Geometric Measure Theory

Geometric Measure Theory

Gláucio Terra

Departamento de Matemática IME - USP

August 12, 2019

Vitali’s Covering Theorem

5–times covering lemma

Definition (2.12)

LetX be a metric space,Fa collection of balls inX andA⊂X. We say thatFis afine cover A, or thatFcovers A finely, ifFis a cover ofA such that,∀x ∈A, inf{diam B|x ∈B∈F}=0.

Corollary (2.13)

Let X be a metric space, A⊂X ,F⊂2^X a family of nondegenerate closed balls of X which covers A finely. Then there exists a disjoint subfamilyG⊂Fsuch that, for all F ⊂Ffinite, A\ ∪_B∈FB⊂ ∪_B∈G\F5B.

5–times covering lemma

Definition (2.12)

LetX be a metric space,Fa collection of balls inX andA⊂X. We say thatFis afine cover A, or thatFcovers A finely, ifFis a cover ofA such that,∀x ∈A, inf{diam B|x ∈B∈F}=0.

Corollary (2.13)

Let X be a metric space, A⊂X ,F⊂2^X a family of nondegenerate closed balls of X which covers A finely. Then there exists a disjoint subfamilyG⊂Fsuch that, for all F ⊂Ffinite, A\ ∪_B∈FB⊂ ∪_B∈G\F5B.

Vitali’s Covering Theorem

5–times covering lemma

Proof.

Since the coverFis fine, we may assume that

sup{diam B|B∈F} ≤1; otherwise, discard the balls inFwith diameter>1, so that the remaining balls still coverAfinely.

TakeG⊂Fas in the previous remark. Letx ∈A\ ∪_B∈FB. SinceF is finite,∪_B∈FBis closed, hence there existsr >0 such that U(x,r)∩ ∪_B∈FB=∅.

SinceFcoversAfinely, there existsB∈Fsuch thatx ∈Band diam B<r, so thatB⊂U(x,r), thusB∩ ∪_B∈FB=∅. TakeB⁰∈G such thatB⁰∩B6=∅(henceB⁰ ∈/ F) and diamB <2 diam B⁰, so thatx ∈B⊂5B⁰. ThenA\ ∪_B∈FB⊂ ∪_B∈G\F5B, as asserted.

5–times covering lemma

Proof.

Since the coverFis fine, we may assume that

sup{diam B|B∈F} ≤1; otherwise, discard the balls inFwith diameter>1, so that the remaining balls still coverAfinely.

TakeG⊂Fas in the previous remark. Letx ∈A\ ∪_B∈FB. SinceF is finite,∪_B∈FBis closed, hence there existsr >0 such that U(x,r)∩ ∪_B∈FB=∅.

SinceFcoversAfinely, there existsB∈Fsuch thatx ∈Band diam B<r, so thatB⊂U(x,r), thusB∩ ∪_B∈FB=∅. TakeB⁰∈G such thatB⁰∩B6=∅(henceB⁰ ∈/ F) and diamB <2 diam B⁰, so thatx ∈B⊂5B⁰. ThenA\ ∪_B∈FB⊂ ∪_B∈G\F5B, as asserted.

Vitali’s Covering Theorem

5–times covering lemma

Proof.

Since the coverFis fine, we may assume that

sup{diam B|B∈F} ≤1; otherwise, discard the balls inFwith diameter>1, so that the remaining balls still coverAfinely.

TakeG⊂Fas in the previous remark. Letx ∈A\ ∪_B∈FB. SinceF is finite,∪_B∈FBis closed, hence there existsr >0 such that U(x,r)∩ ∪_B∈FB=∅.

SinceFcoversAfinely, there existsB∈Fsuch thatx ∈Band diam B<r, so thatB⊂U(x,r), thusB∩ ∪_B∈FB=∅. TakeB⁰∈G such thatB⁰∩B6=∅(henceB⁰ ∈/ F) and diamB <2 diam B⁰, so thatx ∈B⊂5B⁰. ThenA\ ∪_B∈FB⊂ ∪_B∈G\F5B, as asserted.

5–times covering lemma

Proof.

Since the coverFis fine, we may assume that

sup{diam B|B∈F} ≤1; otherwise, discard the balls inFwith diameter>1, so that the remaining balls still coverAfinely.

TakeG⊂Fas in the previous remark. Letx ∈A\ ∪_B∈FB. SinceF is finite,∪_B∈FBis closed, hence there existsr >0 such that U(x,r)∩ ∪_B∈FB=∅.

SinceFcoversAfinely, there existsB∈Fsuch thatx ∈Band diam B<r, so thatB⊂U(x,r), thusB∩ ∪_B∈FB=∅. TakeB⁰∈G such thatB⁰∩B6=∅(henceB⁰ ∈/ F) and diamB <2 diam B⁰, so thatx ∈B⊂5B⁰. ThenA\ ∪_B∈FB⊂ ∪_B∈G\F5B, as asserted.

Vitali’s Covering Theorem

Vitali’s covering theorem

Corollary (Vitali’s covering theorem for the Lebesgue measure;2.14) Let A⊂RⁿandFa collection of nondegenerate closed balls inRⁿ which covers A finely. Then, for every >0, there exists a disjoint subfamilyG⊂Fsuch thatLⁿ(∪G)≤Lⁿ(A) +andLⁿ(A\ ∪G) =0.

Corollary (filling open sets with balls with respect to Lebesgue measure; 2.15)

Let U ⊂Rⁿbe an open set andFa family of nondegenerate closed balls contained in U which covers U finely (for instance, ifFis the family of all nondegenerate closed balls contained in U, or the family of all such balls with diameters bounded by a fixedδ >0). Then there exists a disjoint subfamilyG⊂Fsuch thatLⁿ(U\ ∪G) =0.

Vitali’s covering theorem

Corollary (Vitali’s covering theorem for the Lebesgue measure;2.14) Let A⊂RⁿandFa collection of nondegenerate closed balls inRⁿ which covers A finely. Then, for every >0, there exists a disjoint subfamilyG⊂Fsuch thatLⁿ(∪G)≤Lⁿ(A) +andLⁿ(A\ ∪G) =0.

Corollary (filling open sets with balls with respect to Lebesgue measure; 2.15)

Let U ⊂Rⁿbe an open set andFa family of nondegenerate closed balls contained in U which covers U finely (for instance, ifFis the family of all nondegenerate closed balls contained in U, or the family of all such balls with diameters bounded by a fixedδ >0). Then there exists a disjoint subfamilyG⊂Fsuch thatLⁿ(U\ ∪G) =0.

Steiner Symmetrization

Notation

Notation for sections

ForE ⊂X ×Y and(x₀,y₀)∈X×Y,

E_x0 :={y ∈Y |(x₀,y)∈E}(thex₀-sectionofE);

E_y0 :={x ∈X |(x,y₀)∈E}(they₀-sectionofE).

Steiner Symmetrization

Definition (2.16)

Let(e₁, . . . ,e_n)be the standard basis ofRⁿ and identify

Rⁿ⁻¹≡ he₁, . . . ,e_n−1i,R≡ he_ni, so thatRⁿ≡Rⁿ⁻¹×R. We define the Steiner symmetrizationwith respect toRⁿ⁻¹to be the map

S_en :2^Rn →2^Rn defined by (see figure 1):

Sen(A) := [

{x⁰∈Rⁿ⁻¹|A_x06=∅}

{(x⁰,xn)| |x_n| ≤ 1

2L¹(Ax⁰)}.

Givena∈Sⁿ⁻¹⊂Rⁿ, we define similarly the Steiner symmetrization Sawith respect to the(n−1)-dimensional subspacehai^⊥: take any orthogonal mapφ∈O(n)such thatφ(a) =e_n(henceφ(hai^⊥) =Rⁿ⁻¹) and put S_a:=φ⁻¹◦S_en◦φ.

Steiner Symmetrization

Steiner Symmetrization

Definition (2.16)

Let(e₁, . . . ,e_n)be the standard basis ofRⁿ and identify

Rⁿ⁻¹≡ he₁, . . . ,e_n−1i,R≡ he_ni, so thatRⁿ≡Rⁿ⁻¹×R. We define the Steiner symmetrizationwith respect toRⁿ⁻¹to be the map

S_en :2^Rn →2^Rn defined by (see figure 1):

Sen(A) := [

{x⁰∈Rⁿ⁻¹|A_x06=∅}

{(x⁰,xn)| |x_n| ≤ 1

2L¹(Ax⁰)}.

Givena∈Sⁿ⁻¹⊂Rⁿ, we define similarly the Steiner symmetrization Sawith respect to the(n−1)-dimensional subspacehai^⊥: take any orthogonal mapφ∈O(n)such thatφ(a) =e_n(henceφ(hai^⊥) =Rⁿ⁻¹) and put S_a:=φ⁻¹◦S_en◦φ.

Steiner Symmetrization

Figure:Steiner Symmetrization

Steiner Symmetrization

Steiner Symmetrization

Proposition (properties of Steiner symmetrization; 2.17)

Let a∈Sⁿ⁻¹.

i) ∀A⊂Rⁿ,diam S_a(A)≤diam A.

ii) If A⊂RⁿisLⁿ-measurable, then so isS_a(A)and Lⁿ(A) =Lⁿ S_a(A)

.

Steiner Symmetrization

Lemma (2.18)

Let f :Rⁿ→[0,∞]beLⁿ-measurable. Then

hyp f :={(x,t)∈Rⁿ×[0,∞)|t≤f(x)} ⊂Rⁿ⁺¹isLⁿ⁺¹-measurable.

The isodiametric inequality;Lⁿ=Hⁿ

The isodiametric inequality

Theorem (isodiametric inequality; 2.19)

The Lebesgue measure of any subset ofRⁿ is at most the measure of an euclidean ball with the same diameter. That is, for all A⊂Rⁿ,

Lⁿ(A)≤α(n)diam A 2

n

.

Exercise (2.20)

Show an example of a set A⊂Rⁿwhich is not contained in any ball with diameterdiam A.

The isodiametric inequality

Theorem (isodiametric inequality; 2.19)

The Lebesgue measure of any subset ofRⁿ is at most the measure of an euclidean ball with the same diameter. That is, for all A⊂Rⁿ,

Lⁿ(A)≤α(n)diam A 2

n

.

Exercise (2.20)

Show an example of a set A⊂Rⁿwhich is not contained in any ball with diameterdiam A.

The isodiametric inequality;Lⁿ=Hⁿ

L

= H

Theorem (2.21)

For allδ∈(0,∞]and n∈N,Hⁿ=Hⁿ_δ =LⁿinRⁿ.

Corollary (2.22) H-dimRⁿ=n.

Proof.

Apply the stability with respect to countable unions of the Hausdorff dimension toRⁿ =∪_k∈NC_k, where eachC_k is a nondegenerate cube with finite Lebesgue measure, i.e. 0<Hⁿ(C_k)<∞, so that∀k ∈N, H-dimC_k =n.

L

= H

Theorem (2.21)

For allδ∈(0,∞]and n∈N,Hⁿ=Hⁿ_δ =LⁿinRⁿ.

Corollary (2.22) H-dimRⁿ=n.

Proof.

Apply the stability with respect to countable unions of the Hausdorff dimension toRⁿ =∪_k∈NC_k, where eachC_k is a nondegenerate cube with finite Lebesgue measure, i.e. 0<Hⁿ(C_k)<∞, so that∀k ∈N, H-dimC_k =n.

The isodiametric inequality;Lⁿ=Hⁿ

L

= H

Exercise (2.23)

If E is a k -dimensional subspace of a normed space X , then H-dimE =k .