Integration and measure theory

UPMC Basic functional analysis

Master 1, MM05E 2011-2012

Integration and measure theory.

Riesz Representation Theorem.LetΩbe an open subset ofR^N, andCc(Ω)be the space of all continuous functions having compact support in Ω. Let L:Cc(Ω)→R be a positive linear functional, i.e.,

L(αf+βg) =αL(f) +βL(g) for allf, g∈ Cc(Ω) and allα, β∈R, L(f)≥0 for allf ∈ C_c(Ω) with f ≥0.

Then there exist aσ-algebraM(containing the Borel σ-algebraB(Ω)) and a unique measureµ onMsuch that

L(f) = Z

Ω

f dµ, (1)

for every f ∈ Cc(Ω). Moreover, for each compact set K⊂Ω,µ(K)<∞, and for everyE∈M,

µ(E) = inf{µ(V) :E⊂V, V open}, (2)

and every open setE, and every E∈Mwithµ(E)<∞,

µ(E) = sup{µ(K) :K⊂E, K compact}. (3)

1) Show the uniqueness : if µ1andµ2 are two measures satisfying the conclusion of the theorem, show thatµ1=µ2

(hint : use Urysohn’s theorem).

2) For every open setV ⊂Ω, we define

µ(V) := sup{L(f) :f ∈ Cc(Ω), kfk_∞≤1, supp(f)⊂V}. (4) Show that ifV1⊂V2, thenµ(V1)≤µ(V2).

3) We extend µto any arbitraryE⊂Ωby setting

µ(E) := inf{µ(V) :E⊂V, V open}. (5)

Show that both definitions (4) and (5) are coincide on open sets, and that property (2) is satisfied.

4) Show thatµis an increasing set function : ifE₁⊂E₂, thenµ(E₁)≤µ(E₂).

5) LetMF be the family of all setsE⊂Ωsuch thatµ(E)<∞and

µ(E) = sup{µ(K) :K⊂E, Kcompact}.

Finally, let Mbe the class of allE⊂Ωsuch thatE∩K∈M_F for any compactK. Show that ifK is compact, then K∈MF, and

µ(K) = inf{L(f) :f ∈ Cc(Ω; [0,1]), f = 1onK}.

6) Show that any open set V satisfies (3), and that ifµ(V)<∞, thenV ∈M_F.

7) Show thatµis finitely subadditive on open sets : ifV1andV2 are open sets,µ(V1∪V2)≤µ(V1) +µ(V2).

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8) Show thatµis countably subadditive : for every setsEn ⊂Ωforn∈N^∗, then µ

∞

[

n=1

E_n

!

≤

∞

X

n=1

µ(E_n).

9) Show thatµis finitely additive on compact sets : ifK1 andK2are disjoint compact sets, µ(K1∪K2) =µ(K1) + µ(K₂).

10) Show that ifE=S∞

n=1En, whereEn are pairwise disjoint elements ofMF for alln∈N^∗, then, µ(E) =

∞

X

n=1

µ(En).

Prove that if µ(E)<∞, thenE∈M_F.

11) Show that if E ∈ MF and ε > 0, there is a compact set K and an open set V such that K ⊂ E ⊂ V and µ(V \K)< ε.

12) Deduce that ifAandB∈MF, thenA\B,A∪B andA∩B ∈MF.

13) Show that M is a σ-algebra which contains all Borel sets and all sets E such that µ(E) = 0. Deduce that MF ={E ⊂M:µ(E)<∞}.

14) Show thatµis a measure onMsatisfying (2) and (3).

15) Prove that in order to show the representation property (1) it is enough to check the inequalityL(f)≤R

Ωf dµ for any f ∈ Cc(Ω).

16) Letf ∈ Cc(Ω),K:= supp(f)and[a, b]be an interval which contains the image off. Forε >0, lety₀, . . . , y_n∈R be such that y0< a < y1< . . . < yn =b, andmax_1≤i≤n(yi−y_i−1)< ε. Define

Ei :={x∈Ω :y_i−1< f(x)≤yi} ∩K.

Show thatEi are disjoint Borel sets whose union isK.

17) Show thatL(f)≤R

Ωf dµfor anyf ∈ Cc(Ω).

Regularity of Radon measures

1) With the notations of the Riesz Representation Theorem, show that for every measurable set E ∈M and every ε >0, there exist a closed setC and an open setV such thatC⊂E⊂V andµ(V \C)< ε.

2) Letλbe a Radon measure in an open setΩ⊂R^N. a) Show that

λ(E) = inf{λ(V) :E⊂V, V open} for every Borel setE⊂Ω, and

λ(E) = sup{λ(K) :K⊂E, K compact}for every Borel set E⊂Ωwithλ(E)<∞.

b) Show that for every Borel setE ⊂Ω and anyε >0, there exist a closed setC and an open setV such that C⊂E⊂V and

λ(V \C)< ε.

Hint : Consider the positive linear form L(ϕ) :=R

Ωϕ dλfor all ϕ∈ Cc(Ω).

Existence of the Lebesgue measure.There exists aσ-algebraL(R^N)(containing the Borelσ-algebraB(R^N)) and a unique measureL^N on L(R^N)such that

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1. L^N([0,1]^N) = 1;

2. For anyx∈R^N and anyE∈ L(R^N),L^N(x+E) =L^N(E).

The measure L^N is called the Lebesgue measure, andL(R^N)is theσ-algebra of all Lebesgue measurable sets.

1) Show that there exist aσ-algebra L(R^N)containingB(R^N), and a measureL^N onL(R^N)satisfying (2), (3) and such that

Z

R^N

f(x)dx= Z

R^N

f dL^N,

where the first integral is the Riemann integral off, and the second one is the Lebesgue integral off with respect to the measureL^N.

2) Show thatE∈ L(R^N)if and only if there exist anFσ setA (a countable union of closed sets) and aGδ setB (a countable intersection of open sets) such thatA⊂E⊂BandL^N(B\A) = 0. Deduce thatL(R^N) ={E∪Z, withE∈ B(R^N)and L^N(Z) = 0}.

3) Leta < b be real numbers. Construct a sequence of functionsϕn ∈ Cc(R)such that ϕn= 1in (a+ 1/n, b−1/n) and ϕn = 0inR\[a, b].

4) Show that ifa_i< b_i for alli∈ {1, . . . , N}, then

L^N

N

Y

i=1

(ai, bi)

!

=

N

Y

i=1

(bi−ai)≤ L^N

N

Y

i=1

[ai, bi]

! .

5) Show that for anyi∈ {1, . . . , N} and anya∈R,L^N({xi=a}) = 0. Deduce that

L^N

N

Y

i=1

[a_i, b_i]\

N

Y

i=1

(a_i, b_i)

!

= 0

and that

L^N

N

Y

i=1

(ai, bi)

!

=

N

Y

i=1

(bi−ai) =L^N

N

Y

i=1

[ai, bi]

! .

6) Show that for anyx∈R^N and any open setV ⊂R^N,L^N(x+V) =L^N(V).

7) Show that the Borelσ-algebra B(R^N)is stable by translation : ifE∈ B(R^N)andx∈R^N, then x+E∈ B(R^N).

Deduce that L^N(x+E) =L^N(E)for anyE∈ B(R^N).

8) Using question 3, deduce that for any x∈R^N and any E∈ L(R^N), thenL^N(x+E) =L^N(E).

9) Show that any open set can be covered by countably many closed cubes with pairwise disjoint interior.

10) Show that ifµis a Radon measure invariant by translation, then there exists a constantc >0such thatµ=cL^N. Deduce that ifµ([0,1]^N) = 1, thenµ=L^N.

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