that u ∈ C∞(Rn) in such a way that

u(x + he ) − u(x) + ∂u i −−−−→h→0 . h ∂xi First, we notice that it suﬃces to prove the thesis for an orthonormal basis of directions, e.g. {e1, . . . , en}. A simple computation yields to Z kτiu − ukL1( n) = |u(x + hei) − u(x)| dx ≤ R n R

∂u ≤ |h|, ∂xi 1 n L (R ) which is enough to conclude using the density of C∞(Rn) in BV(Rn).

” ⇐= ” Vice versa, it suﬃces to prove that for every weakly diﬀerentiable summable function u it turns out that

u(x + he ) − u(x) + i −−−−→h→0 D u h i in the sense of distributions.

Theorem 9.14. Let Ω be either Rn or a bounded subset of Rn with Lipschitz boundary ∂Ω. If (un)n∈N ⊂ BV(Ω) is a bounded sequence, then there exists a subsequence (nk)k∈N such that 1 unk → u ∈ BV(Ω) in L (Ω),

Dunk * Du weakly-∗ (in the sense of measures).

Proof. The sequence of measures (Dun) is bounded by assumption; hence, up to sub- n∈N sequences, it converges in the weak sense of measures to a measure µ in M (Ω; Rn). By Sobolev Embedding Theorem 9.12 the immersion

BV(Ω) ,→ L1(Ω)

1 is compact, and thus the sequence (un)n∈N ⊂ L (Ω) is relatively compact, which means that - up to subsequences - it converges strongly (with respect to the L1 norm) to an element u. To conclude the proof, it remains to prove that

µ = Du.

The same argument used in the proof of Theorem 9.10 works here; therefore it is left to the reader to ﬁll in the details. 118 9.3. FUNCTIONAL PROPERTIES

Trace Operator. In the last paragraph of this section, we investigate the trace operator, and we state the main theorem (which is somewhat equivalent to the one valid for Sobolev spaces), and we briefly explain why in the finite perimeter setting, we cannot introduce boundary condition using the trace operator. Theorem 9.15 (Trace Operator). The trace operator, defined by

1 1 T : C Ω −→ L (∂Ω) , u 7−→ u ∂Ω, is bounded with respect to the k · kBV(Ω) norm, and it can be uniquely extended to a linear operator T : BV(Ω) −→ L1 (∂Ω) . Remark 9.10. Recall that the space C1 Ω is dense in BV(Ω) in the weak sense (see Theorem 9.10). This embedding is the main diﬀerence with the Sobolev spaces setting we mentioned above and, as we will see later, it is also the reason why the trace operator is not used to assign boundary conditions. Remark 9.11. Recall that the trace operator deﬁned on W 1, p(Ω) can be used to assign 1, p boundary conditions as follows. Let (un)n∈N ⊂ W (Ω) be a minimizing sequence for a functional F (u): W 1, p(Ω) −→ R, and assume that

T un = g.

1, p If un * u weakly in W (Ω), then the Sobolev trace theorem implies that T u = g, which is exactly what we would like to happen when trying to ﬁnd a solution for a minimum problem.

On the other hand, if (un)n∈N is a bounded sequence in BV(Ω), then - up to subsequences 1 - un converges to u strongly in L (Ω). However, in general, it is not true that T un converges (in any reasonable sense) to T u (as we shall prove in the following examples).

Example 9.2. Let Ω := (0, 1). Consider the sequence (un)n∈N ⊂ BV ((0, 1)) that is deﬁned as it is portrayed in Figure 9.2. Clearly, for every n ∈ N, it turns out that

T un(0) = 0 and T un(1) = 1, but un converges pointwise to u(x) ≡ 1 on Ω, which means that a boundary condition is not preserved under the limit (T u(0) = 1). The reader may check by herself that

un * 1 in BV(Ω), that is,  1 un → u in L (Ω),

kunkBV(Ω) ≤ M ∀ n ∈ N.

n Example 9.3. Let Ω ⊆ R , and consider the sequence of subsets (Ek)k∈N ⊂ P (Ω) deﬁned as it is portrayed in Figure 9.3. Moreover, assume that for any n ∈ N

(a) Ek is smooth,

(b) Ek ⊂ Ω, and

n−1 n−1 (c) the measure H (∂Ek) is of the same order of H (∂Ω). 119 9.3. FUNCTIONAL PROPERTIES

Figure 9.2: First Counterexample

1 The sequence given by the characteristic functions uk := Ek ∈ BV(Ω) is uniformly bounded since the following properties hold:

n (a) kukkL1(Ω) ≤ H (Ω);

n−1 n−1 (b) kDukk = H (∂Ek) → H (∂Ω) as n goes to +∞.

Clearly, the limit of the sequence uk is the characteristic function of Ω, that is, the constant function 1 ∈ BV(Ω). The reader may easily prove that

T un = 0 but T u = 1, which means that T un does not converge in any reasonable sense to T u. 120 9.3. FUNCTIONAL PROPERTIES

Figure 9.3: Second Counterexample Chapter 10

Finite Perimeter Sets

In this ﬁnal chapter, we introduce the notion of ﬁnite perimeter set, and we also present a variational setting that allows us to prove the existence of a solution to the Plateau’s problem (for suitable boundary conditions) and the capillarity problem. In the second half of the chapter, we introduce the essential and the reduced boundary, and we prove the Structure Theorem due to De Giorgi and Federer.

10.1 Main Deﬁnitions and Elementary Properties

In this section, we present the definition of finite perimeter set, and we exploit the theory of bounded variation functions introduced in the previous chapter to derive approximation and compactness results. Definition 10.1 (Finite Perimeter Set). Let E ⊆ Rn. We say that E has finite perimeter if and only if the characteristic function has bounded variation, that is, 1 n E ∈ BV (R ) .

Recall that the functional associated to 1E is given by Z

Λ1E (ϕ) := div(ϕ)(x) dx, E and it is bounded with respect to the uniform norm. The perimeter of E is deﬁned by the operator norm of the functional Λ1E (ϕ), that is,

Per(E) := kΛ1E k∗ . Deﬁnition 10.2 (Relative F.P.S.). Let Ω ⊆ Rn be an open set, and let E ⊆ Ω be a subset. We say that E has ﬁnite perimeter in Ω if and only if

1E ∈ BV (Ω) .

The perimeter of E in Ω is deﬁned by the operator norm of the functional associated to 1E, that is,

PerΩ(E) = kΛ1E k∗ , ∞ where Λ1E denotes the restriction of the functional to the space Cc (Ω). 122 10.1. MAIN DEFINITIONS AND ELEMENTARY PROPERTIES

Remark 10.1. Let E ⊆ Ω ⊆ Rn be a smooth subset (e.g., assume that the boundary of E is at least of class C1). The reader may prove that the perimeter of E in Ω is simply given by the following formula: n−1 PerΩ(E) = H (∂E ∩ Ω) . (10.1) Hint. A similar argument to the one used in the Example 9.1 works here. In particular, we notice that the derivative of the characteristic function is given by

n−1 D (1E) = νi1∂E∩Ω ·H .

Remark 10.2. It is important to notice that, in general, the formula

n−1 PerΩ(E) = H (∂E) is not true, not even if E is smooth. The intuitive idea is clear: Since Ω is an open set, the boundary of E may overlap with the boundary of Ω as it happens in Figure 10.1.

Figure 10.1: The boundary of the set E partially overlaps with the boundary of Ω. Therefore only the magenta part of ∂E contributes to the perimeter of E in Ω.

Compactness Results. In this paragraph, we state two compactness results for ﬁnite perimeter sets, both of which derive from the functional properties of bounded variations functions. Notation. Let Ω be an open set, and let E,F ⊆ Ω be two subsets. The distance between E and F is the n-Lebesgue measure of the symmetric diﬀerence, that is,

n d(E,F ) := k1E − 1F kL1(Ω) = L (E 4 F ) . 123 10.2. APPROXIMATION THEOREM VIA COAREA FORMULA

n Theorem 10.3 (Compactness, I). Let (Ek)k∈N ⊂ P(R ) be a uniformly bounded sequence of ﬁnite perimeter sets, that is,

0 Per(Ek) < +∞ and Ek ⊆ Ω ⊂⊂ Ω, where Ω ⊂ Rn is a bounded open subset. Then there exist a ﬁnite perimeter set E ⊆ Ω and an increasing subsequence (nk)k∈N such that

k→+∞ d(Ek,E) −−−−−→ 0.

Moreover, the perimeter is a lower semi-continuous functional, that is

lim inf (Per(Ek)) ≥ Per(E). k→+∞

Remark 10.3. The assumption ”(Ek)k∈N is a uniformly bounded sequence” is crucial here, otherwise one can exhibit the easy counterexample 1 E := B(k, ), k 2 which is a collection given by integer translations of a ball centered at zero with radius one half.

n Theorem 10.4 (Compactness, II). Let (Ek)k∈N ⊂ P(R ) be a uniformly bounded sequence of ﬁnite perimeter sets, that is,

n L (Ek) , Per(Ek) ≤ c < +∞.

Then there exist a ﬁnite perimeter set E ⊆ Ω and an increasing subsequence (nk)k∈N such that k→+∞ n |(Ek 4 E) ∩ B(x, r)| −−−−−→ 0, ∀ B(x, r) ⊆ R . Moreover, the perimeter is a lower semi-continuous functional, that is

lim inf (Per(Ek)) ≥ Per(E). k→+∞

10.2 Approximation Theorem via Coarea Formula

The primary goal of this section is to prove that any ﬁnite perimeter set can be approximated via a sequence of smooth (e.g. C∞ boundary) sets. In order to prove this assertion, we will need to use the coarea formula presented in Section 7.4.

Theorem 10.5. Let E ⊆ Rn be a ﬁnite perimeter set. Then there exists a sequence of ∞ smooth (boundary of class C ) sets (Ek)k∈N such that

 d Ek −→ E,

n−1 k→+∞ H (∂Ek) = Per(Ek) −−−−−→ Per(E), where d denotes the symmetric distance introduced in the previous section. 124 10.2. APPROXIMATION THEOREM VIA COAREA FORMULA

Proof. Let ρ be a molliﬁer kernel. Denote by ρ the -rescaling, which is deﬁned in the usual way: 1 x ρ (x) := ρ . n Let u := ρ ∗ 1E be the convolution. For every s ∈ [0, 1] and > 0 the set

Es, := {x ∈ Ω | u(x) > s } is open and smooth, provided that we pick an s such that it is not a critical value (which is always possible by Sard’s Lemma).

Step 1. We claim that, if δ ≤ s ≤ 1 − δ, then

L1(Ω) d u −−−−→ u =⇒ Es, −→ E. (10.2) Indeed, a straightforward application of the Fubini-Tonelli’s theorem proves that Z 1 ku − ukL1(Ω) = d (E,Es, ) ds. 0 0 On the other hand, for every s < s it turns out that Es0, ⊇ Es, ; hence we can easily estimate the L1-norm of the diﬀerence as follows: Z 1 1 ku − ukL (Ω) ≥ |E \ Es, | ds ≥ (1 − s) · |E \ Es, | , s where \ denotes the usual diﬀerence between two sets.

1 In particular, if we take s strictly less than 1, then ku−ukL (Ω) → 0 implies |E \ Es, | → 0, which is exactly what we wanted to prove.

Step 2. The function u is smooth (by convolution). Hence we can apply the coarea formula (7.11) with h ≡ 1 and f = u to obtain the following relation: Z Z 1 n−1 −1 |∇u(x)| dx = H u (s) ds, n R 0 −1 1 where u (s) = ∂Es, if s is a regular value . One can easily prove that 1 1 Z 1 Z 1−δ Per(E) ≥ Per(Es, ) ds ≥ − Per(Es, ) ds, 1 − 2δ 1 − 2δ 0 δ for δ > 0 small enough; hence there exists a non-null (with respect to the Lebesgue measure) set of regular values in the interval [δ, 1 − δ] such that 1 1 Per(E) ≥ Per(Es, ) =⇒ lim sup Per(Es, ) ≤ Per(E) 1 − 2δ →0+ 1 − 2δ for every regular value s ∈ [δ, 1 − δ]. Recall that the perimeter is a lower semi-continuous functional; hence we only need to reﬁne the estimate and rule δ out of the equation.

Exercise 10.1. State the approximation theorem in a smooth bounded domain Ω ⊂ Rn.

1 −1 n In particular, the formula holds with u (s) = ∂Es, for L -almost every s ∈ [0, 1] since the set of singular values is negligible by Sard’s Lemma. 125 10.3. EXISTENCE RESULTS FOR PLATEAU’S PROBLEM

10.3 Existence Results for Plateau’s Problem

Introduction. Let Ω ⊂ Rn be a bounded smooth subset of Rn. Assume that Γ is the boundary of a regular hypersurface G ⊂ ∂Ω, that is,

Γ = ∂G.

The Plateau’s problem consists of ﬁnding the minimal hypersurface Σ contained in Ω such that its boundary is given by Γ, that is, satisfying the boundary condition ∂Σ = Γ.

Finite Perimeter Setting. Let E be a subset of Ω satisfying the following condition:

∂E ∩ ∂Ω = G.

If Σ is a regular hypersurface given by the intersection ∂E ∩ Ω, then the following relations holds: Per(E) = Hn−1 (∂E ∩ ∂Ω) + Hn−1(Σ). (10.3) This identity allows us to infer that minimizing the functional Hn−1(Σ) is equivalent to minimizing the functional Per(E) among all ﬁnite perimeter sets E such that

T (1E) = 1G.

Figure 10.2: Plateau’s Problem in the F.P.S. formulation. 126 10.3. EXISTENCE RESULTS FOR PLATEAU’S PROBLEM

Remark 10.4. In the previous sections, we proved that the perimeter is a nice functional, which is also semi-lower continuous. Hence the equivalent formulation in the f.p.s. setting should be enough to demonstrate the existence of a solution. But, as the problem is formulated now, many things could go wrong:

1) Assume that one can ﬁnd a solution of the minimum problem (10.3). How can one recover a surface Σ? 2) The minimum of the functional (10.3) actually exists? 1 1 3) If (En)n∈N is a minimizing sequence, then the trace T ( En ) may not converge to G, as proved in the previous chapter. 4) Topological issues. The hypersurface G with boundary Γ may not even exist.

Figure 10.3: In this Plateau’s problem, the minimizing surface Σ makes an extra contact 1 1 with the boundary of Ω, which means that T ( En ) does not converge to G.

Alternative Formulation. The Plateau’s problem needs to be reformulated diﬀerently (in the f.p.s. setting) to avoid the issues listed above (except the topological ones, which are way more delicate to deal with).

Let Ω ⊂ Rn be a bounded smooth subset of Rn. Assume that Γ is the boundary of a regular hypersurface G ⊂ ∂Ω, that is, Γ = ∂G. 127 10.3. EXISTENCE RESULTS FOR PLATEAU’S PROBLEM

n Let E0 be a fixed subset contained in the complement of Ω, that is, E0 ⊂ R \ Ω. The Plateau’s problem, again, constist of finding the minimal hypersurface Σ contained in Ω such that ∂Σ = Γ. In the finite perimeter setting, the idea is to look for the minimum point(s) of the functional

n−1 Per(E) − H ∂E0 \ Ω (10.4) among all ﬁnite perimeter sets E satisfying the following condition:

n E \ Ω = E0 up to L -null sets.

Figure 10.4: In this picture, one can see how the trace issues are no longer a treat using this alternative formulation of the Plateau’s problem.

Plateau’s problem. In this brief paragraph, we summarize the classical formulation of the Plateau’s problem and we reinterpret it into the ﬁnite perimeter set framework.

C1) Given Ω be a bounded smooth subset of Rn. C2) Given Γ be a (n − 2)-dimensional surface contained in ∂Ω. C3) Goal: Find a (n − 1)-dimensional surface Σ such that ∂Σ = Γ and Σ minimizes the area functional Hn−1(Σ). 128 10.4. CAPILLARITY PROBLEM

In the ﬁnite perimeter set framework, the Plateau’s problem can be formulated in the following way:

F1) Given Ω be a bounded smooth subset of Rn. n F2) Given G ⊂ ∂Ω such that there exists a ﬁnite perimeter set E0 in R contained in the complement of Ω, that is, n 1 1 E0 ⊆ R \ Ω and T ( E0 ) = G up to Ln-null sets.

F3) Goal: Find a ﬁnite perimeter set E in Rn such that E \ Ω is - up to Ln-null sets - equal to E0, such that E is a minimum point of the perimeter functional.

In conclusion, a natural question arises: For which G there exists E0 satisfying the property F2)? Surprisingly, it is enough to ask that G is any Borel set as a consequence of the following theorems. Theorem 10.6. The trace operator

1 1 T : C Ω −→ L (∂Ω) , u 7−→ u ∂Ω is surjective. Remark 10.5. In general, the trace operator introduced above does not admit a linear right inverse. Theorem 10.7. Let u be a characteristic function in L1 (∂Ω). Then there exists a ﬁnite perimeter set E ⊆ Rn such that T (1E) = u.

In particular, the only problems one needs to deal with in Plateau’s problem are of topological nature (see, e.g., Figure 10.5).

Conclusion. To conclude this chapter, we sketch the framework and the solution of a Plateau’s problem. The reader may refer to Figure 10.6 for a better understanding of what is going on, but the main idea is to take Γ equal to the disjoint union of two circumferences (lying on the planes π1 and π2 respectively), and G equal to the disjoint union of the two closed disks. The set Ω is convex, but once again the condition on the trace is not closed. Therefore this example shows how important is the introduction of a diﬀerent formulation where the operator T does not appear.

The solution of the minimum problem depends on the distance d between the planes π1 and π2, and it is either a catenoid or a disjoint union of disks.

10.4 Capillarity Problem

Introduction. Let Ω ⊂ Rn be an open bounded subset of Rn. In this section, we shall discuss the capillarity problem equilibrium when the container (Ω) is ﬁxed and the volume 129 10.4. CAPILLARITY PROBLEM

Figure 10.5: The surface Γ is not the boundary of any hypersurface G contained in the torus Ω. of the liquid is fixed. More precisely, the drop of liquid (denoted by E), contained in Ω, is at equilibrium if and only if it is a critical point (local minima) of the capillarity energy functional F (E) := Hn−1 (∂E ∩ Ω) + σHn−1 (∂E ∩ ∂Ω) , (10.5) where σ ∈ R is a constant, subject to the constraint that the volume is fixed, e.g. |E| = m. In the Appendix, we shall prove that, if we set the first variation of the functional F equal to zero, then we find that the mean curvature H is constant. More precisely, it is equal 2 f to the Lagrange multiplier associated with the constraint on Σ , and θ := θY is constant on Γ and equal to the so-called Young angle, i.e. the solution to the following trigonometric equation: cos(θY ) = −σ. Remark 10.6. Suppose that the drop of liquid is subject to the action of the gravitational force. Then one needs to slightly modify the functional F to take into account the gravity in the following way: Z G (E) := Hn−1 (∂E ∩ Ω) + σHn−1 (∂E ∩ ∂Ω) + ~g · x. (10.6) E

2We denote by Σf the free portion of the surface, that is, the portion of the surface that does not lie on ∂Ω. 130 10.4. CAPILLARITY PROBLEM

Figure 10.6: Example of Plateau’s problem

The reader may compute the ﬁrst variation of the functional G and set it equal to zero. It is not hard to check that the mean curvature H is not constant anymore.

Finite Perimeter Formulation. In the f.p.s. framework, we can translate the capillarity problem associated to the functional F (subject to the volume constraint) as follows:

F1) Given Ω bounded smooth subset of Rn. F2) Goal: Find a ﬁnite perimeter set E ⊆ Ω in Rn, with ﬁxed volume |E| = m, which minimizes the functional

n−1 c F (E) = PerΩ(E) + σH (Σ ) , (10.7)

where 1Σc = T (1E) denotes the portion of the surface Σ that lies on the boundary of Ω. Proposition 10.8. Suppose that σ∈ / [−1, 1]. Then the minimization problem (10.7), with ﬁxed volume, is ill-posed. Remark 10.7. More precisely, we shall prove that the functional (10.7) is not lower semi- continuous with respect to the distance d whenever |σ| > 1.

Proof. Let us consider the sequence of subsets (E1/n)n∈N portrayed in Figure 10.8, and suppose that E1/n is smooth for every n ∈ N. If |σ| > 1, then one can prove that the 131 10.4. CAPILLARITY PROBLEM

Figure 10.7: Capillarity Problem functional F is not lower semi-continuous using this sequence, that is,

+ f c →0 f c F (E1/n) = Σ + |Σ | + o(1) −−−−→ Σ ± |Σ | < F (E).

In other words, the relaxation of the functional F is given by   Σf + |Σc| , if σ > 1, (E) = Ff  Σf − |Σc| if σ < −1, and this is enough to prove the claim above since

f c σ > 1 =⇒ Ff(E) < F (E) = Σ + σ |Σ | , and similarly f c σ < −1 =⇒ Ff(E) > F (E) = Σ + σ |Σ | .

Claim. If σ ∈ [−1, 1], then the functional (10.7) is lower semi-continuous.

The proof of this claim cannot be tackled with the tools introduced in this section since we need to use more sophisticated methods. We will come back to this result in the next chapter. 132 10.5. FINITE PERIMETER SETS: STRUCTURE THEOREM

Figure 10.8: Left: Sequence when σ > 1. Right: Sequence when σ < −1.

10.5 Finite Perimeter Sets: Structure Theorem

The primary goal of this section is to state and prove the structure theorem for ﬁnite perimeter set, due to both De Giorgi and Federer (it appeared in diﬀerent papers, but we shall merge them to obtain a very reasonable result.)

Lemma 10.9. Let E ⊂ Rn be a bounded subset of Rn such that ∂E is a Lipschitz boundary. If νe is the external normal to E, then

n−1 D(1E) = −νe 1∂E ·H and, in particular, it turns out that

Per(E) = Hn−1 (∂E) . (10.8)

Proof. The identity follows immediately from the divergence theorem for Lipschitz bound- aries. Example 10.1. In general, the Hn−1 measure of the boundary ∂E does not coincide with the perimeter of the set Per(E). For example, if E is any Ln-null set, then it is easy to prove that Per(E) = Per(∅) = 0. On the other hand, the interior of E is empty and hence

Hn−1 (∂E) = Hn−1 E , and this last quantity is, in general, strictly positive. 133 10.5. FINITE PERIMETER SETS: STRUCTURE THEOREM

There is a result which links the notion of the perimeter, and the Hausdorﬀ measure of the boundary for every Borel set E Borel, but the proof is rather involved. Here we state the result and sketch the proof of a slightly weaker statement.

Theorem 10.10. For every E ⊂ Rn Borel, it turns out that Per(E) ≤ Hn−1 (∂E) . (10.9)

Theorem 10.11. There is a constant c > 1 such that

Per(E) ≤ c ·Hn−1 (∂E) (10.10) for every E ⊂ Rn Borel.

Sketch of the Proof. We can assume without loss of generality that E is a closed and bounded Borel set with ﬁnite Hausdorﬀ measure of the boundary, that is,

Hn−1 (∂E) < +∞.

δ Step 1. Let δ > 0 be a positive number, and let {Bi }i∈N be a collection of open balls of uniformly bounded radii ri ≤ δ, which is an optimal cover of the boundary ∂E with respect to the spherical Hausdorﬀ measure. By compactness of ∂E, it turns out that

N(δ) [ δ ∂E ⊆ Bi , i=1 where N(δ) is a positive integer. In particular, we have the estimate

N(δ) X n−1 n−1 n−1 (2ri) ≤ Hs (∂E) ≤ C ·H (∂E), i=1 as a consequence of the fact that the Hausdorﬀ measure is equivalent to the Hausdorﬀ spherical measure. Let N(δ) [ δ Eδ := E ∪ Bi , i=1

d + and notice that Eδ −→ E as δ → 0 , that is,

δ→0+ |Eδ 4 E| = |Eδ \ E| −−−−→ 0, since we assumed E to be a closed set.

Step 2. Suppose that the balls intersect transversally. Then the boundary ∂Eδ is Lip- schitz, which means that the perimeter formula (10.8) holds. Moreover, the perimeter functional is a lower semi-continuous function, and therefore

n−1 Per(E) ≤ lim inf Per(Eδ) = lim inf H (∂Eδ) . δ→0+ δ→0+ 134 10.5. FINITE PERIMETER SETS: STRUCTURE THEOREM

On the other hand, the boundary of Eδ is covered by the ﬁnite family of δ-balls, which means that N ! n−1 n−1 [ δ H (∂Eδ) ≤ H ∂Bi ≤ i=1

N X n−1 δ ≤ H ∂Bi ≤ i=1

N 0 X n−1 n−1 ≤ C (2ri) ≤ c ·H (∂E) , i=1 where c := C · C0 is the sought constant.

Exercise 10.2. Let E ⊂ R be a ﬁnite perimeter set. Prove that E is, up to L1-null sets, the union of ﬁnitely many intervals.

Lemma 10.12. Let E ⊆ R be a ﬁnite perimeter set. There exists a representative of E, also denoted by E, such that formula (10.8) holds.

1 Proof. The function E belongs to BV(R) by assumption. Therefore, by Lemma 9.7 there exists a constant c ∈ R such that

1E(x) = c + µ ([−∞, x]) , where µ denotes the weak derivative D(1E). The reader may prove that, if µ({x}) = 0 1 for every x ∈ R, then this is a continuous representative of E that satisﬁes the formula (10.8).

Proposition 10.13. Let E ⊂ Rn and n ≥ 2. Then there exists a Borel set E such that no representative of E satisﬁes the perimeter formula (10.8), that is, the inequality is strict for every representative E0 of E.

2 Proof. Let {xn}n∈N ⊂ R be a dense sequence of points, and let (rn)n∈N be a sequence of positive real numbers satisfying the condition X n · rn < +∞. (10.11) n∈N

Step 1. Let X L := rn < +∞ n∈N and set Bn := B(xn, rn). We define inductively a sequence of finite perimeter sets given by the symmetric differences, that is,  E0 = B0,

En = En−1 4 Bn. 135 10.5. FINITE PERIMETER SETS: STRUCTURE THEOREM

2 We claim that (1E ) ⊂ BV( ) is a Cauchy sequence with respect to the k · kBV-norm. n n∈N R If the claim holds, then we can infer that

1 1 L 1 En −−→ E, and also that 1 D(1E) = ± ν 1Σ ·H , where [ Σ = ∂B(xn, rn) n∈N is a 1-rectiﬁable surface. In fact, it is enough to notice that

n [ 1 ∂En = ∂Bi =⇒ H (∂En) ≤ 2πL, i=0 and 1 1 1 D En = ± νn ∂En ·H .

Step 2. We now prove the claim. By deﬁnition of symmetric diﬀerence it turns out that

n 1 1 1 1 X En = B0 + B1\E0 − B1∩E0 + ··· =: ui, i=0 and therefore it is equivalent to show that

+∞ X n kuikBV(R ) < +∞. i=0

1 The L -norm of ui may be roughly estimated by the area of the ball Bi, which means that

2 1 n kuikL (R ) = π · ri . In a similar way, the total variation of the derivative may be estimated by

1 kDuik = 2π · ri + 2H (∂Ei−1 ∩ Bi) ≤ 2π(i + 1)ri, and therefore the assumption (10.11) proves the claim. 1 2 In particular, the support of the measure D E is the whole plane R , and hence every representative of E has boundary containing the support of 1E, that is,

2 ”∂Ee = R .”

We ﬁnally infer that no representative Ee of E has H1-ﬁnite boundary.

Main Result. In this ﬁnal paragraph, we are ﬁnally ready to state and prove the so-called structure theorem due to De Giorgi and Federer. The proof is quite involved. Therefore we will need a lot of work to get through it. 136 10.5. FINITE PERIMETER SETS: STRUCTURE THEOREM

Deﬁnition 10.14 (Essential Boundary). Let E be a Borel set in Rn. The essential boundary of E in Rn is deﬁned by

∂∗E := {x ∈ ∂E | Θn (E, x) 6= 0, 1 or Θn (E, x) does not exist} , where Θn denotes the n-dimensional Hausdorﬀ density. Remark 10.8. The essential boundary is, in general, strictly contained in the boundary. For example, if ∂E is locally the graph of a function f that admits a cuspid p, then p ∈ ∂E \∂∗E.

Definition 10.15 (Normal Vector Field). If Σ is a (n−1)-rectifiable set, then η :Σ −→ Rn is a normal vector field if |η(x)| = 1 for every x ∈ Σ and

η(x) ⊥ Tan(Σ, x) for Hn−1-almost every x ∈ Σ.

Theorem 10.16 (De Giorgi-Federer). Let E be a ﬁnite perimeter set in Rn (or in Ω). Then the following assertions hold:

n−1 (1) The essential boundary ∂∗E is H -ﬁnite.

(2) The essential boundary ∂∗E is (n − 1)-rectiﬁable. (3) There exists a normal vector ﬁeld η which is ”inner normal” in the following sense:

n + n + L E 4 (x + Mη(x)) ∩ Bx, r r as r → 0 (10.12)

n−1 for H -almost every x ∈ ∂∗E. (4) The derivative of the characteristic function of E is given by 1 1 n−1 D( E) = ∂∗E η ·H (10.13) Remark 10.9. The assertion (10.12) is completely equivalent to the following one:

1 n 1 Lloc(R ) + (E − x) −−−−−→ + M . (10.14) r r→0 η(x) Corollary 10.17. Under the assumptions of the Theorem 10.16 it turns out that 1 Θ (E, x) = n 2

n−1 for H -almost every x ∈ ∂∗E. Corollary 10.18. Under the assumptions of the Theorem 10.16 it turns out that

n−1 Per(E) = H (∂∗E) < ∞.

In a diﬀerent paper, Federer proved that also a sort of converse of this theorem holds true, but we will only state it here since the proof is far beyond the reach of this course.

n n−1 Theorem 10.19 (Federer). Let E ⊆ R be a Borel set such that H (∂∗E) < +∞. Then E is a ﬁnite perimeter set.

n Exercise 10.3. Let E ⊆ R be a set with empty essential boundary, that is ∂∗E = ∅. Prove that either |E| = 0 or |R \ E| = 0. 137 10.5. FINITE PERIMETER SETS: STRUCTURE THEOREM

Figure 10.9: Intuitive idea of the statement of the De Giorgi-Federer structure theorem. + The picture is slightly misleading: the vector space Mη(x) is the whole space ”under” the tangent to ∂∗E at x.

Deﬁnition 10.20 (Reduced Boundary). Let E be a ﬁnite perimeter set in Rn. The reduced boundary of E in Rn, denoted by ∂∗E, is the set of all x ∈ Rn such that the Radon-Nikodym density d(D1 ) η(x) := E (x) d|D1E| exists and has norm equal to one.

Remark 10.10. Equivalently, if we denote by τ · µ the weak derivative of 1E, then η(x) is the L1-approximate limit, with |τ| = 1, of τ at Hn−1-almost every x, that is,

Z + − |τ(y) − η(x)| dµ(y) −−−−→r→0 0. B(x, r)

Example 10.2. In general, the reduced boundary is a proper subset of the boundary. For example, if E is a rhombus, then the four vertexes are not in the reduced boundary ∂∗E. n n 1 Theorem 10.21 (De Giorgi). Let E ⊆ R be a ﬁnite perimeter set in R , and set D( E) := τ · µ. Then the following assertions hold true:

(1) The Radon-Nikodym density η(x) coincides µ-almost everywhere with τ(x). Moreover, the measure µ is supported in the reduced boundary, that is,

n ∗ µ (R \ ∂ E) = 0. (10.15)

(2) The reduced boundary ∂∗E is (n − 1)-rectiﬁable. 138 10.5. FINITE PERIMETER SETS: STRUCTURE THEOREM

(3) The measure µ is the restriction of the (n − 1)-dimensional Hausdorﬀ measure to the reduced boundary, that is, µ = Hn−1 ¬ ∂∗E. (10.16)

(4) The Radon-Nikodym density η(x) is normal to the reduced boundary at x for every x ∈ ∂∗E, that is, ⊥ ∗ η(x) = Tan∗ (∂ E, x)

where Tan∗ (−, ·) denotes the approximate tangent plane. (5) The normal vector ﬁeld η satisﬁes the following property:

n + n + L E 4 (x + Mη(x)) ∩ Bx, r r as r → 0 (10.17)

for every x ∈ ∂∗E. Equivalently,

1 n 1 Lloc(R ) + (E − x) −−−−−→ + M . (10.18) r r→0 η(x) Corollary 10.22. Under the assumptions of the Theorem 10.21 it turns out that 1 Θ (E, x) = n 2 for every x ∈ ∂∗E.

The proof of De Giorgi’s Theorem 10.21 will follow reasonably quickly from a sequence of many technical lemmas that we will patently state and prove here. In order to ease the statements of the following results, we will ﬁx a point x ∈ ∂∗E and denote it by 0 unless otherwise stated.

Lemma 10.23. There exists an universal constant c := c(n) > 0 such that

n−1 µ(Br) ≤ c · r . (10.19)

In particular, it turns out that there exists an universal constant c0 := c0(n) > 0 such that

0 n−1 µ(Br) ≤ c ·H (∂Br ∩ E) . (10.20)

Proof. Set Er := E ∩ Br.

Step 1. We claim that, for almost every r > 0, the weak derivative of the characteristic function satisﬁes an additive formula:

1 1 1 1 1 D( Er ) = Br · D( E) + E · D( Br ). (10.21) 1 We notice that the integral of η(0) with respect to the measure D( Er ) is given by Z 1 η(0) dD( Er ) = 0, n R 139 10.5. FINITE PERIMETER SETS: STRUCTURE THEOREM since η(0) is constant. On the other hand, it follows from (10.21) that Z Z Z 1 1 η(0) dD( Er ) = η(0)η(x) dµ(x) + η(0)dD( Br )(x) = n R Br E Z Z n−1 = η(0)η(x) dµ(x) + η(0)νinn(x) dH (x), Br ∂Br ∩E where νinn(x) denotes the inner normal vector to the sphere at x. In particular, it turns out that Z Z n−1 η(0)η(x) dµ(x) = − η(0)νinn(x) dH (x) ≤ Br ∂Br ∩E

n−1 n−1 ≤ H (∂Br) ∼ αnr , where the inequality follows from the fact that both νinn and η are vectors of norm less or equal than 1. In conclusion, we notice that for r → 0+ the left-hand side of the inequality above is asymptotically equivalent to µ(Br), which means that there exists a r0 > 0 small enough such that Z 1 n−1 n−1 µ(Br) ≤ η(0)η(x) dµ(x) ≤ H (∂Br) ∼ c(n) · r , 2 Br for almost every 0 < r < r0.

Step 2. We now sketch the proof of the claim (10.21). More precisely, we show that the formula holds for every f.p.s. E and almost every r > 0 - even when 0 does not belong to the reduced boundary -.

Step 2.1. First, we notice that (10.21) holds for every smooth f.p.s. E that is also transversal to the boundary of the ball Br. Indeed, one can easily prove that ∂Er = ∂Br ∩ E ∪ Br ∩ ∂E .

Step 2.2. Let En be an approximation of E by smooth sets, that is

L1 En −−−→loc E.

n Choose r > 0 in such a way that ∂Er is transversal to the sphere ∂Br for every n ∈ N. The previous step proves that En satisﬁes (10.21) for every n ∈ N, which means that we can pass the identity to the limit3. Lemma 10.24. There are two universal constants c := c(n), c0 := c0(n) > 0 such that for every r > 0 small enough n 0 n c · r ≤ |E ∩ Br| ≤ c · r , (10.22) and n c 0 n c · r ≤ |E ∩ Br| ≤ c · r . (10.23)

3We will not give any detail, but one needs to be careful to the notion of convergence that comes into play here. 140 10.5. FINITE PERIMETER SETS: STRUCTURE THEOREM

Remark 10.11. It is easy to prove that the complement of a f.p.s. is also a f.p.s. with the same weak derivative up to null sets. In particular, the inequality (10.22) implies the inequality (10.23) by passing to the complement.

Proof. Set Er := E ∩ Br.

Step 1. We claim that the following isoperimetrical inequality holds: For every r > 0 there exists an universal constant c := c(n) > 0 such that

n−1 v(r) n ≤ c · Per(Er), (10.24) where v(r) denotes the Lebesgue measure of Er. If the inequality holds, then it is easy to see that n−1 n 1 v(r) ≤ c1 · kD( Er )k ≤

n−1 ≤ c2 PerBr (E) + H (E ∩ ∂Br) ≤

n−1 ≤ c3 ·H (E ∩ ∂Br) = c3 · v˙(r), since 1 PerBr (E) = |D( E)|(Br) = µ(Br), n−1 and clearly µ(Br) and H (E ∩ ∂Br) are comparable quantities as a consequence of the previous result.

Step 2. The symbolv ˙ denotes the classic derivative of v since a Lipschitz function is almost everywhere diﬀerentiable (by Rademacher Theorem 7.7). The reader may prove, as an exercise, that the formula used above holds:

n−1 v˙(r) = H (E ∩ ∂Br) for almost every r > 0. (10.25)

Step 3. In conclusion, we notice that v(r) is an almost everywhere diﬀerentiable function which satisﬁes the following inequality:

n−1 v(r) n ≤ c3 · v˙(r).

A straightforward computation yields to

v(r) ≥ c(n) · rn, which is exactly the nontrivial part of (10.22). The trivial one, on the other hand, follows immediately from the worst estimate possible:

0 n v(r) ≤ |Br| ∼ c (n) · r . 141 10.5. FINITE PERIMETER SETS: STRUCTURE THEOREM

Step 4. It remains to prove the the claimed isoperimetrical inequality (10.24); surprisingly, it follows easily from a far more general theorem. Precisely, one can prove that there exists a universal constant c := c(n) such that for any given F bounded f.p.s., it turns out that

n−1 |F | n ≤ c(n) · Per(F ).

Recall that the immersion ∗ BV(Ω) ,→ L1 (Ω), ∗ n where 1 = n−1 , is continuous (and compact). Therefore, there exists a universal constant c0 := c0(n) > 0 such that 0 kuk n ≤ c · kuk . L n−1 (Ω) BV(Ω) The reader may prove that one can always replace the BV(Ω)-norm with the total variation kDuk if u is, for example, compactly supported. The isoperimetrical inequality follows immediately by taking u := 1F for every bounded f.p.s. F .

1 1 Lemma 10.25. Let Er := r (E − x) = r E. Then

1 Lloc + Er −−−→ Mη(0), (10.26) or, equivalently, it turns out that

(E 4 M + ) ∩ B rn as r → 0+. (10.27) η(0) r

Proof. Denote by ηr and µr the rescaling of η and µ respectively, in such a way that 1 D( Er ) = ηr · µr.

Step 1. By Lemma 10.23 it turns out that

µ(B ) µ (B ) = r ≤ c(n) =⇒ µ (B ) ≤ c(n) · Rn−1 (10.28) r 1 rn−1 r R for every R > 0. Similarly, by Lemma 10.24 it turns out that

0 c1(n) ≤ |Er ∩ B1| ≤ c1(n), from which it follows that

n 0 n c1(n) · R ≤ |Er ∩ BR| ≤ c1(n) · R (10.29) for every R > 0. The compactness property of the f.p.s. (see Theorem 10.3), together with the inequality (10.28), imply that, up to subsequences,

L1 1 loc 1 Er −−−→ E0

1 locally 1 D ( Er ) −−−−→ D ( E0 ) 142 10.5. FINITE PERIMETER SETS: STRUCTURE THEOREM

∗ for some f.p.s. E0. Since 0 belongs to the reduced boundary ∂ E, we have that

Z + − |η(x) − η(0)| dµ(x) −−−−→r→0 0, Br which means that for every R > 0

Z r→0+ − |ηr(x) − η(0)| dµr(x) −−−−→ 0. BR

We may assume without loss of generality that η(0) = e1, where {e1, . . . , en} is the standard orthonormal basis of Rn. It turns out that 1 D ( E0 ) = η(0) · µ0, i 1 from which we infer that D ( E0 ) is zero for every i = 2, . . . , n.

1 Step 2. By Lemma 10.26 it turns out that E0 has a representative which depends only on the ﬁrst variable x1, and thus we can infer that

+ E0 = x0 + Mη(0)

n for some x0 ∈ R . It remains to prove that x0 is necessarily equal to zero.

We argue by contradiction: suppose that x0 is not 0. Then one can ﬁnd a radius R > 0 small enough such that the Lebesgue measure of the intersection E0 ∩ BR is zero, in contradiction with the inequality (10.29). Therefore

1 + Lloc + E0 = Mη(0) and Er −−−→ E0 = Mη(0), which is exactly what we wanted to prove.

1 n Lemma 10.26. Let f ∈ Lloc(R ) be a function such that Dif ≡ 0 ∀ i = 2, . . . , n.

1 Then there exists fe ∈ Lloc(R) such that

n n fe(x1) = f(x1, . . . , xn) for L -almost every x = (x1, x2, . . . , xn) ∈ R .

1 1 Lemma 10.27. Let Er := r (E − x) = r E. Then

locally n−1 1 1 + 1 D( Er ) −−−−→ D( ) = η(0) η(0)⊥ ·H . (10.30) Mη(0)

Lemma 10.28. For every r > 0 it turns out that

n−1 µ(Br) ∼ αn−1r . (10.31)

Proof. We divide the argument into two steps to ease the notation. 143 10.5. FINITE PERIMETER SETS: STRUCTURE THEOREM

Step 1. First, we deduce from Lemma 10.27 that

locally 1 1 + |D( Er )| −−−−→ D( ) . (10.32) Mη(0)

Indeed, if η(0) = e1, then

Z r→0+ − |ηr(x) − η(0)| dµr(x) −−−−→ 0, ∀ R > 0 BR implies that 1 + 1 ∂ Er r→0 ∂ E0 −−−−→ , ∂x1 ∂x1 + which is exactly what we wanted to prove since E0 = Mη(0).

Step 2. From (10.32) it follows that4

µ(B ) r = µ (B ) −−−→Hn−1(η(0)⊥ ∩ B ) = α rn−1 r 1 1 n−1 for almost every r > 0. To conclude the proof, we need the following auxiliary lemma.

Lemma 10.29. Let (λn)n∈N be a sequence of vector measures on X. Assume that λn := τn · µn converges in the sense of measures (weakly-∗) to a measue λ := τ · µ, and assume that there exists a constant vector v such that

Z n→+∞ |τn − v|(x) dµn(x) −−−−−→ 0. X

Then µn converges to µ and λ = v · µ almost everywhere.

We have ﬁnally introduced all the technical tools to prove the statements of the De Giorgi’s theorem presented above.

Proof of Theorem 10.21.

(1) The first statement about the density and the support of µ is an immediate consequence of the argument used in the next point. (2) We sketch two different proofs of the fact that the reduced boundary is (n − 1)- rectifiable. The first proof is due to De Giorgi, while the second one relies more on the sequence of technical lemmas above.

(a) For every r0 > 0 and δ0 > 0 we consider n o S := x ∈ ∂∗E (E 4(x + M + )) ∩ B(x, r) ≤ δ rn for every r ≤ r . . r0, δ0 η(x) 0 0

4Here the reader needs to be careful. The convergence is not true in general, but one can prove that it n−1 ⊥ is enough to ask that H (η(0) ∩ B1) = 0. 144 10.5. FINITE PERIMETER SETS: STRUCTURE THEOREM

The idea is to prove that each Sr0, δ0 is (n − 1)-rectifiable. In fact, we know that the countable union of d-rectifiable set is also d-rectifiable, and hence the thesis follows from the fact that

∗ [ ∂ E = S 1 . n , m 2 (n, m)∈N (b) The support of the measure µ is contained in the reduced boundary ∂∗E; hence by Lemma 10.28 we can infer that

n−1 ∗ µ (B(x, r)) ∼ αn−1r for µ-almost every x ∈ ∂ E. We claim that there exists F ⊆ spt(µ) such that

n−1 n−1 ∗ c1 1F ·H ≤ µ ≤ c2 1F ·H and µ (∂ E \ F ) = 0. Indeed, it is enough to notice that µ (E \ ∂∗E) = 0 as the support of µ is contained in the reduced boundary. Moreover, the (n−1)-dimensional upper density is zero at almost every point of F , that is,

Θ∗(µ, x) = 0 for Hn−1-almost every x ∈ F,

and this implies that  Hn−1 (∂∗E \ F ) = 0, Hn−1 (∂∗E 4 F ) = 0. The measure µ is thus equivalent to the restriction of the Hn−1 measure to the reduced boundary, that is, there are two constants c1, c2 > 0 such that

0 1 n−1 0 1 n−1 n−1 ¬ ∗ c1 ∂∗E ·H ≤ µ ≤ c2 ∂∗E ·H =⇒ µ ∼ H ∂ E. We deduce that the blowup of µ at x is given by

n−1 1η(x)⊥ ·H ,

which means that η(x)⊥ is the approximate tangent plane of ∂∗E at x. Moreover, the lower density is bounded from below; therefore, by Corollary 8.24, we can ﬁnally infer that ∂∗E is (n − 1)-rectiﬁable.

n−1 (3) In the previous point we have proved that µ is equivalent to 1∂∗E ·H , which means that there exists a function g such that

n−1 µ = 1∂∗E g ·H . Therefore it is enough to show that g is equal to 1 Hn−1-almost everywhere. The (n − 1)-dimensional upper density of the Hausdorﬀ measure is given by

∗ ∗ n−1 ∗ Θn−1(∂ E, x) = 1 for H -almost every x ∈ ∂ E, while the (n − 1)-dimensional upper density of µ is given by

∗ ∗ Θn−1(µ, x) = 1 for µ-almost every x ∈ ∂ E. In particular, it follows that g = 1 almost everywhere (with respect to either µ or Hn−1). 145 10.5. FINITE PERIMETER SETS: STRUCTURE THEOREM

(4) This assertion also follows immediately from the second proof of (2). (5) This assertion simply summarize the content of Lemma 10.25.

We are ﬁnally ready to prove the De Giorgi-Federer Theorem 10.16. The reduced boundary is a subset of the essential boundary, therefore everything follows from the previous result if we are able to prove that n−1 ∗ H (∂∗E \ ∂ E) = 0.

Proposition 10.30 (Isoperimetrical Inequality). Let E ⊆ Rn be a ﬁnite perimeter set in B(x, r). Then there exists an universal constant c := c(n) > 0 such that5

n−1 n (|E| ∧ |B(x, r) \ E|) ≤ c · PerB(x, r)(E). (10.33)

Proof. The Sobolev Embedding Theorem implies that the immersion

∗ BV(B(x, r)) ,→ L1 (B(x, r)),

∗ n where 1 = n−1 , is continuous. Therefore, there exists a universal constant c := c(n) > 0 (which does not depend on the radius and the center of the ball) such that

kuk n ≤ kuk . L n−1 (B(x, r)) BV(B(x, r))

We replace the BV(B(x, r)) norm with the equivalent one

Z − kDuk + u . B(x, r) It turns out that Z ku − − uk n ≤ c(n) kDuk. L n−1 (B(x, r)) B(x, r) By symmetry, we may assume without loss of generality that |E| < |B(x, r) \ E|. Then it is easy to prove that |E| 1 1 (x) − > , if x ∈ E, E |B(x, r)| 2 and the thesis follows from the inequality above since

Z 1 n−1 c(n) kDuk ≥ ku − − uk n ≥ |E| n . L n−1 (B(x, r)) B 2

Lemma 10.31. Under the assumptions of the Theorem 10.16, it turns out that

n−1 ∗ µ (B(x, r)) r =⇒ Θn(E, x) ∈ {0, 1}.

In particular, the point x does not belong to the essential boundary ∂∗E.

5The inequality presented here is not sharp! The result can be reﬁned, but it is not necessary for our purposes. 146 10.6. BACK TO THE CAPILLARITY PROBLEM

n Proof. Let x ∈ R be a point such that Θn(E, x) 6= 0, 1 (i.e., x belongs to the essential boundary). Then there exists δ > 0 and a sequence (rn)n∈N converging to 0 such that either |E ∩ B(x, r )| ∃ (r ) : r −−−−−→k→+∞ 0 and nk ≥ δ, nk k∈N k rn nk or |E ∩ B(x, r )| ∃ (r ) : r −−−−−→k→+∞ 0 and nk ≤ α − δ, nk k∈N k rn n nk Either way the function |E ∩ B(x, r)| 3 r 7−→ R rn is continuous; hence there exists λ0 < λ real numbers such that

|E ∩ B(x, r )| λ0 ≤ nk < λ rn nk

k→+∞ for rnk −−−−−→ 0. On the other hand, the isoperimetrical inequality (10.33) implies that

n−1 |E ∩ B(x, r) \ E| n ≤ c · PerB(x, r)(E) = µ (B(x, r)) , from which we derive a contradiction with the assumption µ(B(x, r)) rn−1 by noticing that 0 n−1 n−1 λ r ≤ |E ∩ B(x, r) \ E| n ≤ c · PerB(x, r)(E) = µ (B(x, r)) for every r > 0 small enough.

Lemma 10.32. Under the assumptions of the Theorem 10.16, it turns out that

n−1 ∗ H (∂∗E \ ∂ E) = 0.

Proof. Equivalently, we may prove that for Hn−1-almost every point x∈ / ∂∗E, it turns out that x does not belong to ∂∗E as well. By Lemma 10.31, it suﬃces to prove that

µ (B(x, r)) rn−1, but this follows easily from the fact that

µ = Hn−1 ¬ ∂∗E.

10.6 Back to the Capillarity Problem

We are ﬁnally ready to prove the existence of a solution to the capillarity problem for any σ ∈ [−1, 1] using the new notion of essential boundary. 147 10.6. BACK TO THE CAPILLARITY PROBLEM

Framework. Fix Ω ⊂ Rn regular bounded open set, and ﬁx a volume 0 < m < |Ω|.

The capillarity energy is given by

F (E) := Hn−1(Σf ) + σ Hn−1(Σc), (10.34) where we deﬁne  f Σ := ∂∗E ∩ Ω, c Σ := ∂∗E ∩ ∂Ω.

Theorem 10.33. The functional (10.34) is lower semi-continuous on the set of all ﬁnite perimeter set. Moreover, there exists (at least) a minimum point E for F for every σ ∈ [−1, 1].

Proof. We split the proof into two cases: positive and negative σ.

Case ”1 ≤ σ ≥ 0”: The functional (10.34) can be equivalently written as

n−1 n−1 f F (E) = σ H (∂∗E) + (1 − σ) H Σ = (10.35) = σ · Per(E) + (1 − σ) · PerΩ(E).

In particular, the functional (10.35) is equal to the convex sum of two lower semi-continuous functionals, which means that it is also lower semi-continuous.

The functional F is also coercive. Indeed, let (En)n∈N ⊂ X be a uniformly bounded sequence, that is, there exists a constant M ∈ R such that

F (En) ≤ M, ∀ n ∈ N.

X Then, up to subsequences, there exists a ﬁnite perimeter set E such that En −→ E. If σ is strictly greater than 0, then the perimeters are equibounded

σ · Per(En) ≤ F (En) ≤ M, which means that we can apply a compactness theorem for f.p.s. and conclude that the functional is coercive. On the other hand, if σ = 0, then the coerciveness follows immediately from the following estimate:

n−1 n−1 Per(En) ≤ H (∂Ω) + PerΩ(En) ≤ H (∂Ω) + M < +∞.

Case ”−1 ≥ σ ≤ 0”: The idea is to pass everything to the complement, that is, we consider as a variable for the functional (10.34) the set Ec := Ω \ E instead of E. In particular, we notice that there are nice relations between the surfaces, that is,

 f f Σc = Σ , c c Σc = ∂Ω \ Σ . 148 10.6. BACK TO THE CAPILLARITY PROBLEM

It turns out that

F (E) = σ Hn−1(∂Ω) − σ Hn−1(∂Ω) + F (E) =

n−1 n−1 f n−1 c = σ H (∂Ω) + H (Σc ) + |σ| H (Σc) = (10.36)

c n−1 = F|σ|(E ) + σ H (∂Ω).

Since σ Hn−1(∂Ω) is a constant, we can immediately reduce to the previous case 1 ≥ |σ| ≥ 0. Remark 10.12. The restriction of (10.34) to any smooth subset Ω in X agrees with the classical capillarity energy presented in the previous section, and the relaxation of F to X is given by F itself. Therefore every minimizing sequence, made up of smooth sets, converges to the minimum of F on X . Remark 10.13. Up to dimension 8, the inner regularity theory of the capillarity problem works just ﬁne; the boundary regularity, on the other hand, is quite messy.

Remark 10.14. A similar problem arises when dealing with Plateau’s type problems. More precisely, the surface Σ = (∂∗Emin) ∩ Ω is closed (in Ω), and it is also analytic up to dimension 7. As we shall prove at the end of the course, starting from dimension 8, one can ﬁnd a counterexample to this statement (actually, it is possible to ﬁnd a minimal surface such that it is analytic outside of a closed singular set of dimension n − 8). Chapter 11

Appendix

11.1 First Variation of a Functional

d n Framework. Let Σ be a d-rectifiable and H -finite set in R , and let (Φt)t∈R be a one n n parameter family of diffeomorphisms of R such that Φ0 = idR . Let

dΦt(x) v(x) := , dt t=0 and suppose that Σt := Φt(Σ) is the collection of all the competitors (of Σ) to a certain minimum problem.

Lemma 11.1. Let x ∈ Σ be a point, and let {e1, . . . , ed} be an orthonormal basis of the tangent space Tan(Σ, x). Then it turns out that d Z dH (Σt) d t=0 = divT (v)(x) dH (x), (11.1) dt Σ where divT (−) denotes the divergence along the tangent plane, that is,

d X ∂ (hv, eii) div (v)(x) = (x). T ∂e i=1 i

Proof. The map Φt is a diﬀeomorphism for every t; hence we can apply the area formula (7.6) to Σt, and obtain the following identity: Z d d H (Σt) = JT (Φt)(x) dH (x), (11.2) Σ where JT denotes the tangential Jacobian, that is,

p t JT (Φt) = det ((∇T Φt) (∇T Φt)). Fix x ∈ Σ. By Taylor’s expansion theorem one can prove that  Φt(x) = x + t v(x) + Ox(t),

dΦt(x) = Id + t dv(x) + Ox(t). 150 11.1. FIRST VARIATION OF A FUNCTIONAL

n Let {e1, . . . , en} be an orthonormal basis of R such that {e1, . . . , ed} is an orthonormal basis of the tangent space Tan(Σ, x). Then we can write the relations above as follows: " # ! Idd×d 0 ∇T v ∇ Φ = + t + O (t). T t 0 0 0 x If we refer to the ﬁrst matrix on the right-hand side by J, then it is easy to prove that

t t t (∇T Φt) (∇T Φt)(x) = Idd×d(x) + t J ∇T v(x) + (∇T v(x)) J + Ox(t), from which it follows that p t JT (Φt)(x) = det ((∇T Φt) (∇T Φt)) '

r t t ' 1 + t · tr J ∇T v(x) + (∇T v(x)) J + Ox(t) =

p = 1 + 2t divT (v)(x) + Ox(t) '

' 1 + t divT (v)(x) + Ox(t), where the ﬁrst approximation ' follows from the Taylor’s expansion

det(Idn×n + tA) ' 1 + t Tr(A) + o(t) as t → 0, √ and the last approximation follows from the Taylor’s expansion of 1 + x. In particular, it turns out that Z Z d d JT (Φt)(x) dH (x) = (1 + t divT (v)(x) + Ox(t)) dH (x) = Σ Σ Z d d = H (Σ) + t divT (v)(x) dH (x) + o(t). Σ Notice that the remainder needs a reasonable assumption to behave properly: For example, we can require either Σ bounded or Φt is the identity outside of a compact set K for every t ∈ R. In conclusion, it is enough to take the derivative with respect to the time of (11.2), and evaluate it at t = 0, to obtain exactly the sought formula (11.1). Theorem 11.2 (Divergence Theorem). Let Σ be a compact (d − 1)-dimensional surface with a boundary ∂Σ of class C2. Then Z Z d d−1 ~ d−1 d−2 H (Σt) t=0 = − H(x)v(x) dH (x) + η∂Σ(x)v(x) dH (x), (11.3) dt Σ ∂Σ where H~ (x) is the mean curvature vector of Σ at x and η∂Σ is the outward normal of ∂Σ at x. n Lemma 11.3. Let Σ0 ⊂ R be a hypersurface which minimizes the area (the (n − 1)- dimensional volume) among all Σ such that

Σ 4 Σ0 ⊂⊂ Ω, where Ω is a ﬁxed open set in Rn. Then ~ HΣ0 (x) = 0, ∀ x ∈ Ω ∩ Σ0. 151 11.1. FIRST VARIATION OF A FUNCTIONAL

Proof. Let v be a compactly supported smooth vector ﬁeld on Ω. Then there exists a family 1 n (Φt)t∈I of diﬀeomorphisms with v as initial speed and Φ0 = idR . It turns out that the hypersurfaces (Σt)t∈I := (Φt(Σ0))t∈I are all competitors for the area problem; hence the function n−1 t 7−→ H (Σt) has a (local) minimum at t = 0. From the divergence formula (11.3) we obtain the relation Z d n−1 ~ n−1 H (Σt) t=0 = 0 =⇒ HΣ0 (x)v(x) dH (x) = 0, dt Σ since the boundary term is equal to zero. Therefore, a straightforward application of the fundamental lemma in calculus of variation allows us to infer that ~ HΣ0 (x) = 0, ∀ x ∈ Ω ∩ Σ0.

Remark 11.1. If Σ is a ﬁnite perimeter set, then a similar computation shows that Z divT (v) = 0, ∂∗Σ which means that we cannot choose any smooth vector ﬁeld v when dealing with it.

We can summarize the results obtained in this section with a slightly diﬀerent version of the divergence formula (11.3), which holds for every smooth vector ﬁeld v. Proposition 11.4. Let Σ be a compact (n−1)-dimensional (hyper)surface with a boundary ∂Σ of class C2. Then Z Z Z n−1 n−1 n−2 divT (v)(x) dH (x) = − H~ (x)v(x) dH (x) + η∂Σ(xv(x) dH (x) (11.4) Σ Σ ∂Σ for every2 v ∈ C∞(Rn).

Proof. We can decompose the vector ﬁeld as follows:

v = vN · η∂Σ + vT , where vN is the normal component, and vT is the tangential component. Then one can easily check that the following identity holds:

divT (v) = divT (vN ) ·η∂Σ + vN · divT (η∂Σ) + divT (vT ). =0

We notice that divT (η∂Σ) is the trace of the second fundamental form, which means that

divT (η∂Σ) = |H~ |(x) =: H(x).

1 For example, the reader may consider the family of diffeomorphisms Φt(x) := x + t v(x), which is well-defined in a small neighborhood of 0. 2Here we can take any smooth vector field v because Σ is compact by assumption. If not, then v needs to be a compactly supported vector field. 152 11.1. FIRST VARIATION OF A FUNCTIONAL

The usual divergence theorem for a smooth vector ﬁeld gives us the identity Z Z n−1 n−2 divT (v) dH (x) = v(x)η∂Σ(x) dH (x). Σ ∂Σ If we plug the ﬁrst relation (for the divergence) into the second one (divergence theorem for smooth functions), then it turns out that Z Z Z n−1 n−1 n−2 divT (v)(x) dH (x) = − H~ (x)v(x) dH (x) + η∂Σ(x)v(x) dH (x), Σ Σ ∂Σ which is exactly what we wanted to prove.

Remark 11.2. Let Σ ⊂ Rn be a compact (n − 1)-dimensional surface with a boundary ∂Σ of class C2. If Σ minimizes the (n − 1)-dimensional volume among all Σe such that Σe 4 Σ ∩ ∂Σ = ∅, then the mean curvature H~ Σ is identically equal to 0.

We can state and prove a similar result for finite perimeter sets, but first we need a technical result concerning the closure of BV(Ω) with respect to the action of a diffeomorphism.

Lemma 11.5. Let Ω be a ﬁxed open set in Rn, and let Φ be a diﬀeomorphism of Ω. If u ∈ BV(Ω), then the composition u ◦ Φ−1 also belongs to BV(Ω).

Lemma 11.6. Let Ω be a ﬁxed open set in Rn, and let E be a ﬁnite perimeter set which minimizes the perimeter inside Ω among all Ee f.p.s. in Ω such that

E 4 Ee ⊂⊂ Ω up to null sets. Then it turns out that Z n−1 ∞ divT (v) dH = 0, ∀ v ∈ Cc (Ω). ∂∗E

Proof. Fix v compactly supported smooth vector field on Ω. Then there exists a family 3 n (Φt)t∈I of diffeomorphisms with v as initial speed and Φ0 = idR . Since Φt(x) = x for every x which does not belong to the support of v, Lemma 11.6 proves that (Et)t∈I := (Φt(E0))t∈I is a collection of finite perimeter set such that

∂∗Et = Φt(∂∗E).

In particular, the f.p.s. Et are all competitors for the perimeter problem; hence the function

t 7−→ PerΩ(Et) admits a (local) minimum at t = 0. The thesis follows from a simple identity which is left to the reader: Z d (PerΩ(Et)) n−1 t=0 = divT (v) dH . dt ∂∗E

3 For example, the reader may consider the family of diﬀeomorphisms Φt(x) := x + t v(x), which is well-deﬁned in a small neighborhood of 0. 153 11.1. FIRST VARIATION OF A FUNCTIONAL

Deﬁnition 11.7 (Weak Mean Curvature). Let Σ be a d-rectiﬁable set in Rn. We say that p n d ¬ Σ has mean curvature in the weak sense if and only if there exists ~g ∈ L R , H Σ such that Z Z d d ∞ n divT (v)(x) dH (x) = ~g(x)v(x) dH (x), ∀ v ∈ Cc (R ). Σ Σ Remark 11.3. Let Σ be a compact (n − 1)-dimensional (hyper)surface with a boundary ∂Σ of class C2. If Σ admits a weak mean curvature ~g, then the following properties hold:

(i) The weak mean curvature ~g coincides with the mean curvature H~ almost everywhere. (ii) The boundary of Σ is empty4. Definition 11.8 (k-Varifold). An integral k-dimensional varifold is a couple (Σ, θ), where n 1 n k ¬ Σ is a k-rectifiable set in R , and θ :Σ −→ N is a function in L R , H Σ . Definition 11.9 (Varifold W.M.C.). Let (Σ, θ) be a k-dimensional varifold in Rn. We say p n d ¬ that Σ has mean curvature in the weak sense if and only if there exists ~g ∈ L R , H Σ such that Z Z k k ∞ n θ(x)divT (v)(x) dH (x) = θ(x)~g(x)v(x) dH (x), ∀ v ∈ Cc (R ). Σ Σ Exercise 11.1. Let Σ be a compact (n − 1)-dimensional (hyper)surface with a boundary ∂Σ of class C2, and let θ ∈ BV(Σ; N). Suppose that Z Z θ(x) div(v)(x) dHk(x) = − v(x) dθ(x) Σ Σ

∞ for every tangent vector ﬁeld v ∈ Cc (Σ). Prove that:

(i) The weak mean curvature ~g coincides with the mean curvature H~ almost everywhere. (ii) The boundary of Σ is empty.

(iii) The function θ is locally constant. Exercise 11.2. Let Σ be a compact (n − 1)-dimensional (hyper)surface with a boundary ∂Σ of class C2, and let θ ∈ C∞(Σ; R). Suppose that Z Z θ(x) div(v)(x) dHk(x) = − v(x) dθ(x) Σ Σ

∞ for every tangent vector ﬁeld v ∈ Cc (Σ). Prove or disprove the following formula: ∇ θ ~g = H~ − T . θ 4This follows fairly easily from the integration by parts formula. 154 11.2. REGULARITY IN CAPILLARITY PROBLEMS

11.2 Regularity in Capillarity Problems

Framework. Let Ω ⊂ Rn be an open bounded subset of Rn. In this section, we shall discuss the capillarity problem equilibrium when the container (Ω) is fixed and the volume of the liquid is fixed. More precisely, the drop of liquid (denoted by E), contained in Ω, is at equilibrium if and only if it is a critical point (local minima) of the capillarity energy functional F (E) := Hn−1 (∂E ∩ Ω) + σHn−1 (∂E ∩ ∂Ω) , (11.5) where σ ∈ R is a constant, subject to the constraint that the volume is fixed, e.g. |E| = m.

Mean Curvature. Assume that E is a f.p.s. which minimizes the capillarity energy (11.5) and satisﬁes the constraint on the volume. If E is regular enough, then one can prove that

 f Hf is constant on the free portion of the surface Σ ,

θY is constant on the boundary Γ.

More precisely, the free mean curvature Hf is the Lagrange multiplier associated to the volume constraint, while θY is the solution of the trigonometric equation

cos θY = −σ. (11.6)

The first assertion is easy to prove. Let (Φt)t∈I be a family of diffeomorphisms fixing the boundary, and choose a smooth vector field v in such a way that the volume constraint is not violated, that is, Z the constraint is preserved ⇐⇒ v · η = 0, Σf which means that the first variation of the volume is zero. As a result of the technical tools introduced in the previous section, it turns out that if we set the first variation equal to zero, then we obtain Z Z ∞ H(v · η) = 0, ∀ v ∈ Cc : v · η = 0. Σf Σf The Du Bois-Reymond’s fundamental lemma in calculus of variation allows us to infer that

H Σf = c =⇒ Hf = c, that is, the free mean curvature is constant. The proof of (11.6) is very similar, but we need to consider a family of transformations which moves the boundary. Remark 11.4. If E is not regular enough, then one might translate the problem in the setting of k-varifolds, and prove that the weak mean curvature is constant on Σf . Unfor- tunately, it is not easy at all to give a meaning to θ, and recover something like formula (11.6). 155 11.3. REGULARITY IN THE F.P.S. SETTING

11.3 Regularity in the F.P.S. Setting

Introduction. Let Ω ⊂ Rn be an open bounded subset of Rn, and let E be a ﬁnite perimeter set in Ω minimizing the perimeter PerΩ(−) among all f.p.s. Ee such that

E 4 Ee ⊂⊂ Ω.

Theorem 11.10. The essential boundary ∂∗E is closed in Ω, and there is a representative of E in its equivalence class which satisﬁes the following properties:

(a) The topological boundary ∂E coincides with the essential boundary ∂∗E. (b) The perimeter formula is satisﬁes, that is,

n−1 PerΩ(E) = H (∂E).

Moreover, the representative above can be uniquely characterized as follows:

E = {x ∈ E | Θ(E, x) = 1} .

Lemma 11.11. There exists a universal constant δ0 := δ0(n) > 0 such that for every (x0, r0) ∈ Ω × R satisfying B(x0, r0) ⊂ Ω, it turns out that |E ∩ B(x , r )| r 0 0 ≤ δ =⇒ E ∩ B x , 0 = 0, n 0 0 αnr0 2

|E ∩ B(x , r )| r 0 0 ≥ 1 − δ =⇒ E ∩ B x , 0 = |B(x , r )| , n 0 0 0 0 αnr0 2 almost everywhere.

r0 Truncation Argument. Fix (x0, r0) ∈ Ω × R. For every r ∈ [ 2 , r0], set

Er := E \ B(x0, r).

Step 1. The set E minimizes the perimeter; hence

PerΩ(Er) ≥ PerΩ(E).

The usual isoperimetrical inequality yields to5

1 1− n c(n) |E ∩ B(x0, r)| ≤ PerB(x0, r)(E) =

n−1 = H (∂∗E ∩ B(x0, r)) ≤

n−1 ≤ H (E ∩ ∂B(x0, r))

5The last inequality needs to be justiﬁed. The reader should check very carefully the details since a lot of issues could arise here. 156 11.3. REGULARITY IN THE F.P.S. SETTING

r0 for almost every r ∈ [ 2 , r0]. Recall that 1 1 1 1 n−1 D( Er ) = Ω\B(x0, r) D( E) + ν E∩∂B(x0, r) ·H , from which it follows that ∂∗Er = ∂∗E \ B(x0, r) ∪ (E ∩ ∂B(x0, r)) for almost every admissible r.

6 Step 2. Let v(r) := |E ∩ B(x0, r)| be the n-dimensional Lebesgue measure. Then n−1 v˙(r) = H (E ∩ ∂B(x0, r)) for a.e. admissible r. It follows from the isoperimetrical inequality that 1− 1 c(n) v(r) n ≤ v˙(r), from which it follows that7 1 1 r0 n c(n) r0 v (r ) n − v ≥ . 0 2 n 2 If we rearrange the inequality, then we obtain the following estimate: " 1 # 1 n n 1 r0 n v(r0) c(n) v ≤ αn n − r0. 2 αnr0 2n

In particular, the thesis follows by choosing the universal constant δ0 in such a way that v(r0/2) is either 0 or equal to the volume of the ball, that is, 1 c(n)n δ0 = . αn 2n

Proof of Theorem 11.10. Everything is obvious except the fact that ∂∗E is a closed set; we shall show that the complement is open.

Case 1. Fix a point x0 with density Θ(E, x0) equal to 0. Then there exists r0 > 0 small enough such that |E ∩ B(x0, r0)| n ≤ δ0, αnr0 and thus by Lemma 11.11 it turns out that r E ∩ B x , 0 = 0. 0 2 In particular, it turns out that r Θ(E, x) = 0 ∀ x ∈ E ∩ B x , 0 . 0 2 6Here the reader needs to be extremely careful. A priori v is simply differentiable in the weak sense of distributions, while in the computation above we need to deal with the classical derivative of v. 7To solve the differential inequality one needs to divide by v. If v(r) 6= 0 for a.e. admissible r, then everything is fine. On the other hand, if v(r) = 0 in the middle of the interval, then one needs to check that v is strictly increasing, and thus {r : v(r) = 0} is a null set. 157 11.4. STRUCTURE OF FINITE LENGTH CONTINUA IN RN

Case 2. Fix a point x0 with density Θ(E, x0) equal to 1. Then there exists r0 > 0 small enough such that |E ∩ B(x0, r0)| n ≥ 1 − δ0, αnr0 and thus by Lemma 11.11 it turns out that r r E ∩ B x , 0 = B x , 0 . 0 2 0 2 In particular r Θ(E, x) = 1 ∀ x ∈ E ∩ B x , 0 , 0 2 which means that the complement of ∂∗E is open (i.e., ∂∗E is closed).

Exercise 11.3. What happens if the f.p.s. E minimizes the perimeter PerΩ(−) among all f.p.s. Ee such that E 4 Ee ⊂⊂ Ω, subject to a volume constraint? Is ∂∗E still closed?

11.4 Structure of Finite Length Continua in Rn

The main goal of this section is to prove that a continua (= compact and connected) set K in Rn with ﬁnite length is the image of a surjective path γ such that Length(γ) ≤ 2 ·H1(K) < +∞.

Deﬁnition 11.12. Let I := [a, b]. A continuous path γ : I −→ Rn of ﬁnite length has constant speed c if and only if

Length(γ, J) < c · Length(γ, I) for every J ⊂ I.

Remark 11.5 (Arc Length). Let γ : I −→ Rn be a continuous path of ﬁnite length L. If we set σ(s) := inf {t ∈ [a, b] | Length (γ, [a, t]) ≥ Ls} , then the riparametrization γ ◦ σ : [0, 1] −→ Rn is a continuous path with constant speed L. Remark 11.6 (Tangent Plane). Let K be a continua set in Rn. Then the tangent plane Tan(K, x) is the vector space in the classical sense for H1-almost every x.

Lemma 11.13. Let K be a continua set in Rn with ﬁnite length. Then K is connected by arcs and, more precisely, it is connected by injective paths of ﬁnite length.

Sketch of the Proof. The main idea is to obtain the injective path γ as the uniform limit (with respect to the Hausdorﬀ distance) of a sequence of ﬁnite length paths γδ satisfying certain properties. 158 11.4. STRUCTURE OF FINITE LENGTH CONTINUA IN RN

Step 1. Fix x0, x ∈ K. For every δ > 0 we may consider the (almost) shortest δ-chain of δ δ δ δ points x0, . . . , xn ∈ K and times 0 =: t0 ≤ · · · ≤ tn := 1, where n depends on δ, such that the following properties hold:

δ δ i) The ﬁrst point is x0 := x0, and the ﬁnal point is x := xn for every δ > 0. ii) The distance between two consecutive points is less than δ, that is, δ δ |xi − xi−1| < δ, and the total length is less than the length of K, that is, n X δ δ 1 |xi − xi−1| < H (K). i=1

δ δ δ iii) For every i = 1, . . . , n it turns out that γ reaches the point xi at the time ti . iv) The Lipschitz constant of γδ is less or equal than 4 times its length.

The reader may prove that the curve γ can be obtained as the uniform limit of the sequence δ (γ )δ>0.

Theorem 11.14. Let K be a continua set in Rn with finite length. Then there exists a surjective path γ : [0, 1] −→ K such that Length(γ) ≤ 2 ·H1(K) < +∞. In particular, every continua H1-finite set is rectifiable.

Proof. The assertion is an immediate consequence of Lemma 11.13. Remark 11.7 (Degree). The degree of γ at x, which will be denoted by deg(γ, x), is well- 1 ∞ n n deﬁned for H -almost every x. In particular, for every vector ﬁeld v ∈ Cc (R ; R ), it turns out that Z v(γi(s))γ ˙ i(s) ds = 0 for every i ∈ N, which means that Z ∞ n n v(γ(s))γ ˙ (s) ds = 0, ∀ v ∈ Cc (R ; R ).

∞ Since v(γn) converges uniformly to v(γ) andγ ˙ n converges toγ ˙ in Lw∗ (i.e., in the weak-∗ topology of L∞), then the area formula implies that Z γ˙ (s) Z X 0 = v(γ(s)) J(γ)(s)ds = v(x) τ(s) · deg(γ, x) dH1(x), |γ˙ (s)| K s∈γ−1(x) where γ˙ (s) τ(s) := |γ˙ (s)| indicates the orientation of the curve. In particular, it turns out that τ(x) · deg(γ, x) = 0 for H1-almost every x, which means that the cardinality of the set γ−1(x) is even for H1- almost every x. 159 11.5. LOWER SEMI-CONTINUITY: GOLAB THEOREM

11.5 Lower Semi-Continuity: Golab Theorem

In this brief section, we use the structure theorem of a continua set to prove the famous Golab compactness result.

n Theorem 11.15 (Golab). Let (Ki)i∈N be a sequence of continua sets in R . If Ki converges to a continua set K with respect to the Hausdorﬀ distance, then

1 1 lim inf H (Ki) ≥ H (K). i→+∞

Proof. By Theorem 11.14 for every i ∈ N there exists a surjective path γi of ﬁnite length that parametrizes the corresponding continua set Ki. Up to subsequences, it turns out that γi converges to some path γ, which parametrizes K. The preimage γ−1(x) is nonempty, and its cardinality is even (see Remark 11.7). There- fore, one can easily infer that

−1 1 γ (x) ≥ 2 at H -almost every x ∈ K, which means that 1 H1(K) ≤ · Length(γ). 2 The length is a semi-lower continuous functional

Length(γ) ≤ lim inf Length(γi), i→+∞ and by Theorem 11.14 it turns out that

1 Length(γ) ≤ lim inf Length(γi) ≤ 2 lim inf H (Ki). i→+∞ i→+∞

If we combine the two inequalities, then we obtain the thesis:

1 1 1 H (K) ≤ · Length(γ) ≤ lim inf H (Ki). 2 i→+∞

11.6 Simons’ Cone

In this ﬁnal section, we show that the minimality of the functional Hn−1(− ∩ K) may be achieved via a calibration. We shall apply this result to prove the minimality of the Simons’ Cone, which is a singular surface, in dimension n = 8.

Theorem 11.16 (Minimality through Calibration). Let Σ0 be a compact (n−1)-dimensional n hypersurface in R with boundary ∂Σ0 oriented by a continuous unit normal η0. Assume n that there exists a calibration for Σ0, that is, a vector ﬁeld v, deﬁned on the space R , with the following properties:

(i) The vector ﬁeld v coincides with η0 on Σ0. 160 11.6. SIMONS’ CONE

(ii) The norm of v is less than or equal to 1. (iii) The divergence of v is identically zero.

n−1 Then Σ0 is the hypersurface minimizing the functional H (−) among all the (n − 1)- dimensional hypersurfaces Σ with the same boundary, oriented in the same way.

Proof. A straightforward computation shows that Z n−1 n−1 H (Σ0)= v · η0 dH = Σ0 Z n−1 = v · η0 dH ≤ Σ

≤ Hn−1(Σ), where the blue equality follows from (i), and the red follows from (ii), (iii), and the divergence theorem.

Theorem 11.17 (Minimality through Calibration, II). Let Σ0 be a complete (n − 1)- dimensional hypersurface in Rn without boundary. Assume that there exist a Borel set E n and a calibration for Σ0, that is, a vector ﬁeld v, deﬁned on the space R , with the following properties:

(i) The vector ﬁeld v coincides with η0 on Σ0. (ii) The norm of v is less than or equal to 1. (iii) The divergence of v is less than or equal to zero in E, and greater than or equal to zero in the complement Rn \ E.

(iv) The surface Σ0 is the boundary of E, and η0 is the outwards normal.

n−1 Then Σ0 is the hypersurface minimizing the functional H (− ∩ K) among all the (n − 1)- dimensional hypersurfaces Σ which agrees with Σ outside of a compact set K.

Simons’ Cone. The surface

4 4 ΣS := (x, y) ∈ R × R | |x| = |y| admits a calibration v. In particular, the Simons’ cone is the minimal surface (with respect to the (n − 1)-dimensional Hausdorﬀ measure) in dimension n = 8, and it is easy to prove that the origin (0, 0) is a singular point (as stated in the ﬁnal remark of the previous chapter). We will not present a proof here, but the calibration is given by ∇f(x, y) |x|4 − |y|4 v(x, y) := , where f(x, y) = , f(x, y) 4 and the Borel set E is given by

4 4 E := (x, y) ∈ R × R | |x| < |y| . Index

approximate tangent cone, 99 outer measure, 16 approximate tangent plane, 92 push-forward, 71 approximately continuous function, 53 support, 50 tightness, 14 Besicovitch Covering Lemma, 40 total variation,9 Besicovitch Covering Theorem, 42 vector-valued,9 bounded variation, 105 weak-∗ convergence, 11 alternative definition, 108 open set condition (OSC), 63 Carathéodory measurable, 16 Carathéodory Theorem, 16 purely unrectifiable set, 88 cone, 92 Rademacher Theorem, 77 essential boundary, 131 rectifiable set, 85 reduced boundary, 132 fine cover, 37 Riesz Theorem, 10 finite perimeter set, 117 relative, 117 self-similar set, 62 set diameter, 21 Hausdorff dimension, 27 Sobolev critical exponent, 112 Hausdorff measure, 23 Sobolev Embedding Theorem, 112 upper density, 45 space-filling curve,8 Hausdorff spherical measure, 26 strong tangent plane, 92 Hutchinson Theorem, 63 tangent bundle, 91 Lipschitz boundary, 109 tangent cone, 97 Lipschitz map, 75 trace theorem, 114 Ascoli-ArzelàTheorem, 75 Kirszbraun Theorem, 76 universally measurable, 21 Mc Shame Theorem, 76 varifold, 147 measure,9 Vitali Covering Lemma, 35 asymptotically doubling, 44 Vitali Covering Theorem, 36 Banach-Alaoglu, 11 convergence in variation, 15 weak mean curvature, 147 density, 44 varifold, 147 doubling, 36 integralgeometric, 69 invariant, 72 locally finite, 36 mass,9 orthogonal, 50

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