Programme GMT 2018 19.Pdf (325.34

GEOMETRIC MEASURE THEORY Academic Year 2018/19 Francesco Serra Cassano

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

1. Recalls and complements of measure theory. 1.1. Measures and outer measures, approximation of measures. Recalls of some important notions and results of abstract measure theory concerning outer measures and measures. Approximation of Radon measures on l.c.s. metric spaces (◦): let (X, d) be a separable, locally compact metric space and ϕ (respectively µ) be a Radon outer measure (respectively Radon measure) on X. Then (i) for each B ⊂ X, ϕ(B) = inf{ϕ(U): U ⊃ B,U open} 1 2

(respectively, for each B ∈ B(X), µ(B) = inf{µ(U): U ⊃ B,U open}); (ii) for each B ∈ Mϕ, ϕ(B) = sup{ϕ(K): K ⊂ B,K compact} (respectively, for each B ∈ B(X), µ(B) = sup{µ(K): K ⊂ B,K compact}). 1.2. Convergence and approximation of measurable functions: Severini- Egoroff’s and Lusin’s theorems. Severini-Egoroff’s theorem :let (X, M, µ) be a measure space with µ finite. Suppose fh : X → R (h = 1, 2,... ) and f : X → R are measurable functions that are finite µ-a.e.on X. Also, suppose that (fh)h converges pointwise µ-a.e. to f. Then for each > 0 there exists a set A ∈ M such that µ(X \ A) < and fh → f uniformly on A, that is sup |fh(x) − f(x)| → 0 as h → ∞ . . x∈A Lusin’s theorem let µ be a Radon outer measure on a locally compact, separable metric space X . Let f : X → R be a µ-measurable function such that there exists a Borel set A ⊂ X with µ(A) < ∞, f(x) = 0 ∀ x ∈ X \ A and |f(x)| < ∞ µ − a.e. x ∈ X. 0 Then, for each > 0, there exists g ∈ Cc (X) such that µ ({x ∈ X : f(x) 6= g(x)}) < . Moreover g can be chosen such that sup |g(x)| 6 sup |f(x)| . x∈X x∈X A consequence of Lusin’s theorem is the following. Corollary (◦): Let µ be a Radon outer measure on a locally compact, separable metric space X and let f : X → R be a µ-measurable function. Then there exist a Borel function g : X → R such that f = g µ-a.e. on X. 1.3. Absolutely continuous and singular measures. Radon-Nikodym and Lebesgue decomposition theorems. Definitions of absolutely continuous and mutually singular measures in a measure space (X, M). Recalls of the Radon-Nikodym and Lebesgue’s decomposition theorems. 1.4. Signed vector measures. Definition and notions concerning signed measures. Examples of signed measures. Lebesgue decomposition theorem for signed measures : Let (X, M, µ) be a measure space with µ σ-finite, and ν : M → R be a σ-finite signed measure . Then there are two signed measures νac, νs : M → R such that

(LD) νac << µ, νs ⊥ µ, ν = νac + νs , and there exists a measurable function w : X → R such that either w+ or w− is integrable with respect to µ such that Z (RN) νac(E) = w dµ ∀ E ∈ M . E 3

Moreover both decomposition (LD) and representation (RN) are unique. Definitions and notions concerning vector signed measures. Theorem (Properties of the total variation of a signed vector measure) : (i) Let ν be a signed vector measure on (X, M). Then its total variation |ν| is a positive measure. (ii) If ν is a vector measure, then |ν| is a positive finite measure, that is |ν|(X) < ∞. Remark: the above theorem shows that for any real measure ν, its positive and negative part are positive finite measures, hence the decomposition ν = ν+ − ν− holds; it is known as the Jordan decomposition of ν. Remark: It is immediate to check that Rm-valued vector measures can be added and multiplied by real numbers, hence they form a real vector space; moreover, an easy consequence of the above theorem is that the total variation is a norm on the space of measures, which turns out to be a Banach space .

Example: Given a measure space (X, M, µ) and a vector function w = (w1, . . . , wm): m X → , with each wi : X → (i = 1, . . . , m) measurable functions such that either R R m wi,+ or wi,− is integrable. Let us deﬁne the vector set function µw : M → R deﬁned as follows Z Z Z (1.1) µw(E) = w dµ := w1 dµ, . . . , wm dµ E ∈ M . E E E

Then it is easy to see that µw is a signed vector measure and its total variation is computed in the following proposition. Proposition (◦) Let (X, M, µ) be a measure space and let w = (w1, . . . , wm): X → m R , with each wi : X → R (i = 1, . . . , m) measurable functions such that either wi,+ or wi,− is integrable. Consider the vector signed measure µw in (1.1). Then Z (1.2) |µw|(E) = |w| dµ ∀E ∈ M . E

Definitions of integrals on a measure space (X, M) and with respect to a Rm-vector measure. Definitions of absolute continuity and singularity for vector signed measures. Lebegue decomposition theorem for vector signed measures (◦): let ν and µ be m respectively a R -valued σ-finite measure and a σ-finite positive measure on a measure space (X, M). Then there is a decomposition of ν such that

(1.3) ν = νac + νs , m where νac and νs are still R -valued signed measures on (X, M) with νac << µ and νs ⊥ µ. The decomposition is unique. Moreover there exists a unique vector function m w = (w1, . . . , wm): X → R with either wi,+ or wi,− (i = 1, . . . , m) integrable functions w.r.t. µ such that Z (1.4) νac(E) = µw(E) = w dµ ∀E ∈ M . E 4

Corollary (Polar decomposition for vector measures): let ν be a Rm -valued measure on the measure space (X, M). Then there exists a unique measurable vector function m wν : X → R with |wν(x)| = 1 |ν| a.e. x ∈ X such that ν = |ν|wν , that is Z ν(E) = wν d|ν| ∀E ∈ M . E . 1.5. Spaces Lp(X, µ) and their main properties. Riesz representation theorem. Deﬁnition of of Lp(X, µ) and Lp-norm. p Fisher-Riesz’s theorem : (L (X, µ), k · kLp ) is a B.s. if 1 6 p 6 ∞. Moreover L2(X, µ) turns out to be a Hilbert space with respect to the scalar product Z 2 (f, g)L2 := f g dµ f, g ∈ L (X, µ) . X As a consequence of the proof of Riesz- Fisher’s theorem we have the following p p useful result. Theorem : let (fh)h ⊂ L (X, µ) and f ∈ L (X, µ) with 1 6 p 6 ∞. Suppose that

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

(i) fhk (x) → f(x) µ − a.e. x ∈ X;

(ii) |fhk (x)| 6 g(x) µ − a.e. x ∈ X, ∀ k. 0 Theorem (H¨olderinequality) (◦): let p and p be conjugate exponents, 1 6 p < ∞, that is ( p if 1 < p < ∞ p0 := p − 1 ∞ if p = 1 Let f ∈ Lp(X, µ) and g ∈ Lp0 (X, µ). Then f g ∈ L1(X, µ) and

kfgkL1(X,µ) 6 kfkLp(X,µ) kgkLp0 (X,µ) Riesz representation theorem for the dual of Lp (◦): if 1 < p < ∞, then the mapping T : Lp0 (X, µ) → (Lp(X, µ))0, defined by Z p hT (u), fi(Lp(X,µ))0×Lp(X,µ) := u f dµ ∀f ∈ L (X, µ) , X is an isometric isomorphism, that is, T is a linear, one-to-one, onto mapping and p0 kT (u)k(Lp(X,µ))0 = kukLp0 (X,µ) ∀u ∈ L (X, µ) . If p = 1, the same conclusion holds under the additional assumption that µ is σ-finite. We will mean this feature by means of the identification (1.5) Lp0 (X, µ) ≡ (Lp(X, µ))0 . Remark 1.1. Identification (1.5) may fail in the other cases. 5

Theorem (Approximation in Lp by continuous functions ) : let (X, B(X), µ) be 0 a measure space with (X, d) l.c.s. and µ Radon measure. Then Cc (X) is dense in p (L (X), k · kLp ), provided that 1 6 p < ∞.

0 m Deﬁnition of continuity for a linear functional L :(Cc (X)) → R and its characterization by means of sequences. Riesz representation theorem : let (X, d) be a separable, locally compact metric 0 m space and let L :(Cc (X)) → R be a continuous linear functional. Then there exist a Radon measure µ : B(X) → [0, ∞] and a Borel measurable vector function m−1 wL : X → S such that Z 0 m (1.6) L(u) = (wL, u)Rm dµL ∀u ∈ (Cc (X)) , X that is, L = wLµ, and µL is characterized by the following identity: for each open set A ⊂ X m (1.7) µL(A) = sup {L(u): u ∈ (Cc(X)) , sptu ⊂ A, kuk∞ 6 1} . Moreover representation (1.6) is unique.

Definitions of Radon vector measure and finite Radon vector measure. Spaces of m m measures (Mloc(X)) and (M(X)) . The Riesz representation theorem provides a characterization of the measures m m spaces M(X) and (Mloc(X)) as dual spaces of suitable spaces of continuous functions. Corollary (Characterization of Mloc(X)): let (X, d) be a locally compact separable metric space and define 0 0 0 (1.8) (Cc (X)) := L : Cc (X) → R : L is linear and continuous . Let us define the map 0 0 (1.9) I : Mloc(X) → (Cc (X)) I(ν) := Lν . Then I is an isomorphism (between vector spaces). 0 Proposition: Let ν ∈ M(X), and let Lν : Cc (X) → R be the functional Z 0 L(u) := f dν ∀ u ∈ Cc (X) . X . Then 0 kL k 0 0 := sup |L (u)| : u ∈ C (X), kuk 1 = |ν|(X) . ν (Cc (X),k·k∞) ν c ∞ 6 Theorem (Characterization of M(X)): let (X, d) be a locally compact separable metric space. Let I the map in (1.9). Then 0 0 (i) I(M(X)) = (C0 (X), k · k∞) ; 0 0 (iI) I : M(X) → (C0 (X), k·k∞) is a topological isomorphism, that is an algebraic isomorphism, continuous with its inverse.

0 0 Corollary(◦): let (X, d) be a compact metric space. Then (C (X), k · k∞) is iso- metrically isomporphic to M(X). 6

1.6. Operations on measures. Definition of support for a positive or signed vector measure. Definition of restriction of an outer measure, positive or vector signed measure to a set. Theorem : (i) If ν is an outer measure on X, then so is ν E. Moreover every ν-measurable set is also ν E-measurable. (ii) If ν is a Borel regular outer measure on X and E is ν-measurable with ν(E) < ∞ , then ν E is a Radon outer measure. (iii) If ν is a positive or vector signed measure on a measure space (X, M) and E ∈ M, so is ν E. Definition of push-forward for a positive or vector measure. Example: Push-forward of the classical length measure. Theorem (Cavalieri’s principle): let (X, M) be a measure space, µ a positive measure on it and u : X → [0, ∞] be measurable. Let [0, ∞) 3 t 7→ µ({u > t}) denote the distribution function of u, that is, (1.10) µ({u > t} = µ ({x ∈ X : u(x) > t}) if t ∈ [0, ∞) . Let θ : [0, ∞) → [0, ∞) be (strictly) increasing such that θ(0) = 0, θ : [0,T ] → [0, ∞) is absolutely continuous for each T ∈ [0, ∞). Then Z Z ∞ (1.11) (θ ◦ u) dµ = θ0(t) µ({u > t}) dt . X 0 p In particular, if θ(t) = t with p > 1, then Z Z ∞ up dµ = p tp−1 µ({u > t}) dt . X 0 1.7. Weak*-convergence of measures. Regularization of Radon measures in Rn. m Definition of locally weakly* convergence of measures in (Mloc(X)) and of weakly* convergence in (M(X))m. Proposition (Locally weak* convergence vs. weak* convergence): assume that (νh)h, ν ⊂ Mloc(X) . Then they are equivalent: ∗ (i) νh * ν and suph |νh|(X) < ∞; (ii)( νh)h, ν ⊂ M(X) and (νh)h weakly* converges to ν.

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

Theorem (Characterziation of the locally weak* convergence of positive Radon measures): let (µh)h and µ be positive Radon measures on (X, B(X)).Then the following are equivalent. ∗ (i) µh * µ as h → ∞. 7

(ii) If K compact and A open, then

(1.12) µ(K) > lim sup µh(K) , h→∞

(1.13) µ(A) lim inf µh(A) . 6 h→∞

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

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

(1.14) lim xh = x, xh ∈ sptµh ∀ h ∈ N . h→∞ Proposition (Characterization of the narrow convergence of positive Radon measures) : let (µh)h be a sequence of positive, ﬁnite Radon measures on (X, B(X)) and assume the existence of a positive, ﬁnite Radon measure µ such that

(1.15) lim µh(X) = µ(X) and lim inf µh(A) µ(A) h→∞ h→∞ > for every A ⊂ X open set. Then Z Z 0 (NC) lim u dµh = u dµ ∀ u ∈ Cb (X) h→∞ X X 0 where Cb (X) denotes the class of all bounded continuous function u : X → R. In particular (µh)h weakly* converges to µ. Moreover if (NC) holds so does (1.15), that is (NC) and (1.15) are equivalent. m Theorem : let (νh)h and ν be R -valued Radon vector measures, that is ν : B(X) → Rm, and let µ a positive Radon measure on a l.c.s. metric space (X, d). ∗ (i) If νh*ν, then for every open set A ⊂ X

(1.16) |ν|(A) lim inf |νh|(A) . 6 h→∞ ∗ ∗ (ii) If νh*ν and |νh|*µ, then (1.17) |ν|(B) 6 µ(B) ∀ E ∈ B(X) .

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

ν(E) = lim νh(E) . h→∞ ∗ (iii) If νh*ν and limh→∞ |νh|(X) = |ν|(X) < ∞, then (NC) holds with µh = |νh| ∗ and µ = |ν|. In particular |νh|*|ν|. m Theorem (Weak*-compactness): if (νh)h is a sequence of R - valued ﬁnite Radon m measures on the l.c.s. metric space X, that is (νh)h ⊂ (M(X)) , with suph |νh|(X) < ∞, then it has a weakly*- converging subsequence. Moreover, the map ν 7→ |ν| is lower semicontinuous with respect the weak*-convergence. m Corollary (Local weak* compactness): let (νh)h be a sequence of R - valued Radon m measures on the l.c.s. metric space X,(νh)h ⊂ (Mloc(X)) , such that

sup{|νh|(K): h ∈ N} < ∞ for every compact K ⊂ X; then it has a locally weakly*-converging subsequence. 8

2. Differentiation of Radon measures 2.1. Covering theorems and Vitali-type covering property for measures on Rn. Deﬁnition of a cover and of a ﬁne cover in a metric space.

Theorem (Vitali covering theorem): let G be a family of closed balls in Rn with D = sup {d(B): B ∈ G} < ∞ . Then there exists a (pairwise) disjoint family F ⊆ G, which is at most countable, such that ˆ ∪B∈GB ⊂ ∪B∈F B. where Bˆ is an enlargement of B, that is Bˆ = 5B. Theorem (Vitali covering property for the Lebesgue measure): Let G be a family of closed balls in Rn, which is a ﬁne cover of a (possibly non measurable) set A ⊂ Rn in Rn. Then there exists a disjoint subfamily F ⊂ G, at most countable, such that Ln (A \ ∪F) = 0 , where Ln denotes the n-dimensional Lebesgue outer measure. Example: Vitali covering property does not hold for all Radon measures in Rn

Theorem (Besicovitch’s covering theorem): there are integers P (n)and Q(n) de- pending only on n with the following properties. Let A be a bounded subset of Rn, and let G be a family of closed balls such that each point of A is the centre of some ball of G. (i) There is a ﬁnite or countable subfamily F ⊂ G which covers A and every point of Rn belongs to at most P (n) balls of F, that is, X χA 6 χB 6 P (n) . B∈F

(ii) There are subfamilies F1,..., FQ(n) ⊂ G covering A such that each Fi is disjoint, that is, Q(n) A ⊂ ∪i=1 (∪Fi) and 0 0 0 B ∩ B = ∅ for B,B ∈ Fi with B 6= B . Theorem (Vitali covering property for Radon measures): let ϕ be a Radon o. m. in Rn, A ⊂ Rn (even not ϕ-measurable) and G a family of closed balls. Assume that G is cover of A and (2.1) inf { r : B(x, r) ∈ G } = 0 ∀ x ∈ A. Then there is a disjoint subfamily F ⊂ G, at most countable, such that ϕ (A \ ∪F) = 0 . 9

2.2. Derivatives of Radon measures on Rn. Lebesgue-Besicovitch differentiation theorem for Radon measures on Rn. Definition of upper/lower derivatives and of derivative of a positive Radon measure with respect to an other one. Theorem (Differentation for positive Radon measures): let ν and µ be positive Radon measures on Rn. (i) The derivative Dµν(x) exists and is finite (that is Dµν(x) ∈ [0, ∞)) for µ-a.e. x ∈ Rn. n (ii) The function Dµν : R → [0, +∞] is Borel measurable, by defining Dµν = ∞ on the possible µ-negligible set where it does not exist. (iii) Let n (2.2) A := {x ∈ R : ∃ Dµν(x) ∈ [0, ∞)} . For all Borel sets B ⊂ Rn Z (2.3) Dµν dµ = ν(A ∩ B) 6 ν(B) , B with equality if ν << µ. In this case dν dν D ν(x) = (x) = ac (x) µ-a.e. x ∈ n . µ dµ dµ R dλ denoting the Radon-Nikodym derivative of λ with respect to µ. dµ n (iv) ν << µ if and only if Dµν(x) < ∞ ν-a.e. x ∈ R . Theorem (Lebesgue-Besicovitch differentiation theorem): let µ be a positive Radon n 1 n measure on R and let f ∈ Lloc(R , µ). Then Z n ∃ lim f(y) dµ(y) = f(x) µ − a.e. x ∈ R , r→0 B(x,r) that is, by definition, there exists a µ-negligible set N ⊂ Rn (i.e. µ(N) = 0) such that Z n (?) ∃ lim f(y) dµ(y) = f(x) ∀ x ∈ R \ N. r→0 B(x,r)

Theorem (of Lebesgue points): let µ be a positive Radon measure on (Rn, B(Rn)) 1 n and let f ∈ Lloc(R , µ). Then µ-a.e. x ∈ spt(µ) there exists 1 Z (LP) lim |f(y) − f(x)| dµ(y) = 0 . r→0 µ(B(x, r)) B(x,r) Corollary (Density of a set): let µ be a positive Radon measure on Rn and let E ⊂ Rn be a measurable set. Then ( µ(E ∩ B(x, r)) 1 for µ-a.e. x ∈ E ∃ lim = , r→0 µ(B(x, r)) 0 for µ-a.e. x ∈ Rn \ E that is, µ-a.e. x ∈ E is a point of density 1 for E and µ-a.e. x ∈ Rn \ E is a point of density 0 for E. 10

Theorem (Lebesgue decomposition in terms of derivatives of measures): let ν and µ n be positive Radon measures on R . Let νac and νs denote respecitively the absolutely continuous and singular parts of ν in the Lebesgue decomposition with respect to µ. Then, for each Borel set B, Z νac(B) = Dµν(x) dµ, νs(B) = ν S(B) = ν(S ∩ B) B where S is the µ-negligible Borel set n S = (R \ spt(µ)) ∪ {x ∈ spt(µ): Dµν(x) = ∞ } Deﬁnition of a regualr diﬀerenziation basis for a positive Radon measure on Rn (?).

Theorem (Diﬀerentiation for positive Radon measures with respect to a diﬀerenti- ation basis): let ν and µ be postive Radon measures on Rn, then there exists

ν(Eh(x)) dνac lim = (x) = Dµνac(x) µ-a.e. x ∈ spt(µ) , h→∞ µ(Eh(x)) dµ whenever (Eh(x))h is a diﬀerentiation basis of µ at x.

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

ν(B(x, r)) m (2.4) ∃ w(x) := lim ∈ R . r→0 µ(B(x, r))

Moreover the Lebsesgue decomposition of ν with respect to µ is given by ν = µw + νs where νs(B) = ν S(B) = ν(S ∩ B) and S is the µ-negligible Borel set n S = (R \ spt(µ)) ∪ {x ∈ spt(µ): Dµ|ν|(x) = ∞ } 3. An introduction to Hausdorff measures.

3.1. Carathéodory’s construction and definition of Hausdorff measures on a metric space and their elementary properties; Hausdorff dimension. Caratheodory’s construction in a metric space (X, d) and set function ψ = ψ(F, ζ): P(X) → [0, ∞].

Theorem (i) ψ is a Borel outer measure. (ii) If F ⊂ B(X), then ψ is a Borel regular outer measure.

s Definition of s-dimensional Hausdorff pre-measure Hδ : P(X) → [0, ∞], measure Hs : P(X) → [0, ∞] and spherical Hausdorff measure Ss : P(X) → [0, ∞] in a separable metric space (X, d).

s s s s s s Comparison between H and S : H (A) 6 S (A) 6 2 H (A) ∀ A ⊂ X.

s Theorem : let s ∈ [0, ∞), αs > 0 and ζ(E) := αs d(E) for E ⊂ X. If 11

(i) F = {F ⊂ X : F closed } or (ii) F = {U ⊂ X : F open }, then ψ(F, ζ) = Hs. Corollary(◦): Hs is a Borel regular outer measure.

s Lemma (H -null sets): Let A ⊂ X, 0 < s < ∞ and 0 < δ 6 ∞. Then the following conditions are equivalent: (i) Hs(A) = 0. s (ii) Hδ(A) = 0. (iii) ∀ > 0 ∃ E1,E2, · · · ⊂ X such that ∞ ∞ X s A ⊂ ∪i=1Ei and d(Ei) < . i=1

Theorem : for 0 < s < t < ∞ and A ⊂ X, (i) Hs(A) < ∞ implies Ht(A) = 0, (ii) Ht(A) > 0 implies Hs(A) = ∞.

Deﬁnition of Hausdorﬀ metric in a separable metric space and its basic properties.

3.2. Recalls of some fundamental results on Lipschitz functions between Euclidean spaces and relationships with Hausdorﬀ measures.

Deﬁnition of Lipschitz function bewteen metric spaces.

Theorem (Whitney’s extension theorem): let C be a closed set in Rn. Let f : C → R and v : C → Rn be continuous functions. Deﬁne f(x) − f(y) − v(y) · (x − y) R(x, y) := ∀ x, y ∈ C, x 6= y . |x − y| Suppose that for all compact sets K ⊂ C (3.1) lim sup {|R(x, y)| : x, y ∈ K, 0 < |x − y| < r } = 0 . r→0 ˆ 1 n ˆ ˆ Then there is f ∈ C (R ) such that f|C = f and ∇f|C = v. Theorem (Mc Shane’s extension theorem): let (X, d) be a metric space and f : ˆ ˆ ˆ E ⊂ X → R be L-Lipschitz. Then there is f : X → R such that f|E = f and f is L-Lipschitz.

Corollary: let f : E → m, E ⊂ (X, d) be an L-Lipschitz function. Then there √ R ˆ m ˆ exists an mL-Lipschitz function f : X → R such that f|E = f.

Theorem (Kirszbraun’s theorem) (◦): let f : E → Rm, E ⊂ Rn, be an L-Lipschitz ˆ n m ˆ function. Then there exists an L-Lipschitz function f : R → R such that f|E = f. 12

Theorem (Rademacher’s theorem[Rad]): let f : Rn → R be locally Lipschitz. Then f is differentiable (in classical sense) Ln-a.e., that is, n n ∃ ∇f(x) := (∂1f(x), . . . , ∂nf(x)) for L -a.e. x ∈ R and f(y) − f(x) − df(x)(y − x) (3.2) lim = 0 y→x |y − x| where df(x): Rn → R denotes the (linear) differential map of f at x defined by n df(x)(v) := ∇f(x) · v ∀ v ∈ R . ∞ n n Moreover ∇f ∈ (Lloc(R )) .

Theorem (Approximation of Lipschitz functions) :Let f : Rn → R be a Lipschitz function. Then for each > 0 there is a g ∈ C1(Rn) such that Ln ({x : f(x) 6= g(x)} ∪ {x : ∇f(x) 6= ∇g(x)}) < . In addition, there is a positive constant c = c(n) such that sup |∇g| 6 c Lip(f) . Rn

Theorem (Hausdorff measures vs. Lipschitz maps): let f : E ⊂ Rn → Rm be a Lipschitz map. Then s s s H (f(E)) 6 Lip(f) H (E) ∀ 0 6 s < ∞ . In particular Hdim(f(E)) 6 Hdim(E) 3.3. Hausdorff measures in the Euclidean spaces; H1 and the classical notion of length in Rn; isodiametric inequality and identity Hn = Ln on Rn. Definition of s-dimensional Hausdorff measure in Rn. s-dimensional Hausdorff measures are invariant by translations and s-honogeneous by dilations in Rn.

Theorem (Classical length and H1): let γ : [0, a] → Rn be a curve and denote Γ = γ([0, a]) its support. Then 1 H (Γ) 6 l(γ) and equality holds if γ is injective.

Proposition: s s n (3.3) H (B(x, r)) = c(s, n) r x ∈ R , 0 < r < ∞ with c(s, n) positive and ﬁnite constant only when s = n; for s > n, c(s, n) = 0; for s < n, c(s, n) = ∞. Corollary : (i) Hs is a (non trivial) Radon measure on Rn if and only s = n. 13

(ii) Hdim(A) = n for each (nonemtpy) open set A ⊂ Rn. In particular Hdim(Rn) = n .

Theorem (Isodiametric inequality): n n n L (A) 6 αn d(A) for A ⊂ R .

n n n n n n Theorem (H ≡ L ): L (A) = Hδ (A) = H (A) for each A ⊂ R , 0 < δ 6 ∞.

Deﬁnition of Cantor sets C(λ) in R if λ ∈ (0, 1/2): their basic properties and self-similar structure.

Theorem (Hausdorff dimension of the Cantor sets in R): let log 2 s = 1 log λ and let αs be the constant in the definition of s-dimensional Hausdorff measure. Then s (i) H (C(λ)) 6 αs < ∞; s (ii) H (C(λ)) > αs > 0, s In particular H (C(λ)) = αs and log 2 Hdim(C(λ)) = 1 . log λ

∗ Deﬁnition of k-dimensional upper/lower density, denoted, respectively, Θk(µ, ·) and n Θ∗k(µ, ·), and density, denoted Θk(µ, ·), for a Radon measure µ on an open set of R . If µ = Hk E, we set ∗ ∗ Θk(E, ·) := Θk(µ, ·), Θ∗k(E, ·) := Θ∗k(µ, ·), Θk(E, ·) := Θk(µ, ·) .

Theorem (Estimates of the upper density of a Radon measure): let Ω ⊂ Rn be an open set and µ a positive Radon measure in Ω. Then, for any t ∈ (0, ∞) and any Borel set B ⊂ Ω the following implications hold: ∗ k (3.4) Θk(µ, x) > t ∀ t ∈ B ⇒ µ > t H B,

∗ k k (3.5) Θk(µ, x) 6 t ∀ t ∈ B ⇒ µ 6 2 t H B. Corollary: let k ∈ [0, n] and assume that E ⊂ Rn is Hk-measurable and Hk(E) < ∞. Then k H (E ∩ B(x, r)) k n (3.6) ∃ Θk(E, x) = lim = 0 for H -a.e. x ∈ \ E ; + ∗ k R r→0 αk r

k −k H (E ∩ B(x, r)) k (3.7) 2 6 lim sup ∗ k 6 1 for H -a.e. x ∈ E. r→0+ αk r 14

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

Deﬁnition of Jacobian function Jf : Rn → [0, ∞] for a Lipschitz map f : Rn → Rm. n m Theorem (Area formula): let f : R → R be Lipschitz with n 6 m. Then for each Ln-measurable subset A ⊂ Rn Z Z (AF) Jf dLn = H0 A ∩ f −1(y) dHn(y) . A Rm n m Theorem (Area formula for injective maps): let n 6 m and let f : R → R be an injective Lipschitz function and A ⊂ Rn be a measurable set. Then Z (IAF) Hn(f(A)) = Jf dLn A and Hn f(Rn) is a Radon measure on Rm.

Some applications of the area formula: lenght of a curve and area of a graph.

n m Theorem (Change of variables): let n 6 m and f : R → R be Lipschitz. The for each Ln-integrable function g : Rn → R,   Z Z n X n g(x) Jf(x) dL (x) =  g(x) dH (y) . n m R R x∈f −1(y) In particular, if f is injective, Z Z g(x) Jf(x) dLn(x) = g(f −1(y)) dHn(y) . Rn f(Rn)

n m Theorem (Corea formula): let n > m and let f : R → R be a Lipschitz function. The for each Ln-measurable set A ⊂ Rn Z Z Jf(x) dLn(x) = Hn−m(A ∩ f −1(y)) dLm(y) . A Rm Remark: applying the coarea formula to set A := {x ∈ Rn : Jf(x) = 0}, we get that n−m −1 m m (WMS) H {Jf = 0} ∩ f (y) = 0 L -a.e. y ∈ R . This is a weak variant of Morse-Sard’s theorem which asserts −1 m m (MS) {Jf = 0} ∩ f (y) = ∅ L -a.e. y ∈ R , provided that f ∈ Ck(Rn; Rm) for k = 1 + n − m.

Theorem (Change of variables formula): under the same assumptions of the Coarea Formula. Then for each Ln-measurable function g : Rn → R n−m m m (i) g|f −1(y) is H -summable L -a.e. y ∈ R . 15

(ii) Z Z Z g(x) Jf(x) dLn(x) = g(x) dHn−m(x) dLm(y) , Rn Rm f −1(y)

4. Rectifiable sets and blow-ups of Radon measures

Definition of k-dimensional planes and the orthogonal group in Rn. Definition of countably Hk-rectifiable, locally Hk-rectifiable and Hk-rectifiable set in Rn. Example of a rectifibale set: a Lipschitz k-graph.

4.1. Rectiﬁable sets of Rn and their decomposition in Lipschitz images. Deﬁnition of regular Lipschitz image.

Theorem (Decomposition of rectifiable sets): if Γ is countably Hk- rectifiable in Rn n and t > 1, then there exist a Borel set Γ0 ⊂ R , countably many Lipschitz maps k n k fh : R → R and compact sets Eh ⊂ R such that ∞ k Γ = Γ0 ∪ (∪h=1f(Eh)) , H (Γ0) = 0 . Each pair (fh,Eh) defines a regular Lipschitz image, with Lip(fh) 6 t and −1 t |x − y| 6 |fh(x) − fh(y)| 6 t |x − y|, −1 t |v| 6 |Dfh(x)v| 6 t |v|, −k k t 6 Jfh(x) 6 t , k for every x, y ∈ Eh and v ∈ R . 4.2. Approximate tangent planes to rectifiable sets. Theorem (Existence of approximate tangent spaces): let y − x Φ : n → n, Φ (y) := , y ∈ n . x,r R R x,r r R If Γ ⊂ Rn is a locally Hk-rectifiable set, then for Hk-a.e. x ∈ Γ there exists a unique + k- dimensional plane πx such that, as r → 0 , (Φ ) µ Γ − x x,r # = Hk *∗ Hk π , rk r x that is Z Z 1 y − x k k 0 n lim ϕ dH (y) = ϕ(y) dH (y) ∀ ϕ ∈ Cc (R ) . r→0+ k r Γ r πx In particular k H (Γ ∩ B(x, r)) k ∃ Θk(Γ, x) := lim = 1 for H -a.e. x ∈ Γ . + ∗ k r→0 αk r Definition of the approximate tangent plane to a subset at a point 16

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

TxΓ1 = TxΓ2 .

Theorem (Besicovitch-Marstrand-Mattila): let E a Borel set with Hk(E) < ∞. Then the following are equivalent: (i) E is Hk-rectifiable; k (ii) there exists Θk(E, x) = 1 for H -a.e. x ∈ E. 4.3. Blow-ups of Radon measures on Rn and rectifiability. Definition of cone K(π, t) in Rn. Theorem (Rectifiability criterion): if Γ ⊂ Rn is a compact set, π is a k-dimensional plane in Rn, and there exist δ and t positive with (4.1) Γ ∩ B(x, δ) ⊂ x + K(π, t) ∀ x ∈ Γ , k k n then Γ is H -rectifiable, since there exist finitely many Lipschitz maps fh : R → R k (h = 1,...,N) and compact sets Fh ⊂ R with N Γ = ∪h=1fh(Fh) . Theorem (Rectifiability by convergence of the blow-ups): If µ is a Radon measure on Rn, Γ is a Borel set in Rn, µ is concentrated on Γ (that is µ = µ Γ), and, for n every x ∈ Γ, there exists a k-dimensional plane πx in R such that (Φ ) µ x,r # *∗ Hk π as r → 0+, rk x then µ = Hk Γ and Γ is locally Hk-rectifiable. Definition of purely unrectifiable set, Example of an unrectifiable set. 17

Results with proofs that will be a part of the interview: the student has to agree with the teacher two results, which do not belong to the same chapter of the programme.

• 1. Riesz representation theorem: let (X, d) be a separable, locally compact 0 m metric space and let L :(Cc (X)) → R be a continuous linear functional. Then there exist a Radon measure µ : B(X) → [0, ∞] and a Borel measurable m−1 vector function wL : X → S such that Z 0 m (?) L(u) = (wL, u)Rm dµL ∀u ∈ (Cc (X)) , X

that is, L = wLµ, and µL is characterized by the following identity: for each open set A ⊂ X m µL(A) = sup {L(u): u ∈ (Cc(X)) , sptu ⊂ A, kuk∞ 6 1} . Moreover representation is unique. • 2. Theorem (Characterization of the locally weak* convergence of positive Radon measures): let (µh)h and µ be positive Radon measures on (X, B(X)).Then the following are equivalent. ∗ (i) µh * µ as h → ∞. (ii) If K compact and A open, then

µ(K) > lim sup µh(K) , h→∞

µ(A) lim inf µh(A) . 6 h→∞

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

µ(E) = lim µh(E) . h→∞ ∗ Moreover, if µh*µ as → ∞, then for every x ∈ sptµ there exists (xh)h ⊂ X with lim xh = x, xh ∈ sptµh ∀ h ∈ N . h→∞ m • 3. Theorem (Weak*-compactness): if (νh)h is a sequence of R - valued ﬁnite m Radon measures on the l.c.s. metric space X, that is (νh)h ⊂ (M(X)) , with suph |νh|(X) < ∞, then it has a weakly*- converging subsequence. Moreover, the map ν 7→ |ν| is lower semicontinuous with respect the weak*-convergence. • 4. Theorem (Vitali covering theorem) Let G be a family of closed balls in Rn with D = sup {d(B): B ∈ G} < ∞ . Then there exists a (pairwise) disjoint family F ⊆ G, which is at most countable, such that ˆ ∪B∈GB ⊂ ∪B∈F B. where Bˆ is an enlargement of B, that is Bˆ = 5B. 18

• 5. Theorem (Vitali covering property for the Lebesgue measure): Let G be a family of closed balls in Rn, which is a fine cover of a (possibly non measurable) set A ⊂ Rn in Rn. Then there exists a disjoint subfamily F ⊂ G, at most countable, such that Ln (A \ ∪F) = 0 , where Ln denotes the n-dimensional Lebesgue outer measure. • 6. Theorem (Vitali covering property for Radon measures): let ϕ be a Radon o. m. in Rn, A ⊂ Rn (even not ϕ-measurable) and G a family of closed balls. Assume that G is cover of A and inf { r : B(x, r) ∈ G } = 0 ∀ x ∈ A. Then there is a disjoint subfamily F ⊂ G, at most countable, such that ϕ (A \ ∪F) = 0 . • 7. Theorem (Differentation for positive Radon measures):let ν and µ be positive Radon measures on Rn. (i) The derivative Dµν(x) exists and is finite (that is Dµν(x) ∈ [0, ∞)) for µ-a.e. x ∈ Rn. n (ii) The function Dµν : R → [0, +∞] is Borel measurable, by defining Dµν = ∞ on the possible µ-negligible set where it does not exist. (iii) Let n A := {x ∈ R : ∃ Dµν(x) ∈ [0, ∞)} . For all Borel sets B ⊂ Rn Z Dµν dµ = ν(A ∩ B) 6 ν(B) , B with equality if ν << µ. In this case dν dν D ν(x) = (x) = ac (x) µ-a.e. x ∈ n . µ dµ dµ R dλ denoting the Radon-Nikodym derivative of λ with respect to µ. dµ n (iv) ν << µ if and only if Dµν(x) < ∞ ν-a.e. x ∈ R . • 8. Theorem (Rademacher’s theorem[Rad]): let f : Rn → R be locally Lips- chitz. Then f is differentiable (in classical sense) Ln-a.e., that is, n n ∃ ∇f(x) := (∂1f(x), . . . , ∂nf(x)) for L -a.e. x ∈ R and f(y) − f(x) − df(x)(y − x) lim = 0 , y→x |y − x| where df(x): Rn → R denotes the (linear) differential map of f at x defined by n df(x)(v) := ∇f(x) · v ∀ v ∈ R . ∞ n n Moreover ∇f ∈ (Lloc(R )) . (The suggested proof is in [EG, Theorem 2, Sect. 3.1.2].) 19

• 9. Theorem (Classical length and H1): let γ : [0, a] → Rn be a curve and denote Γ = γ([0, a]) its support. Then 1 H (Γ) 6 l(γ) and equality holds if γ is injective. n n n n n n • 10. Theorem (H ≡ L ): L (A) = Hδ (A) = H (A) for each A ⊂ R , 0 < δ 6 ∞. n • 11. Theorem (Area formula for injective maps): let n 6 m and let f : R → Rm be an injective Lipschitz function and A ⊂ Rn be a measurable set. Then Z Hn(f(A)) = Jf dLn A and Hn f(Rn) is a Radon measure on Rm. • 12. Theorem (Existence of approximate tangent spaces): let y − x Φ : n → n, Φ (y) := , y ∈ n . x,r R R x,r r R If Γ ⊂ Rn is a locally Hk-rectifiable set, then for Hk-a.e. x ∈ Γ there exists a + unique k- dimensional plane πx such that, as r → 0 , (Φ ) µ Γ − x x,r # = Hk *∗ Hk π , rk r x that is Z Z 1 y − x k k 0 n lim ϕ dH (y) = ϕ(y) dH (y) ∀ ϕ ∈ Cc (R ) . r→0+ k r Γ r πx In particular k H (Γ ∩ B(x, r)) k ∃ Θk(Γ, x) := lim = 1 for H -a.e. x ∈ Γ . + ∗ k r→0 αk r • 13. Theorem (Rectifiability criterion): if Γ ⊂ Rn is a compact set, π is a k-dimensional plane in Rn, and there exist δ and t positive with Γ ∩ B(x, δ) ⊂ x + K(π, t) ∀ x ∈ Γ , k then Γ is H -rectifiable, since there exist finitely many Lipschitz maps fh : k n k R → R (h = 1,...,N) and compact sets Fh ⊂ R with N Γ = ∪h=1fh(Fh) . • 14. Theorem (Rectifiability by convergence of the blow-ups): If µ is a Radon measure on Rn, Γ is a Borel set in Rn, µ is concentrated on Γ (that is µ = n µ Γ), and, for every x ∈ Γ, there exists a k-dimensional plane πx in R such that (Φ ) µ x,r # *∗ Hk π as r → 0+, rk x then µ = Hk Γ and Γ is locally Hk-rectifiable. 20

Final examination procedure : the ﬁnal examination will be an interview with the student. Before the interview, the student must agree with the teacher two results fron the above list, with their proofs.Then the student will write a report on those results, which must be sent to the teacher two/three days before the interview. The main part of the interview will focus on this report. A second part will deal with some notions and results related to the ﬁrst one, but without proofs. 21

References [A] L. Ambrosio, Corso introduttivo alla Teoria Geometrica della Misura ed alle Superfici min- ime,Appunti dei corsi tenuti da docenti della Scuola, Scuola Normale Siperiore, Pisa,1997. [A2] L. Ambrosio, La teoria dei perimetri di Caccioppoli-De Giorgi e i suoi più recenti sviluppi. Rend. Lincei Mat. Appl. 21 (2010), 275–286. [AFP] L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, Oxford University Press, 2000. [AG] L. Ambrosio, R. Ghezzi, Sobolev and bounded variation functions on metric measure spaces, Geometry, Analysis and Dynamics on sub-Riemannian manifolds, vol. II, D. Barilari, U. Boscain, M. Sigalotti edts., EMS Series of Lecture Notes in Mathematics, 2016. [AK] L. Ambrosio, B. Kirchheim, Rectifiable sets in metric and Banach spaces. Math. Ann. 318, (2000), 527-555. [AT] L. Ambrosio, P. Tilli, Topics on analysis in metric spaces. Oxford University Press,Oxford, 2004. [Ba] S. Banach, Sur un theorèmede M. Vitali, Fund. Math. 5 (1924), 130–136. [Be1] A.S. Besicovitch, A general form of the covering principle and relative differentiation of addi- tive functions, Proc.Cambridge Philos.Soc. 41 (1945), 103–110. [Be2] A.S. Besicovitch, A general form of the covering principle and relative differentiation of addi- tive functions II, Proc. Cambridge Philos. Soc.41 (1946), 1–10. [Be3] A.S. Besicovitch, On the fundamental geometrical properties of linearly measurable plane sets of points, Math. Ann. 98 (1928), 422–464. [Be4] A.S. Besicovitch, On the fundamental geometrical properties of linearly measurable plane sets of points II, Math. Ann. 115 (1938), 296–329. [Be5] A.S. Besicovitch, On the fundamental geometrical properties of linearly measurable plane sets of points III, Math. Ann. 116 (1939), 349–357. [B] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011. An Italian version is also available: H. Brezis, Analisi Funzionale- Teoria e applicazioni, Liguori, Napoli, 1986. [C] C. Carathéodory , Vorlesungen uber reelle Funktionen, Leipzig, 1927. [C2] C. Carathéodory,Uber¨ das lineare Mass von Punktmengen, eine Verallgemeinerung des Längen- begriffs, Nach. Ges. Wiss. Gottingen (1914), 406–426. [Ch] J. Cheeger Differentiability of Lipschitz functions on metric measure spaces, Geom. Func. Anal. 9 (1999),428–517. [Co] D. L. Cohn, Measure Theory, Birkhäuser,1980. [DG1] E. De Giorgi, Su una teoria generale della misura (r − 1)-dimensionale in uno spazio ad r dimensioni. Ann.Mat.Pura Appl.(4), 36, (1954), 191–213. [DG2] E. De Giorgi, Nuovi teoremi relativi alle misure (r − 1)-dimensionali in uno spazio ad r dimensioni,Ricerche Mat. 4, (1955), 95–113. [DG3] E. De Giorgi, Sulla proprietà isoperimetrica dell’ipersfera nella classe degli insiemi aventi frontiera orientata di misura finita, Memorie Acc. Naz. Lincei Ser. VIII 5, (1958), 33–44. [DGCP] E. De Giorgi, F. Colombini, L.C. Piccinini, Frontiere orientate di misura minima e questioni collegate. Scuola Normale Superiore, Pisa, 1972. [dG] M. de Guzmán, Differentiation of Integrals in Rn, Lecture Notes in Math., Springer-Vrerlag, 1975. [EG] L.C. Evans, R. Gariepy, Lecture Notes on Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992. [Fa] K. Falconer, Fractal Geometry. Mathematical Foundations and Applications, Second Edi- tion,Chichester, John Wiley and Sons, 2003. [Fe] H. Federer, Geometric Measure Theory, Springer, 1969. [F] G.B. Folland , Real Analysis. Modern techniques and their applications, Second Edition, Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. [GZ] R. F. Gariepy, W. P. Ziemer, Modern Real Analysis, PWS Publishing Company, Boston, 1994. 22

[GO] B. R. Gelbaum, J. M. H. Olmsted Counterexamples in Analysis, Dover Publications,Inc., New York, 2003. [GH] M. Giaquinta, S. Hildebrandt, Calculus of Variations I,Springer, Berlin Heidelberg, 2004. [G] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser,Boston, 1984. [G2] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, Singapore, 2003. [HOH] H. Hanche-Olsen, H. Holge, The Kolmogorov-Riesz compactness theorem, Expo. Math. 28 (2010), no. 4, 385–394. [Ha] F. Hausdorff, Dimension und äusseres Mass, Math. Ann. 79 (1919), 157–179. [He] J. Heinonen Lectures on analysis on metric spaces, Springer, New York, 2001. [He2] J. Heinonen Lectures on Lipschitz analysis, Report. University of JyväskyläDepartment of Mathematics and Statistics, 100. University of Jyväskylä,Jyväskylä,2005. ii+77 pp. [LeR] E. Le Donne, S. Rigot, Besicovitch covering property on graded groups and applications to measure differentiation, Preprint 2015. [Le] H. Lebesgue, Lecons sur l’integration et la recherche de fonctions primitives, Deuxieme Edition, Gauthier-Villars, Paris, 1926. [Le2] H. Lebesgue, Sur l’intégration des fonctions discontinues, Ann. Sci. Ecole´ Norm. Sup. (3) 27 (1910), 361–450. [Mag] F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems, An Introduction to Geometric Measure Theory , Cambridge University Press, 2012. [MM] U. Massari, M. Miranda Minimal Surfaces of Codimension One, North-Holland, Amsterdam, 1984. [Ma] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, 1995. [McS] E.J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), 837–842. [Mi] M. Miranda, Un Teorema di esistenza e unicità per il problema dell’area minima in n variabili. Ann.Sc.Norm.Sup. Pisa III, 19 (1965), 233–249. [Mo] F. Morgan, Geometric Measure Theory, A beginner’s Guide, Academic Press, 1988. [Mor] A.P. Morse, Perfect blankets, Trans. Amer. Math. Soc. 6 (1947), 418–442. [Ni] O. Nikodym, Sur une génésalizationdes intégrales de M. J. Radon, Fund. Math. 15 (1930), 131–179. [Pe] J. Pérez, A New Golden Age of Minimal Surfaces, Notices of the AMS 64 (2017), 347–358. [P] A. Pietsch, History of Banach Spaces and Linear Operators, Birkhäuser,Boston, 2007. [Pre] D. Preiss, Dimension of metrics and differentiation of measures, General topology and its relations to modern analysis and algebra, V (Prague, 1981), Sigma Ser. Pure Math., vol. 3, Heldermann, Berlin, 1983, 565–568. [Pre2] D. Preiss, Geometry of measures in Rn: distributions. rectifiability, and densities, Ann. of Math. 125 (1987), 537–643. [Rad] H. Rademacher,Uber partielle und totale Differenziarbarkeit I, Math. Ann. 79 (1919), 340–359. [Ra] J. Radon, Theorie und Anwendungen der Theorie der absolut additiven Mengenfunktionen, Sitzungsber. Kaiserl. (Osterreich)¨ Akad. Wiss. Math.- Nat. Kl. , Abtteilung IIa 122, (1913), 1295–1438. [Ro] H. L. Royden, Real Analysis, Macmillan Publishing Company, New York, 1988. [R1] W. Rudin, Real and Complex Analysis, Third Edition, McGraw-Hill Book Co.,New York,1987. [R2] W. Rudin, Functional Analysis, McGraw-Hill Book Co.,New York,1973. [Sch] A.R. Schep, Addendum to ”A still one more proof of the Radon-Nikodym Theorem”, 2006, available at www.math.sc.edu/ schep/Radon-update.pdf [Ser] R. Serapioni Notes for the course in Geometric Measure Theory-Master degree in Mathematics- University of Trento. [SC] F. Serra Cassano Notes of Advanced Analysis-a.y. 2016/17-Master degree in Mathematics- University of Trento. [SC2] F. Serra Cassano, Some topics of geometric measure theory in Carnot groups, Geometry, Anal- ysis and Dynamics on sub-Riemannian manifolds, vol. I, D. Barilari, U. Boscain, M. Sigalotti edts., EMS Series of Lecture Notes in Mathematics, 2016. 23

[Ti] J. Tiˇser, Vitali covering theorem in Hilbert space, Trans. Amer. Math. Soc. 355 (2003), 3277– 3289. [T] F. Treves, Topological vector spaces, distributions and kernels, Academic Press,New York,1967. [Vitali] G. Vitali, Sulle funzioni integrabili, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. 40 (1904-05), 1021–1034. [Vitali2] G. Vitali, Sui gruppi di punti e sulle funzioni di variabili reali, Atti Accad. Sci. Torino 43 (1908), 75–92.