Geometric Measure Theory

(1)

Geometric Measure Theory

Gláucio Terra

Departamento de Matemática IME - USP

August 18, 2019

(2)

Densities

Up to the end of this section we fix a metric space(X,d).

(3)

Densities

Upper and lower n-dimensional densities

Definition (3.1)

LetA⊂X,x ∈X,n>0 real andµa measure onX. We define:

1) then-dimensional upper density of A at x with respect toµ:

Θ^∗n(µ,A,x) := lim sup

r→0

µ A∩B(x,r)

α(n)rⁿ ∈[0,∞].

2) then-dimensional lower density of A at x with respect toµ:

Θⁿ_∗(µ,A,x) := lim inf

r→0

µ A∩B(x,r)

α(n)rⁿ ∈[0,∞].

IfΘ^∗n(µ,A,x) = Θⁿ_∗(µ,A,x), we denote their common value by Θⁿ(µ,A,x)and call itdensity of A at x with respect toµ.

ForA=X, we use the notationsΘ^∗n(µ,x),Θⁿ_∗(µ,x)andΘⁿ(µ,x).

(4)

Densities

Upper and lower n-dimensional densities

Definition (3.1)

r→0

µ A∩B(x,r)

α(n)rⁿ ∈[0,∞].

r→0

µ A∩B(x,r)

α(n)rⁿ ∈[0,∞].

(5)

Densities

Upper and lower n-dimensional densities

Definition (3.1)

r→0

µ A∩B(x,r)

α(n)rⁿ ∈[0,∞].

r→0

µ A∩B(x,r)

α(n)rⁿ ∈[0,∞].

(6)

Densities

Upper and lower n-dimensional densities

Definition (3.1)

r→0

µ A∩B(x,r)

α(n)rⁿ ∈[0,∞].

r→0

µ A∩B(x,r)

α(n)rⁿ ∈[0,∞].

(7)

Densities

Remark (3.2)

With the notation above:

1) Note thatΘ^∗n(µ,A,x) = Θ^∗n(µ

x

^A,^x⁾^and

Θⁿ_∗(µ,A,x) = Θⁿ_∗(µ

x

^A,^x^).

2) IfU⊂X is an open set andx ∈U,

Θ^∗n(µ,A,x) = Θ^∗n(µ

x

Û,Â,^x⁾ând^Θⁿ^∗^(µ,Â,^x^{) = Θ}ⁿ^∗^(µ

x

^U,^A,^x).

(8)

Densities

Lemma (3.3)

Ifµis a locally finite Borel measure on X , A⊂X , x ∈X and n>0real, then, the definitions ofΘ^∗n(µ,A,x)orΘⁿ_∗(µ,A,x)do not change if we use open balls instead of closed balls.

Proposition (3.4)

Ifµis a locally finite Borel measure on X , A⊂X and n>0real, then the functions X →[0,∞]given by x ∈X 7→Θ^∗n(µ,A,x)and

x ∈X 7→Θⁿ_∗(µ,A,x)are Borelian.

Corollary (3.5)

Ifµis a locally finite Borel measure on X , A⊂X and n>0real, then the set Y :={x ∈X |Θ^∗n(µ,A,x) = Θⁿ_∗(µ,A,x)}is Borel measurable andΘⁿ(µ,A,·) :Y →[0,∞]is Borelian.

(9)

Densities

Lemma (3.3)

Proposition (3.4)

Corollary (3.5)

(10)

Densities

Lemma (3.3)

Proposition (3.4)

Corollary (3.5)

(11)

Densities

Comparison density theorem

Theorem (3.6)

Letµbe a Borel measure on a metric space X , n>0real, t ≥0and A⊂A₁⊂X . If∀x ∈A,Θ^∗n(µ,A₁,x)≥t then tHⁿ(A)≤µ(A₁).

(12)

Densities

Upper density theorem

Theorem (3.7)

Letµbe a Borel regular measure on a metric space X , n>0real and B ∈σ(µ)withµ(B)<∞. ThenΘ^∗n(µ,B,x) =0forHⁿ-a.e. x ∈X \B.

Exercise (3.8)

Ifµis an openσ-finite Borel regular measure on a metric space X , the thesis in the previous theorem holds for all B ∈σ(µ), i.e. the

hypothesis ofµ(B)being finite may be dropped.

(13)

Densities

Upper density theorem

Theorem (3.7)

Letµbe a Borel regular measure on a metric space X , n>0real and B ∈σ(µ)withµ(B)<∞. ThenΘ^∗n(µ,B,x) =0forHⁿ-a.e. x ∈X \B.

Exercise (3.8)

Ifµis an openσ-finite Borel regular measure on a metric space X , the thesis in the previous theorem holds for all B ∈σ(µ), i.e. the

hypothesis ofµ(B)being finite may be dropped.

(14)

Densities

Density theorem for the Lebesgue measure

Corollary (3.9)

If B⊂RⁿisLⁿ-measurable, thenΘⁿ(Lⁿ,B,x)exists forLⁿ-a.e.

x ∈Rⁿ,Θⁿ(Lⁿ,B,x) =1forLⁿ-a.e. x ∈B andΘⁿ(Lⁿ,B,x) =0for Lⁿ-a.e. x ∈Rⁿ\B.

(15)

Differentiation Theorems

Upper and lower densities of a measure relative another

Definition (3.12)

LetX be a metric space,µandνmeasures onX, andx ∈X. We define theupperandlower density ofµrelative toν at x by, respectively:

Θ^∗ν(µ,x) := lim sup

r→0

µ B(x,r)

ν B(x,r) ∈[0,∞],

Θ^ν_∗(µ,x) := lim inf

r→0

µ B(x,r)

ν B(x,r) ∈[0,∞],

where we adopt the extended arithmetic rules ⁰₀ :=0, ^∞_∞ :=0.

IfΘ^∗ν(µ,x) = Θ^ν_∗(µ,x), we say that thedensity ofµrelative toν at x exists and denote it byΘ^ν(µ,x) := Θ^∗ν(µ,x) = Θ^ν_∗(µ,x).

(16)

Upper and lower densities of a measure relative another

Definition (3.12)

LetX be a metric space,µandνmeasures onX, andx ∈X. We define theupperandlower density ofµrelative toν at x by, respectively:

Θ^∗ν(µ,x) := lim sup

r→0

µ B(x,r)

ν B(x,r) ∈[0,∞],

Θ^ν_∗(µ,x) := lim inf

r→0

µ B(x,r)

ν B(x,r) ∈[0,∞],

where we adopt the extended arithmetic rules ⁰₀ :=0, ^∞_∞ :=0.

IfΘ^∗ν(µ,x) = Θ^ν_∗(µ,x), we say that thedensity ofµrelative toν at x exists and denote it byΘ^ν(µ,x) := Θ^∗ν(µ,x) = Θ^ν_∗(µ,x).

(17)

Remark (3.13)

IfX =Rⁿ,A⊂Rⁿ,x ∈Rⁿandµa measure onRⁿ, then-dimensional upper and lower densities ofAatx with respect toµ, defined in 1, are special cases of the previous definition: Θ^∗n(µ,A,x) = Θ^∗^Lⁿ(µ

x

^A,^x)

andΘⁿ_∗(µ,A,x) = Θ^L_∗ⁿ(µ

x

^A,^x^).

(18)

Lemma (3.14)

Ifµandν are locally finite Borel measures on a metric space X , and x ∈X , then the definitions ofΘ^∗ν(µ,x)orΘ^ν_∗(µ,x)do not change if we use open balls instead of closed balls.

Proposition (3.15 and 3.16)

Letµandν be locally finite Borel measures on a metric space X . Suppose that X is separable or thatν is finite on all closed balls of X . Then the functions X →[0,∞]given by x ∈X 7→Θ^∗ν(µ,x)and x ∈X 7→Θ^ν_∗(µ,x)are Borelian.

Corollary (3.17)

Letµandν be locally finite Borel measures on a metric space X , with X separable orν finite on all closed balls of X . Then the set

Y :={x ∈X |Θ^∗ν(µ,x) = Θ^ν_∗(µ,x)}is Borel measurable and Θ^ν(µ,·) :Y →[0,∞]is Borelian.

(19)

Lemma (3.14)

Corollary (3.17)

(20)

Lemma (3.14)

Corollary (3.17)

(21)

Recall

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 (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.

(22)

Recall

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 (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.

(23)

Symmetric Vitali property (SVP)

Definition (3.18)

LetX be a metric space,Fa collection of balls inX andA⊂X. We say thatFis astrongly fine cover A, or thatFcovers A finely in the strong sense, ifFis a cover ofAsuch that,∀x ∈A,

inf{r >0|B(x,r)∈F}=0.

It is clear that every strongly fine cover ofAis a fine cover ofAin the sense of definition 12, but the converse does not hold.

Definition (3.19)

We say that a measureµon a metric spaceX satisfies thesymmetric Vitali property (SVP)if, for allA⊂X withµ(A)<∞and for allF strongly fine cover ofAby nondegenerate closed balls, there exists a countable disjoint subfamilyG⊂Fsuch thatµ(A\ ∪G) =0.

(24)

Symmetric Vitali property (SVP)

Definition (3.18)

inf{r >0|B(x,r)∈F}=0.

Definition (3.19)

(25)

Symmetric Vitali property (SVP)

Definition (3.18)

inf{r >0|B(x,r)∈F}=0.

Definition (3.19)

(26)

Symmetric Vitali property (SVP)

Remark (3.20)

1) It is clear that, if a measureµon a metric spaceX has SVP, so does any restriction ofµ, i.e.∀Y ⊂X,µ

x

^Y ^{has SVP.}

2) If a measureµon a metric spaceX isσ-finite and has SVP, thenµ is concentrated on its support, i.e. µ(X\sptµ) =0.

Proof: LetX =∪_k_∈_NA_k, with∀k ∈N,A_k ∈σ(µ)andµ(A_k)<∞.

For eachk ∈N, the family of nondegenerate closed balls F={B(x,r)|x ∈X \sptµ,r >0, µ B(x,r)

=0}covers A_k \sptµfinely in the strong sense. Hence, there exists a countable disjoint subfamilyGk ⊂Fsuch that

µ (A_k\sptµ)\ ∪Gk

=0; sinceµ(∪Gk) =0, we conclude that µ(A_k \sptµ) =0. ThereforeX \sptµ=∪_k∈_N(A_k \sptµ)has µ-measure zero.

(27)

Symmetric Vitali property (SVP)

Remark (3.20)

x

^Y ^{has SVP.}

(28)

Symmetric Vitali property (SVP)

Remark (3.20)

x

^Y ^{has SVP.}

(29)

Symmetric Vitali property (SVP)

Remark (3.20)

x

^Y ^{has SVP.}

(30)

Doubling property implies SVP

Proposition (3.21)

Let X be a separable metric space andµa finite Borel regular measure on X . Assume thatµsatisfies the doubling property:

∃C>0,∀B ⊂X nondegenerate closed ball, µ(5B)≤Cµ(B).

Thenµhas the symmetric Vitali property.

(31)

Besicovitch covering theorem

Theorem (3.24)

For each n∈N, there exists a natural constant N =N(n), depending only on n, which satisfies the following property: ifFis any family of nondegenerate closed balls inRⁿwithsup{diam B|B∈F}<∞and A is the set of centers of the balls inF, then existG1, . . . ,GN such that, for 1≤i ≤N,Gi is a disjoint subfamily ofFand∪^N_i=1Gi covers A.

Corollary (3.25)

Letµbe a Borel measure inRⁿ, A⊂Rⁿwithµ(A)<∞andFa family of nondegenerate closed balls which covers A finely in the strong sense. Then, for any open set U ⊃A, there exists a countable disjoint subfamilyG⊂Fsuch that∪G⊂U andµ(A\ ∪G) =0.

(32)

Besicovitch covering theorem

Theorem (3.24)

For each n∈N, there exists a natural constant N =N(n), depending only on n, which satisfies the following property: ifFis any family of nondegenerate closed balls inRⁿwithsup{diam B|B∈F}<∞and A is the set of centers of the balls inF, then existG1, . . . ,GN such that, for 1≤i ≤N,Gi is a disjoint subfamily ofFand∪^N_i=1Gi covers A.

Corollary (3.25)

Letµbe a Borel measure inRⁿ, A⊂Rⁿwithµ(A)<∞andFa family of nondegenerate closed balls which covers A finely in the strong sense. Then, for any open set U ⊃A, there exists a countable disjoint subfamilyG⊂Fsuch that∪G⊂U andµ(A\ ∪G) =0.

(33)

Proposition (Borel measures on subsets ofRⁿsatisfy SVP; 3.23) Let X be a metric subspace ofRⁿ andµa Borel measure on X . Then µsatisfies the symmetric Vitali property.

(34)

General comparison density theorem

Theorem (3.26)

Letµandν be openσ-finite Borel regular measures on a metric space X such thatνhas the symmetric Vitali property, t ≥0and A⊂X . If

∀x ∈A,Θ^∗ν(µ,x)≥t then tν(A)≤µ(A).

Corollary (3.27)

Letµandν be openσ-finite Borel regular measures on a metric space X such thatνhas the symmetric Vitali property. ThenΘ^∗ν(µ,x)<∞ forν-a.e. x ∈X .

(35)

General comparison density theorem

Theorem (3.26)

Letµandν be openσ-finite Borel regular measures on a metric space X such thatνhas the symmetric Vitali property, t ≥0and A⊂X . If

∀x ∈A,Θ^∗ν(µ,x)≥t then tν(A)≤µ(A).

Corollary (3.27)

Letµandν be openσ-finite Borel regular measures on a metric space X such thatνhas the symmetric Vitali property. ThenΘ^∗ν(µ,x)<∞ forν-a.e. x ∈X .

(36)

General upper density theorem

Theorem (3.28)

Letµbe a Borel regular measure on a metric space X ,ν an open σ-finite Borel regular measure on X with the symmetric Vitali property, and A∈σ(µ)withµ(A)<∞. ThenΘ^∗ν(µ

x

^A,^x^{) =}⁰^for^ν-a.e.

x ∈X \A.

Theorem (general density theorem; 3.29)

Letµbe an openσ-finite Borel regular measure on a metric space X with symmetric Vitali property and A∈σ(µ). Then the density Θ^µ(µ

x

^A,^·)^coincidesµ-a.e. on X withχ_A, i.e.

Θ^µ(µ

x

^A,^x^{) = lim}_r_→0^µ_µ^A^∩_B_(x,^B(x_r^,₎^r⁾ ⁼

(1 forµ-a.e. x ∈A, 0 forµ-a.e. x ∈X \A.

(37)

General upper density theorem

Theorem (3.28)

Letµbe a Borel regular measure on a metric space X ,ν an open σ-finite Borel regular measure on X with the symmetric Vitali property, and A∈σ(µ)withµ(A)<∞. ThenΘ^∗ν(µ

x

^A,^x^{) =}⁰^for^ν-a.e.

x ∈X \A.

Theorem (general density theorem; 3.29)

Letµbe an openσ-finite Borel regular measure on a metric space X with symmetric Vitali property and A∈σ(µ). Then the density Θ^µ(µ

x

^A,^·)^coincidesµ-a.e. on X withχ_A, i.e.

Θ^µ(µ

x

^A,^x^{) = lim}_r_→0^µ_µ^A^∩_B_(x,^B(x_r^,₎^r⁾ ⁼

(1 forµ-a.e. x ∈A, 0 forµ-a.e. x ∈X \A.

(38)

Lusin’s theorem

Theorem (1.112)

Letµbe a Borel regular measure on a metric space X (respectively, a Radon measure on a locally compact Hausdorff space X ), Y a

separable metric space, f : dom f ⊂X →Y aµ-measurable map.

Then, for each A∈σ(µ)withµ(A)<∞and for each >0, there exists a closed (respectively, compact) set C ⊂A such thatµ(A\C)< and f|_Cis continuous.

(39)

General Lebesgue differentiation theorem

Corollary (3.30)

Letµbe an openσ-finite Borel regular measure on a metric space X with symmetric Vitali property and f :X →Caµ-measurable function satisfying one of the following conditions:

i) f ∈L¹(µ)or

ii) X is separable and f ∈L¹_loc(µ), i.e. ∀x ∈X ,∃r >0, R

B(x,r)|f|dµ <∞.

Then, forµ-a.e. x ∈X :

rlim→0

1 µ B(x,r)

Z

B(x,r)

fdµ=f(x).

(40)

Lebesgue Points

Corollary (3.31)

Let X be a separable metric space,µan openσ-finite Borel regular measure on X with symmetric Vitali property,1≤p<∞and f ∈L^p_loc(µ), i.e.∀x ∈X,∃r >0,R

B(x,r)|f|^pdµ <∞. Then, forµ-a.e.

x ∈X ,

lim

r→0

1 µ B(x,r)

Z

B(x,r)

|f(y)−f(x)|^pdµ(y) =0. (1)

Definition (Lebesgue Points; 3.32)

With the same notation from the previous corollary, a pointx ∈X for which (1) holds is calledLebesgue point of f with respect toµ.

(41)

Lebesgue Points

Corollary (3.31)

Let X be a separable metric space,µan openσ-finite Borel regular measure on X with symmetric Vitali property,1≤p<∞and f ∈L^p_loc(µ), i.e.∀x ∈X,∃r >0,R

B(x,r)|f|^pdµ <∞. Then, forµ-a.e.

x ∈X ,

lim

r→0

1 µ B(x,r)

Z

B(x,r)

|f(y)−f(x)|^pdµ(y) =0. (1)

Definition (Lebesgue Points; 3.32)

With the same notation from the previous corollary, a pointx ∈X for which (1) holds is calledLebesgue point of f with respect toµ.

(42)

Lebesgue points with noncentered balls

Corollary (3.33)

Let1≤p<∞and f ∈L^p_loc(Lⁿ). Then, for each Lebesgue point x of f with respect toLⁿ(in particular, forLⁿ-a.e. x ∈Rⁿ),

B↓{x}lim 1 Lⁿ(B)

Z

B

|f(y)−f(x)|^pdLⁿ(y) =0,

where the limit is taken over all closed balls B containing x with diam B→0.

(43)

Absolute continuity and mutual singularity

Definition (3.34)

Letµandν be Borel measures on a topological spaceX. We say that:

1) µisabsolutely continuouswith respect toν(notation: µν) if

∀A⊂X,ν(A) =0 impliesµ(A) =0.

2) µandν aremutually singular(notation: µ⊥ν) if there exists A∈BX such thatµis concentrated onAandνis concentrated on X \A.

(44)

Lebesgue decomposition theorem

Lemma (3.36)

Letµbe aσ-finite Borel measure andνa Borel regular measure on a metric space X . Then there exists B ∈BX such thatν is concentrated on B^c andµ

x

^B^c ^ν^{, so that}

µ=µ

x

^B⁺^µ

x

^B^c^, ^µ

x

^B^⊥^{ν, µ}

x

^B^c ^ν. ^(LD)

Moreover:

1) B∈BX satisfying (LD) is unique up toµ-null sets, i.e. if B⁰∈BX

also satisfies (LD), then Ba

B⁰ isµ-null.

2) the decomposition (LD) is unique in the sense that, ifν=µs+µa

withµ_s⊥ν andµ_aν, thenµ_s =µ

x

^{B and}^µ^a⁼^µ

x

^B^c^.

(45)

Lebesgue decomposition theorem

Lemma (3.36)

Letµbe aσ-finite Borel measure andνa Borel regular measure on a metric space X . Then there exists B ∈BX such thatν is concentrated on B^c andµ

x

^B^c ^ν^{, so that}

µ=µ

x

^B⁺^µ

x

^B^c^, ^µ

x

^B^⊥^{ν, µ}

x

^B^c ^ν. ^(LD)

Moreover:

1) B∈BX satisfying (LD) is unique up toµ-null sets, i.e. if B⁰∈BX

also satisfies (LD), then Ba

B⁰ isµ-null.

2) the decomposition (LD) is unique in the sense that, ifν=µs+µa

withµ_s⊥ν andµ_aν, thenµ_s =µ

x

^{B and}^µ^a⁼^µ

x

^B^c^.

(46)

Comparison theorem for lower densities

Theorem (3.38)

Letµandν be openσ-finite Borel regular measures on a metric space X , t ≥0and A⊂X with∀x ∈A,Θ^ν_∗(µ,x)≤t.

i) Ifµhas SVP, thenµ(A)≤tν(A).

ii) Ifνhas SVP and B is given by the previous lemma, so that (LD) holds, thenµ(A\B)≤tν(A).

(47)

Differentiation theorem for Borel measures on metric spaces

Theorem (3.39)

Letµandν be openσ-finite Borel regular measures on a metric space X . Suppose that X is separable or thatν is finite on closed balls of X .

i) The set Y :={x ∈X |Θ^∗ν(µ,x) = Θ^ν_∗(µ,x)}is Borel measurable andΘ^ν(µ,·) :Y →[0,∞]is Borelian.

ii) Ifνhas SVP, Y_f :={x ∈Y |Θ^ν(µ,x)<∞}is a Borel measurable subset of X whose complement isν-null.

iii) If bothµandν have SVP,µ(Y^c) =ν(Y^c) =0.

(48)

Lebesgue-Besicovitch-Radon-Nikodym differentiation theorem

Theorem (3.40)

Letµandν be openσ-finite Borel regular measures on a metric space X . Suppose that X is separable or thatν is finite on closed balls of X , and thatν has SVP.

i) Letµ=µs+µabe the Lebesgue decomposition ofµwith respect toν, i.e. µs=µ

x

^{B and}^µ^a⁼^µ

x

^B^c^{, where B}^∈^B^X is given by lemma 29. Then, for all A∈BX,

µ_a(A) = Z

A

Θ^ν(µ,x)dν(x),

so that, for all A∈BX,µ(A) =R

AΘ^ν(µ,x)dν(x) +µs(A).

ii) Ifµalso has SVP, in lemma 29 we can take B⁰={x ∈X |Θ^ν(µ,x) =∞}in place of B.

(49)

Corollary (3.41)

With the same hypothesis from the previous theorem,Θ^ν(µ,·) coincidesν-a.e. with the Radon-Nikodym derivative ^d(µ_d(ν|^a^|^BX⁾

BX) .