Geometric Measure Theory

Geometric Measure Theory

Gláucio Terra

Departamento de Matemática IME - USP

September 30, 2019

General hypothesis

We fix a locally compact Hausdorff spaceX, which will be assumed σ-compact, unless otherwise specified.

Notation We denote by

C_c(X,Rⁿ)the space of continuous functionsf :X →Rⁿ with sptf compact;

C₀(X,Rⁿ)the space of continuous functionsf :X →Rⁿwhich vanish at infinity, i.e. such that∀ >0,∃K ⊂X compact such that kfk< onX \K.

C_b(X,Rⁿ)the space of bounded continuous functionsf :X →Rⁿ.

R

ⁿ

-valued Radon measures

Definition (4.1)

We say that a linear functionalµ:C_c(X,Rⁿ)→Ris anRⁿ-valued Radon measureonX if, for each compactK ⊂X, the restriction ofµto C^K_c(X,Rⁿ) :={f ∈C_c(X,Rⁿ)|sptf ⊂K}, endowed withk·k_u, is linear continuous; that is, if∃C_K ≥0 such that

sup{µ·f |f ∈C^K_c(X,Rⁿ),kfk_u ≤1} ≤C_K. (LF cont) If the condition above holds with a constantC ≥0 which does not depend onK, i.e. if µis linear continuous on C_c(X,Rⁿ)endowed with k·k_u, we callµafiniteRⁿ-valued Radon measureonX.

R

ⁿ

-valued Radon measures

Definition (4.1)

We say that a linear functionalµ:C_c(X,Rⁿ)→Ris anRⁿ-valued Radon measureonX if, for each compactK ⊂X, the restriction ofµto C^K_c(X,Rⁿ) :={f ∈C_c(X,Rⁿ)|sptf ⊂K}, endowed withk·k_u, is linear continuous; that is, if∃C_K ≥0 such that

sup{µ·f |f ∈C^K_c(X,Rⁿ),kfk_u ≤1} ≤C_K. (LF cont) If the condition above holds with a constantC ≥0 which does not depend onK, i.e. if µis linear continuous on C_c(X,Rⁿ)endowed with k·k_u, we callµafiniteRⁿ-valued Radon measureonX.

R

ⁿ

-valued Radon measures

Remark (4.2)

1 The definition adopted for anRⁿ-valued Radon measure onX is equivalent to saying thatµ:C_c(X,Rⁿ)→Ris linear continuous with respect to the natural topological vector space topology on C_c(X,Rⁿ), which is an inductive limit of Fréchet spaces (anLF spacefor short).

2 For those fluent in locally convex spaces: ifX is an open set in some Euclidean space, C^∞_c (X,R)ⁿhas a continuous dense inclusion in C_c(X,Rⁿ)≡C_c(X,R)ⁿ. That means that the dual of C_c(X,R)ⁿ may be identified with a linear subspace of the dual of C^∞_c (X,R)ⁿ, i.e. everyRⁿ-valued Radon measure onX is an Rⁿ-valued Schwartz distribution onX.

R

ⁿ

-valued Radon measures

Remark (4.2)

1 The definition adopted for anRⁿ-valued Radon measure onX is equivalent to saying thatµ:C_c(X,Rⁿ)→Ris linear continuous with respect to the natural topological vector space topology on C_c(X,Rⁿ), which is an inductive limit of Fréchet spaces (anLF spacefor short).

2 For those fluent in locally convex spaces: ifX is an open set in some Euclidean space, C^∞_c (X,R)ⁿhas a continuous dense inclusion in C_c(X,Rⁿ)≡C_c(X,R)ⁿ. That means that the dual of C_c(X,R)ⁿ may be identified with a linear subspace of the dual of C^∞_c (X,R)ⁿ, i.e. everyRⁿ-valued Radon measure onX is an Rⁿ-valued Schwartz distribution onX.

R

ⁿ

-valued Radon measures on open sets of Euclidean spaces

Exercise (4.3)

Let X be an open subset ofR^m and(U_k)k∈Nbe an increasing sequence of relatively compact open subsets of X such that

∪_k∈NU_k =X . Letµ:C^∞_c (X,Rⁿ)→Rbe a linear map such that

∀k ∈N,µ|

C^∞_c (Uk,Rⁿ),k·ku

is continuous. Thenµmay be uniquely extended to a continuous linear mapC_c(X,Rⁿ)→R.

Recall

Notation

LetX be a locally compact Hausdorff space,U ⊂X open andf a function onX.

f ≺U means that 0≤f ≤1,f ∈C_c(X,R)and spt f ⊂U.

Lemma (Urysohn’s lemma for LCH; 4.5)

If X is a locally compact Hausdorff space, U ⊂X open and K ⊂U compact, then there exists f ∈C_c(X,R)such thatχ_K ≤f ≺U.

Theorem (Tietze’s extension theorem for LCH; 4.6)

If X is a locally compact space, K ⊂X compact and f :K →R

continuous, then f admits a continuous extensionef :X →R. Moreover, we may takeef with compact support and, if f is bounded, we may also takeef such thatkefk_u=kfk_u.

Recall

Notation

LetX be a locally compact Hausdorff space,U ⊂X open andf a function onX.

f ≺U means that 0≤f ≤1,f ∈C_c(X,R)and spt f ⊂U.

Lemma (Urysohn’s lemma for LCH; 4.5)

If X is a locally compact Hausdorff space, U ⊂X open and K ⊂U compact, then there exists f ∈C_c(X,R)such thatχ_K ≤f ≺U.

Theorem (Tietze’s extension theorem for LCH; 4.6)

If X is a locally compact space, K ⊂X compact and f :K →R

continuous, then f admits a continuous extensionef :X →R. Moreover, we may takeef with compact support and, if f is bounded, we may also takeef such thatkefk_u=kfk_u.

Recall

Notation

LetX be a locally compact Hausdorff space,U ⊂X open andf a function onX.

f ≺U means that 0≤f ≤1,f ∈C_c(X,R)and spt f ⊂U.

Lemma (Urysohn’s lemma for LCH; 4.5)

If X is a locally compact Hausdorff space, U ⊂X open and K ⊂U compact, then there exists f ∈C_c(X,R)such thatχ_K ≤f ≺U.

Theorem (Tietze’s extension theorem for LCH; 4.6)

If X is a locally compact space, K ⊂X compact and f :K →R

continuous, then f admits a continuous extensionef :X →R. Moreover, we may takeef with compact support and, if f is bounded, we may also takeef such thatkefk_u=kfk_u.

Recall

Theorem (Riesz representation theorem for positive linear functionals;

4.7)

Let X be a locally compact Hausdorff space and L:C_c(X,R)→Ra positive linear functional, i.e. L is linear and L·f ≥0whenever f ≥0.

Then there exists a unique Radon measureηon X which represents L, i.e. ∀f ∈C_c(X,R), L·f =R

fdη. Moreover, on open setsηis given by

η(U) =sup{L·f |f ≺U}.

Remark (4.8)

Every positive linear functional on C_c(X,R)is anR-valued Radon measure onX, i.e. positivity implies continuity on C_c(X,R).

Proof.

GivenK ⊂X compact, takeΦ∈C_c(X,R)given by lemma 3 such that χ_K ≤Φ≺X. For allf ∈C^K_c(X,R)withf 6=0, we have _kf^|f_k^|

u ≤Φ, so that Φ±_kf^f_k

u ≥0 andΦ±_kf^f_k

u ∈C_c(X,R). Hence 0≤L Φ±_kf^f_k

u

=L(Φ)±_kf^L(f_k⁾

u, which implies|L(f)| ≤L(Φ)kfk_u. The continuity condition (LF cont) is then satisfied withC_K :=L(Φ).

Remark (4.8)

Every positive linear functional on C_c(X,R)is anR-valued Radon measure onX, i.e. positivity implies continuity on C_c(X,R).

Proof.

GivenK ⊂X compact, takeΦ∈C_c(X,R)given by lemma 3 such that χ_K ≤Φ≺X. For allf ∈C^K_c(X,R)withf 6=0, we have _kf^|f_k^|

u ≤Φ, so that Φ±_kf^f_k

u ≥0 andΦ±_kf^f_k

u ∈C_c(X,R). Hence 0≤L Φ±_kf^f_k

u

=L(Φ)±_kf^L(f_k⁾

u, which implies|L(f)| ≤L(Φ)kfk_u. The continuity condition (LF cont) is then satisfied withC_K :=L(Φ).

Riesz representation theorem for Radon measures

Theorem (4.9)

Let X be aσ-compact locally compact Hausdorff space and

µ:C_c(X,Rⁿ)→RanRⁿ-valued Radon measure on X . Then there exists a unique Radon measureλon X and a Borel measurable map ν :X →Rⁿunique up toλ-null sets such thatkνk=1λ-a.e. on X and

∀f ∈C_c(X,Rⁿ),

µ·f = Z

hf, νidλ, (1)

whereh·,·idenotes the Euclidean inner product inRⁿ.Moreover, i) ∀U⊂X open,

λ(U) =sup{µ·f |f ∈C_c(X,Rⁿ),kfk ≺U}. (2)

ii) µis a finiteRⁿ-valued Radon measureiffλis a finite Radon measure; if that is the case,kµk_C

0(X,Rⁿ)^∗ =λ(X).

Riesz representation theorem for Radon measures

Theorem (4.9)

Let X be aσ-compact locally compact Hausdorff space and

µ:C_c(X,Rⁿ)→RanRⁿ-valued Radon measure on X . Then there exists a unique Radon measureλon X and a Borel measurable map ν :X →Rⁿunique up toλ-null sets such thatkνk=1λ-a.e. on X and

∀f ∈C_c(X,Rⁿ),

µ·f = Z

hf, νidλ, (1)

whereh·,·idenotes the Euclidean inner product inRⁿ. Moreover, i) ∀U⊂X open,

λ(U) =sup{µ·f |f ∈C_c(X,Rⁿ),kfk ≺U}. (2) ii) µis a finiteRⁿ-valued Radon measureiffλis a finite Radon

measure; if that is the case,kµk_C

0(X,Rⁿ)^∗ =λ(X).

Riesz representation theorem for Radon measures

Remark (4.10)

Note that, in (2), sup{µ·f |f ∈C_c(X,Rⁿ),kfk ≺U}=sup{|µ·f| |f ∈ C_c(X,Rⁿ),kfk ≺U}. Indeed, iff ∈C_c(X,Rⁿ)andkfk ≺U, so does−f, andµ·(−f) =−µ·f, hence eitherµ·f orµ·(−f)coincides with|µ·f|.

Lemma (4.11)

Let X be a locally compact Hausdorff space, f :X →[0,∞)bounded Borelian andµaσ-finite Radon measure onBX. Then

λ:=fµ:BX →[0,∞]given by A7→R

Afdµis a Radon measure on BX.

Total variation and polar decomposition

Definition (4.13)

Letµbe anRⁿ-valued Radon measure on aσ-compact locally

compact Hausdorff spaceX. With the same notation of theorem 7,λis called thetotal variation ofµ, and the pair(ν, λ)is called thepolar decomposition ofµ. Henceforth, we will use the notation|µ|:=λto denote the total variation ofµ, and

µ=ν|µ|

with the meaning that(ν,|µ|)is the polar decomposition ofµ.

Total variation and polar decomposition

Example (4.14)

1 Letµbe a locally finite Borel measure onX. Thenµinduces a positive linear functionalµˆon C_c(X,R), given byµˆ·f :=R

fdµ. If µis a Radon measure, thenµˆ=1·µis the polar decomposition of ˆ

µ.

2 Similarly, letν be a signed measure onBX whose total variation

|ν|is locally finite. Thenνinduces a continuous linear functionalνˆ on C_c(X,R)given byνˆ·f :=R

fdν.

3 LetX =RandIbe the positive linear functional defined on C_c(X,R)by the Riemann integral, i.e.I·f :=Rb

a f(x)dx fora<b such that sptf ⊂[a,b]. The polar decomposition ofI isI=1·Lⁿ. In particular, that could have been taken as the definition of the Lebesgue measure, i.e. it is the total variation of the positive linear functional induced by the Riemann integral.

Total variation and polar decomposition

Example (4.14)

1 Letµbe a locally finite Borel measure onX. Thenµinduces a positive linear functionalµˆon C_c(X,R), given byµˆ·f :=R

fdµ. If µis a Radon measure, thenµˆ=1·µis the polar decomposition of ˆ

µ.

2 Similarly, letν be a signed measure onBX whose total variation

|ν|is locally finite. Thenνinduces a continuous linear functionalνˆ on C_c(X,R)given byνˆ·f :=R

fdν.

3 LetX =RandIbe the positive linear functional defined on C_c(X,R)by the Riemann integral, i.e.I·f :=Rb

a f(x)dx fora<b such that sptf ⊂[a,b]. The polar decomposition ofI isI=1·Lⁿ. In particular, that could have been taken as the definition of the Lebesgue measure, i.e. it is the total variation of the positive linear functional induced by the Riemann integral.

Total variation and polar decomposition

Example (4.14)

1 Letµbe a locally finite Borel measure onX. Thenµinduces a positive linear functionalµˆon C_c(X,R), given byµˆ·f :=R

fdµ. If µis a Radon measure, thenµˆ=1·µis the polar decomposition of ˆ

µ.

2 Similarly, letν be a signed measure onBX whose total variation

|ν|is locally finite. Thenνinduces a continuous linear functionalνˆ on C_c(X,R)given byνˆ·f :=R

fdν.

3 LetX =RandIbe the positive linear functional defined on C_c(X,R)by the Riemann integral, i.e.I·f :=Rb

a f(x)dx fora<b such that sptf ⊂[a,b]. The polar decomposition ofI isI=1·Lⁿ. In particular, that could have been taken as the definition of the Lebesgue measure, i.e. it is the total variation of the positive linear functional induced by the Riemann integral.

Properties of the total variation, part I

Proposition (4.15)

Letµandν beRⁿ-valued Radon measures on aσ-compact locally compact Hausdorff space X and c∈R. Then:

i) |µ+ν| ≤ |µ|+|ν|, with equality if|µ| ⊥ |ν|.

ii) |cµ|=|c||µ|.

Integration with respect to R

ⁿ

-valued Radon measures

Definition (4.16)

Letµbe anRⁿ-valued Radon measure on aσ-compact locally compact Hausdorff spaceX, with polar decompositionµ=ν|µ|.

1 A vector Borelian mapf :X →Rⁿis calledsummable with respect toµiff ∈L¹(|µ|,Rⁿ)≡L¹(|µ|)ⁿ. For suchf, we define

Z

f· dµ:=

Z

hf, νid|µ| ∈R.

2 An scalar Borelian mapf :X →Ris calledsummable with respect toµiff ∈L¹(|µ|). For suchf, we define

Z

fdµ:=

Z

fνd|µ|= Z

fν₁d|µ|, . . . , Z

fνnd|µ|

∈Rⁿ.

Integration with respect to R

ⁿ

-valued Radon measures

Definition (4.16)

Letµbe anRⁿ-valued Radon measure on aσ-compact locally compact Hausdorff spaceX, with polar decompositionµ=ν|µ|.

1 A vector Borelian mapf :X →Rⁿis calledsummable with respect toµiff ∈L¹(|µ|,Rⁿ)≡L¹(|µ|)ⁿ. For suchf, we define

Z

f· dµ:=

Z

hf, νid|µ| ∈R.

2 An scalar Borelian mapf :X →Ris calledsummable with respect toµiff ∈L¹(|µ|). For suchf, we define

Z

fdµ:=

Z

fνd|µ|= Z

fν₁d|µ|, . . . , Z

fνnd|µ|

∈Rⁿ.

Integration with respect to R

ⁿ

-valued Radon measures

Definition (4.16)

Letµbe anRⁿ-valued Radon measure on aσ-compact locally compact Hausdorff spaceX, with polar decompositionµ=ν|µ|.

1 A vector Borelian mapf :X →Rⁿis calledsummable with respect toµiff ∈L¹(|µ|,Rⁿ)≡L¹(|µ|)ⁿ. For suchf, we define

Z

f· dµ:=

Z

hf, νid|µ| ∈R.

2 An scalar Borelian mapf :X →Ris calledsummable with respect toµiff ∈L¹(|µ|). For suchf, we define

Z

fdµ:=

Z

fνd|µ|= Z

fν₁d|µ|, . . . , Z

fνnd|µ|

∈Rⁿ.

Integration with respect to R

ⁿ

-valued Radon measures

Remark (4.17)

1 Note that C_c(X,Rⁿ)⊂L¹(|µ|,Rⁿ)and the integral defined above extendsµ:C_c(X,Rⁿ)→R, i.e.∀f ∈C_c(X,Rⁿ),

Z

f· dµ=µ·f.

2 The integrals defined above satisfy the usual linearity and

convergence properties and the following versions of the triangle inequality:

| Z

f · dµ| ≤ Z

kfkd|µ| and k Z

fdµk ≤ Z

|f|d|µ|, forf ∈L¹(|µ|,Rⁿ)orf ∈L¹(|µ|), respectively.

Integration with respect to R

ⁿ

-valued Radon measures

Remark (4.17)

1 Note that C_c(X,Rⁿ)⊂L¹(|µ|,Rⁿ)and the integral defined above extendsµ:C_c(X,Rⁿ)→R, i.e.∀f ∈C_c(X,Rⁿ),

Z

f· dµ=µ·f.

2 The integrals defined above satisfy the usual linearity and

convergence properties and the following versions of the triangle inequality:

| Z

f · dµ| ≤ Z

kfkd|µ| and k Z

fdµk ≤ Z

|f|d|µ|, forf ∈L¹(|µ|,Rⁿ)orf ∈L¹(|µ|), respectively.

R

ⁿ

-valued measure on a σ-algebra

Definition (4.18)

LetX be a set andMaσ-algebra of subsets ofX. We say that a map µ:M→Rⁿis anRⁿ-valued measure onMif

VM1) µ(∅) =0;

VM2) µisσ-additive, i.e. for all countable disjoint family(An)n∈NinM, µ(∪_n∈NA_n) =X

n∈N

µ(A_n)

Notation

We denote byB^cX the set of Borel subsets ofX which are relatively compact.

R

ⁿ

-valued measure on a σ-algebra

Definition (4.18)

LetX be a set andMaσ-algebra of subsets ofX. We say that a map µ:M→Rⁿis anRⁿ-valued measure onMif

VM1) µ(∅) =0;

VM2) µisσ-additive, i.e. for all countable disjoint family(An)n∈NinM, µ(∪_n∈NA_n) =X

n∈N

µ(A_n)

Notation

We denote byB^cX the set of Borel subsets ofX which are relatively compact.

R

ⁿ

-valued Radon measures as set functions

Definition (4.19)

1 afiniteRⁿ-valued Radon measure set functiononX is an Rⁿ-valued measure onBX.

2 anRⁿ-valued Radon measure set functiononX is a set function µ:B^cX →Rⁿsuch that, for allK ⊂X compact,µ|_BK :BK →Rⁿis anRⁿ-valued measure onBK.

Notation

M(X)ⁿorM(X,Rⁿ)for finiteRⁿ-valued Radon measures onX Mloc(X)ⁿorMloc(X,Rⁿ)forRⁿ-valued Radon measures onX

R

ⁿ

-valued Radon measures as set functions

Definition (4.19)

1 afiniteRⁿ-valued Radon measure set functiononX is an Rⁿ-valued measure onBX.

2 anRⁿ-valued Radon measure set functiononX is a set function µ:B^cX →Rⁿsuch that, for allK ⊂X compact,µ|_BK :BK →Rⁿis anRⁿ-valued measure onBK.

Notation

M(X)ⁿorM(X,Rⁿ)for finiteRⁿ-valued Radon measures onX Mloc(X)ⁿorMloc(X,Rⁿ)forRⁿ-valued Radon measures onX

R

ⁿ

-valued Radon measures as set functions

Definition (4.19)

1 afiniteRⁿ-valued Radon measure set functiononX is an Rⁿ-valued measure onBX.

2 anRⁿ-valued Radon measure set functiononX is a set function µ:B^cX →Rⁿsuch that, for allK ⊂X compact,µ|_BK :BK →Rⁿis anRⁿ-valued measure onBK.

Notation

M(X)ⁿorM(X,Rⁿ)for finiteRⁿ-valued Radon measures onX Mloc(X)ⁿorMloc(X,Rⁿ)forRⁿ-valued Radon measures onX

Remark (4.20)

Eachµ∈M(X,Rⁿ)determines an element ofMloc(X,Rⁿ)by restriction ofµ:BX →RⁿtoB^cX.

SinceX isσ-compact,µis uniquely determined by its restriction to B^c_X, i.e. the associationµ∈M(X,Rⁿ)7→µ|_B^c

X ∈Mloc(X,Rⁿ)is linear 1-1 and allows us to identifyM(X,Rⁿ)with a linear subspace of Mloc(X,Rⁿ).

Remark (4.20)

Eachµ∈M(X,Rⁿ)determines an element ofMloc(X,Rⁿ)by restriction ofµ:BX →RⁿtoB^cX.

SinceX isσ-compact,µis uniquely determined by its restriction to B^c_X, i.e. the associationµ∈M(X,Rⁿ)7→µ|_B^c

X ∈Mloc(X,Rⁿ)is linear 1-1 and allows us to identifyM(X,Rⁿ)with a linear subspace of Mloc(X,Rⁿ).

Induced R

ⁿ

-valued Radon measure set functions

Definition (4.21)

Letµbe anRⁿ-valued Radon measure on aσ-compact locally compact Hausdorff spaceX. TheRⁿ-valued Radon measure set function induced byµis the set functionµˆ:B^c_X →Rⁿdefined, for all A∈B^cX, by

ˆ µ(A) :=

Z

χ_Adµ∈Rⁿ.

Ifµis finite, we defineµˆ:BX →Rⁿby the same formula.

Induced R

ⁿ

-valued Radon measure set functions

Proposition (4.22)

With the notation from the previous definition:

i) µˆis a (finite)Rⁿ-valued Radon measure set function on X ifµis a (finite)Rⁿ-valued Radon measure on X .

ii) The maps I :C_c(X,Rⁿ)^∗→Mloc(X,Rⁿ)and

I :C₀(X,Rⁿ)^∗ →M(X,Rⁿ)defined byµ7→µˆare linear 1-1 and commute with the inclusions, i.e. the following diagram is commutative:

C_c(X,Rⁿ)^∗ Mloc(X,Rⁿ)

C₀(X,Rⁿ)^∗ M(X,Rⁿ)

I

I

Induced R

ⁿ

-valued Radon measure set functions

Remark

IfX is a locally compact separable metric space,

I:C_c(X,Rⁿ)^∗ →Mloc(X,Rⁿ)andI:C₀(X,Rⁿ)^∗→M(X,Rⁿ)are surjective, i.e.

C_c(X,Rⁿ)^∗ ≡Mloc(X,Rⁿ) C₀(X,Rⁿ)^∗ ≡M(X,Rⁿ)

Restrictions of R

ⁿ

-valued Radon measures

Definition (4.31)

LetX be a locally compact separable metric space,µ∈C_c(X,Rⁿ)^∗ an Rⁿ-valued Radon measure andg ∈L¹_loc(|µ|)(in particular, ifg :X →R a bounded Borelian function onX). We define therestriction ofµto g, denoted byµ

x

g, as the continuous linear functional on C_c(X,Rⁿ) given by

µ

x

^{g·f :=Z hfg, νid|µ|}

if(ν,|µ|)is the polar decomposition ofµ.

Notation

Ifλis a positive measure onX andh∈L⁺(λ),

λ

x

^h:=hλ

Restrictions of R

ⁿ

-valued Radon measures

Definition (4.31)

LetX be a locally compact separable metric space,µ∈C_c(X,Rⁿ)^∗ an Rⁿ-valued Radon measure andg ∈L¹_loc(|µ|)(in particular, ifg :X →R a bounded Borelian function onX). We define therestriction ofµto g, denoted byµ

x

g, as the continuous linear functional on C_c(X,Rⁿ) given by

µ

x

^{g·f :=Z hfg, νid|µ|}

if(ν,|µ|)is the polar decomposition ofµ.

Notation

Ifλis a positive measure onX andh∈L⁺(λ), λ

x

^h:=hλ

Restrictions of R

ⁿ

-valued Radon measures

Remark (4.32)

1 The polar decomposition ofµ

x

^gis(gν|g|,|g||µ|), where we define

gν

|g| :=0 on the Borel set{g =0}. In particular,

|µ

x

^g|=|µ|

x

^|g|.

2 Ifµis a positive Radon measure onX (which we identify with the element of C_c(X,R)^∗ whose polar decomposition is(1, µ)) and A∈BX, thenµ

x

^χAcoincides with the positive Radon measure µ

x

A. We extend this notation for an arbitraryµ∈C_c(X,Rⁿ)^∗, i.e.

we use the notationµ

x

^Ain place ofµ

x

^χA. It then follows from the previous item that

|µ

x

^A|=|µ|

x

^A.

Restrictions of R

ⁿ

-valued Radon measures

Remark (4.32)

1 The polar decomposition ofµ

x

^gis(gν|g|,|g||µ|), where we define

gν

|g| :=0 on the Borel set{g =0}. In particular,

|µ

x

^g|=|µ|

x

^|g|.

2 Ifµis a positive Radon measure onX (which we identify with the element of C_c(X,R)^∗ whose polar decomposition is(1, µ)) and A∈BX, thenµ

x

^χAcoincides with the positive Radon measure µ

x

A. We extend this notation for an arbitraryµ∈C_c(X,Rⁿ)^∗, i.e.

we use the notationµ

x

^Ain place ofµ

x

^χA. It then follows from the previous item that

|µ

x

^A|=|µ|

x

^A.

Restrictions of R

ⁿ

-valued Radon measures

Remark (4.32)

3 We may similarly defineµ

x

^{g ∈Cc(X,Rn)∗ forµ∈Cc(X,R)∗ and}

g ∈L¹_loc(|µ|,Rⁿ):

µ

x

^{g :f ∈Cc(X,Rn)7→Z hf,giνd|µ|}

where(ν,|µ|)is the polar decomposition ofµ. Then(_kgk^gν ,kgk|µ|) is the polar decomposition ofµ

x

g. In particular,

|µ

x

^g|=|µ|

x

^kgk.

4 As a final generalization of the restriction operation, we may defineµ

x

^{T ∈Cc(X,Rm)∗ forµ∈Cc(X,Rn)∗and}

T ∈L¹_loc |µ|,L(R^m,Rⁿ)

byf ∈C_c(X,R^m)7→R

hT ·f, νid|µ|, where (ν,|µ|)is the polar decomposition ofµ.

Restrictions of R

ⁿ

-valued Radon measures

Remark (4.32)

3 We may similarly defineµ

x

^{g ∈Cc(X,Rn)∗ forµ∈Cc(X,R)∗ and}

g ∈L¹_loc(|µ|,Rⁿ):

µ

x

^{g :f ∈Cc(X,Rn)7→Z hf,giνd|µ|}

where(ν,|µ|)is the polar decomposition ofµ. Then(_kgk^gν ,kgk|µ|) is the polar decomposition ofµ

x

g. In particular,

|µ

x

^g|=|µ|

x

^kgk.

4 As a final generalization of the restriction operation, we may defineµ

x

^{T ∈Cc(X,Rm)∗ forµ∈Cc(X,Rn)∗and}

T ∈L¹_loc |µ|,L(R^m,Rⁿ)

byf ∈C_c(X,R^m)7→R

hT ·f, νid|µ|, where (ν,|µ|)is the polar decomposition ofµ.

Restrictions of R

ⁿ

-valued Radon measures

Remark (4.32)

Note that, definingT^∗ :X →L(Rⁿ,R^m)byx 7→T(x)^∗, we have,

∀f ∈C_c(X,R^m):

µ

x

^{T ·f =Z hT ·f, νid|µ|=Z f,}_kT^T∗∗·_·^ν_νk^{kT∗·νkd|µ|.}

Fundamental lemma of the Calculus of Variations

Exercise (4.34)

Let X be an open set inR^m. Ifµ:C_c(X,Rⁿ)→Ris anRⁿ-valued Radon measure on X such thatµ·f =0for all f ∈C^∞_c (X,Rⁿ), then µ=0. In particular, if g ∈L¹_loc(L^m|_X,Rⁿ)and

Z

X

hf,gidL^m=0

for all f ∈C^∞_c (X,Rⁿ), then g=0L^m-a.e. on X .

Trace of R

ⁿ

-valued Radon measures

Definition (4.35)

LetX be a locally compact separable metric space andA⊂X a locally compact subspace ofX (i.e the intersection of an open with a closed subset ofX). Ifµis anRⁿ-valued Radon measure onX with polar decomposition(ν,|µ|), we define anRⁿ-valued Radon measureµ|_A on Aby

f ∈C_c(A,Rⁿ)7→

Z

hef, νid|µ|,

whereef :X →Rⁿis the extension off by 0 in the complement ofA.

Trace of R

ⁿ

-valued Radon measures

Proposition (4.36)

With the notation above,µ|_Ais a well-definedRⁿ-valued Radon measure on A and it is finite if so isµ. Moreover, the polar decomposition ofµ|_Ais(ν|_A,|µ|

A).

Continuity of linear maps on C

_c

(X , R

ⁿ

)

Definition (4.37)

LetX andY be locally compact separable metric spaces.

i) We say thatA⊂C_c(X,Rⁿ)isboundedit there existsK ⊂X compact such thatA⊂C^K_c(X,Rⁿ)andAis bounded in the latter space (i.e. it bounded as a subset of the Banach space

C^K_c(X,Rⁿ)).

ii) We say that a sequence(x_n)n∈Nin C_c(X,Rⁿ)converges to x ∈C_c(X,Rⁿ)if there existsK ⊂X compact such that the image of the sequence is contained in C^K_c(X,Rⁿ),x ∈C^K_c(X,Rⁿ)and x_n→x in C^K_c(X,Rⁿ).

iii) We say that a linear mapT :C_c(X,Rⁿ)→C_c(Y,R^m)is continuous if one of the following equivalent conditions hold:

T(A)is bounded wheneverA⊂C_c(X,Rⁿ)is bounded.

T(xn)→0 whenever(xn)_n∈Nis a sequence in C_c(X,Rⁿ)such that xn→0.

Continuity of linear maps on C

_c

(X , R

ⁿ

)

Definition (4.37)

LetX andY be locally compact separable metric spaces.

i) We say thatA⊂C_c(X,Rⁿ)isboundedit there existsK ⊂X compact such thatA⊂C^K_c(X,Rⁿ)andAis bounded in the latter space (i.e. it bounded as a subset of the Banach space

C^K_c(X,Rⁿ)).

ii) We say that a sequence(x_n)n∈Nin C_c(X,Rⁿ)converges to x ∈C_c(X,Rⁿ)if there existsK ⊂X compact such that the image of the sequence is contained in C^K_c(X,Rⁿ),x ∈C^K_c(X,Rⁿ)and x_n→x in C^K_c(X,Rⁿ).

iii) We say that a linear mapT :C_c(X,Rⁿ)→C_c(Y,R^m)is continuous if one of the following equivalent conditions hold:

T(A)is bounded wheneverA⊂C_c(X,Rⁿ)is bounded.

T(xn)→0 whenever(xn)_n∈Nis a sequence in C_c(X,Rⁿ)such that xn→0.

Continuity of linear maps on C

_c

(X , R

ⁿ

)

Definition (4.37)

LetX andY be locally compact separable metric spaces.

i) We say thatA⊂C_c(X,Rⁿ)isboundedit there existsK ⊂X compact such thatA⊂C^K_c(X,Rⁿ)andAis bounded in the latter space (i.e. it bounded as a subset of the Banach space

C^K_c(X,Rⁿ)).

ii) We say that a sequence(x_n)n∈Nin C_c(X,Rⁿ)converges to x ∈C_c(X,Rⁿ)if there existsK ⊂X compact such that the image of the sequence is contained in C^K_c(X,Rⁿ),x ∈C^K_c(X,Rⁿ)and x_n→x in C^K_c(X,Rⁿ).

iii) We say that a linear mapT :C_c(X,Rⁿ)→C_c(Y,R^m)is continuous if one of the following equivalent conditions hold:

T(A)is bounded wheneverA⊂C_c(X,Rⁿ)is bounded.

T(xn)→0 whenever(xn)_n∈Nis a sequence in C_c(X,Rⁿ)such that xn→0.

Transposition

Proposition (4.39)

Let X and Y be locally compact separable metric spaces and T :C_c(X,Rⁿ)→C_c(Y,R^m)a linear map.

i) If T is continuous andµis anR^m-valued Radon measure on Y , thenµ◦T is anRⁿ-valued Radon measure on X .

ii) If T is continuous with respect to theC₀topology (i.e. the topology induced byk·k_u) on both domain and codomain, andµis a finite R^m-valued Radon measure on Y , thenµ◦T is a finiteRⁿ-valued Radon measure on X .

Transposition

Definition (4.40)

With the notation from the previous proposition, we define the transposeofT,T^t :C_c(Y,R^m)^∗→C_c(X,Rⁿ)^∗ in case (i) or T^t:C₀(Y,R^m)^∗ →C₀(X,Rⁿ)^∗in case (ii), byT^t·µ:=µ◦T.

Transposition

Example (4.41)

LetX be a locally compact separable metric space.

1) LetT :X →L(R^m,Rⁿ)be a continuous map. We define

Tˆ :C_c(X,R^m)→C_c(X,Rⁿ)by( ˆT ·f)(x) :=T(x)·f(x). ThenTˆ is linear continuous and its transpose is given byµ7→µ

x

^T.

2) LetU ⊂X open. The inclusion C_c(U,Rⁿ)⊂C_c(X,Rⁿ)(which maps f ∈C_c(U,Rⁿ)to its extension by 0 on the complement ofU) is clearly continuous; its transpose coincides withµ7→µ|_U.

Pushforward

Proposition

Let X and Y be locally compact separable metric spaces and f :X →Y a continuous proper map. Then both

(◦f) :C_c(Y,Rⁿ)→C_c(X,Rⁿ)and(◦f) :C₀(Y,Rⁿ)→C₀(X,Rⁿ)given by g7→g◦f are well-defined and linear continuous.

Definition

With the notation from the previous definition, the transposes

(◦f)^t :C_c(X,Rⁿ)^∗→C_c(Y,Rⁿ)^∗ and(◦f)^t:C₀(X,Rⁿ)^∗→C₀(Y,Rⁿ)^∗ are calledpushforward by f and denoted byf_#:µ7→f_#µ.

Pushforward

Proposition

Let X and Y be locally compact separable metric spaces and f :X →Y a continuous proper map. Then both

(◦f) :C_c(Y,Rⁿ)→C_c(X,Rⁿ)and(◦f) :C₀(Y,Rⁿ)→C₀(X,Rⁿ)given by g7→g◦f are well-defined and linear continuous.

Definition

With the notation from the previous definition, the transposes

(◦f)^t :C_c(X,Rⁿ)^∗→C_c(Y,Rⁿ)^∗ and(◦f)^t:C₀(X,Rⁿ)^∗→C₀(Y,Rⁿ)^∗ are calledpushforward by f and denoted byf_#:µ7→f_#µ.

Pushforward

Proposition

Let X and Y be locally compact separable metric spaces, f :X →Y a continuous proper map andµ∈C_c(X,Rⁿ)^∗with polar decomposition (ν_X,|µ|). Suppose that there exists a Borelian mapν_Y :Y →Rⁿsuch thatνY ◦f =νX. Then the polar decomposition of f_#µis(νY,f_#|µ|). In particular, ifµis a positive Radon measure on X , the pushforward ofµ by f in the sense of definition above coincides with the pushforward in the sense of positive measures.

Weak-star convergence

Definition (4.47)

LetX be a locally compact separable metric space. We say that i) a sequence(µ_k)k∈Nin C_c(X,Rⁿ)^∗isweakly-star convergentto

µ∈C_c(X,Rⁿ)^∗ (notation: µ_k* µ) if, for all^∗ f ∈C_c(X,Rⁿ), R f ·dµ_k →R

f · dµ;

ii) a sequence(µ_k)k∈Nin C₀(X,Rⁿ)^∗isweakly-star convergent in the sense of finite measurestoµ∈C₀(X,Rⁿ)^∗ (notation:µk

∗f

* µ) if, for allf ∈C₀(X,Rⁿ),R

f · dµ_k →R f· dµ.

Weak-star convergence

Definition (4.47)

LetX be a locally compact separable metric space. We say that i) a sequence(µ_k)k∈Nin C_c(X,Rⁿ)^∗isweakly-star convergentto

µ∈C_c(X,Rⁿ)^∗ (notation: µ_k* µ) if, for all^∗ f ∈C_c(X,Rⁿ), R f ·dµ_k →R

f · dµ;

ii) a sequence(µ_k)k∈Nin C₀(X,Rⁿ)^∗isweakly-star convergent in the sense of finite measurestoµ∈C₀(X,Rⁿ)^∗ (notation:µk

∗f

* µ) if, for allf ∈C₀(X,Rⁿ),R

f · dµ_k →R f· dµ.

Weak-star convergence

Remark (4.48)

Both types of convergence above are actually the same notion, i.e.

convergence of sequences with respect to weak star topologies: the first type in the weak-star dual of C_c(X,Rⁿ)and the second in the weak-star dual of C₀(X,Rⁿ).

Weak-star convergence

Proposition (relation between weak-star convergence and weak-star convergence in the sense of finite measures; 4.49)

Let X be a locally compact separable metric space,(µ_k)k∈Na

sequence inC_c(X,Rⁿ)^∗ andµ∈C_c(X,Rⁿ)^∗. The following conditions are equivalent:

i) µ_k* µ^∗ andsup_k∈N|µ_k|(X)<∞.

ii) (µk)k∈Nis a sequence inC₀(X,Rⁿ)^∗,µ∈C₀(X,Rⁿ)^∗ andµk

∗f

* µ.

Weak-star convergence

Proposition (4.50)

Let X and Y be locally compact separable metric spaces and T :C_c(X,Rⁿ)→C_c(Y,R^m)linear continuous. Then

T^t:C_c(Y,R^m)^∗→C_c(X,Rⁿ)^∗ preserves weak-star convergence of sequences. The same holds for weak-star convergence in the sense of finite measures if T is continuous with respect to theC₀topologies.

Foliations by Borel sets for positive Radon measures

Proposition (4.53)

Let X be a locally compact separable metric space,µa positive Radon measure on X and(E_α)_α∈Aa disjoint family of Borel sets in X . Then {α∈A|µ(E_α)>0}is countable.

Characterization of weak-star convergence for positive Radon measures

Theorem (4.54)

Let X be a locally compact separable metric space,(µ_k)k∈Na sequence of positive Radon measures in X andµa positive Radon measure in X . The following conditions are equivalent:

i) µ_k* µ.^∗

ii) For all K ⊂X compact and for all U⊂X open,

µ(K)≥lim supµ_k(K) and µ(U)≤lim infµ_k(U).

iii) For all E ∈B^c_X such thatµ(∂E) =0,µk(E)→µ(E).

Weak convergence and total variation

Proposition (4.57)

Let X be a locally compact separable metric space and(µ_k)k∈Na sequence inC_c(X,Rⁿ)^∗ weakly-star convergent toµ∈C_c(X,Rⁿ)^∗. Then, for every A⊂X open,|µ|(A)≤lim inf|µ_k|(A).

Proposition (4.58)

Let X be a locally compact separable metric space and(µk)k∈Na sequence inC_c(X,Rⁿ)^∗ weakly-star convergent toµ∈C_c(X,Rⁿ)^∗. If

|µ_k|(X)→ |µ|(X)<∞, then|µ_k|*|µ|.^∗f

Weak convergence and total variation

Proposition (4.57)

Let X be a locally compact separable metric space and(µ_k)k∈Na sequence inC_c(X,Rⁿ)^∗ weakly-star convergent toµ∈C_c(X,Rⁿ)^∗. Then, for every A⊂X open,|µ|(A)≤lim inf|µ_k|(A).

Proposition (4.58)

Let X be a locally compact separable metric space and(µk)k∈Na sequence inC_c(X,Rⁿ)^∗ weakly-star convergent toµ∈C_c(X,Rⁿ)^∗. If

|µ_k|(X)→ |µ|(X)<∞, then|µ_k|*|µ|.^∗f

De La Vallée Poussin Theorem

Theorem (4.61)

Let X be a locally compact separable metric space and(µ_k)_k∈Nbe a sequence of finiteRⁿ-valued Radon measures on X such that

sup{|µ_k|(X)|k ∈N}<∞. Then there exists a finiteRⁿ-valued Radon measureµon X and a subsequence(µ_kj)_j∈Nof(µ_k)_k∈Nsuch that µ_kj* µ. Moreover,^∗f |µ|(X)≤lim inf|µ_kj|(X).

De La Vallée Poussin Theorem

Corollary (4.63)

Let X be a locally compact separable metric space and(µ_k)k∈Nbe a sequence ofRⁿ-valued Radon measures on X such that, for any K ⊂X compact,sup{|µ_k|(K)|k ∈N}<∞. Then there exists an Rⁿ-valued Radon measureµon X and a subsequence(µ_kj)j∈Nof (µ_k)k∈Nsuch thatµ_kj* µ.^∗