Chapter 3 L p -space

Preliminaries on measure and integration

σ-algebra Ω ≠ψ is a set Σis a family of subsets of Ω with Σis called σ-algebra of subsets of Ω

measure space Ω ≠ψ is a set μ:Σ→[0, ∞] satisfies Σis aσ-algebra of subsets of Ω (Ω, Σ, μ) is called a measure space

measurable function p.1 f:Ω→R is measurable if (Ω, Σ, μ) is a measure space The family of measurable functions is a real vector space.

measurable function p.2 if The family is closed under limit, i.e. is a sequence of measurable functions, which converges pointwise to a finite-valued function f, then f is measurable.(see Exercise 1.1 and 1.3)

Exercise 1.1 If f, g are measurable, then (Ω, Σ, μ) is a measure space f+g is also measurable. Hint: for all

Exercise 1.3 Let f 1,f 2,… be measurable and (Ω, Σ, μ) is a measure space and f(x) is finite for each Hint: for all Show that f is measurable

indicator function for (Ω, Σ, μ) is a measure space χ A is the indicator function of A χ A is measurable

<W> denotes the smallest vector subspace containing W in a vector space.

Simple function p.1 are called simple functions Elements of

Simple function p.2 if the right hand side has a meaning, where A simple function can be expressed as are different values and A i = {f =α i }, we define then

Simple function p.3 In particular is meaningful if f is simple and nonnegative, although it is possible that

Integration for f ≧ 0,measurable For f ≧ 0, measurable,define

f +,f - If f is measurable, then f + and f - are measurable

Integration for measurable function p.1 For any measurable function f,define if R.H.S has a meaning

Integration for measurable function p.2 and is finite if and only if both are finite f is called integrable

Integration for measurable function p.3 is integrable f is integrable if and only if

limsupA n, liminf A n Ω is a set and {A n } is a sequence of subsets of Ω. Define

limA n then we say that the limit of the sequence {A n } exists and has the common set as the limit which is denoted by If

A n : monotone increasing then If

A n : monotone decreasing then If

Lemma 2.1 be monotone increasing, then If (Ω, Σ, μ) is a measure space

Lemma 2.2 be monotone decreasing, then If (Ω, Σ, μ) is a measure space

Egoroff Theorem Let {f n } be a sequence of measurable function and f n →f with finite limit on (Ω, Σ, μ) is a measure space then for any ε>0, there is such that μ(A\B)<ε and f n →f uniformly on B

Monotony Convergence Theorem Let {f n } be a nondecreasing sequence of nonnegative measurable functions Suppose (Ω, Σ, μ) is a measure space is a finite valued,then

Theorem (Beppo-Levi) Let {f n } be a increasing sequence of integrable functions such that f n ↗ f. Then f is integrable and (Ω, Σ, μ) is a measure space

Fatous Lemma Let {f n } be a sequence of extended real-valued measurable functions which is bounded from below by an integrable function. Then (Ω, Σ, μ) is a measure space

Remark Let {f n } be a sequence of extended real-valued measurable functions and f n ≦ 0. Then (Ω, Σ, μ) is a measure space

Lebesque Dominated Convergence Theorem If f n,n=1,2,…, and f are measurable functions and f n →f a.e. Suppose that a.e. with g being an integrable function.Then (Ω, Σ, μ) is a measure space

Corollary If f n,n=1,2,…, and f are measurable functions and f n →f a.e. Suppose that a.e. with g being an integrable function.Then (Ω, Σ, μ) is a measure space

5 The space L p (Ω,Σ,μ)

For measurable function f, let is called the essential sup-norm of f.

Exercise 5.1 Show that

Conjugate exponents If are such that then they are called conjugate exponents

Theorem 5.1 (Hölder’s Inequality) p.1 If are conjugate exponents, then

Theorem 5.1 (Hölder’s Inequality) p.2 More generally, if f 1,f 2, …,f k are functions s.t with then

Theorem 5.1 (Hölder’s Inequality) p.3 and

Theorem 5.2 (Minkowske Inequality) If f, g be measurable, whenever f+g is meaningful a.e. on Ω then

is the family of all measurable function f with From Minkowski Inequjality, it is readily seen that L p (Ω, Σ, μ) L p (Ω, Σ, μ) is a vector space.

Theorem L p is a normed vector space with for

then if and only if f=0 a.e. on Ω Let

then L p (Ω, Σ, μ) is a vector space which consists of equivalent classes of L p (Ω, Σ, μ) L p (Ω, Σ, μ) L p (Ω, Σ, μ) =L p (Ω, Σ, μ)/N w.r.t the equivalent relation ~definded by f~g if and only if f=g a.e. on Ω

is a Banach space. Theorem 5.3 (Fisher) L p (Ω, Σ, μ) with norm

Theorem then there is a subsequence such that

f is called an essential bounded function if Exercise 5.4 L ∞ (Ω, Σ, μ) is a Banach space.

Outer Measure

Ω ≠ψ: a set μis called outer measure on Ω if

μ-M easurable p.1 A subset A of Ω is called μ-measurable if for any i.e. for any

Exercise 1.1 Let μ:2 Ω →[0,+ ∞ ] be defined by (μis called the counting measure of Ω and every subset of Ω is μ-measurable. Show that μis an outer measure

Exercise 1.2 p.1 Let S be a subset of 2 Ω h aving the following properties:

Exercise 1.2 p.2 Define μ : 2 Ω →[ 0, +∞] by What are theμ-measurable subsets of Ω Show that μis an outer measure

Define υ : 2 Ω →[ 0, +∞] by What are theυ-measurable subsets of Ω Show that υis an outer measure

Exercise 1.3 Suppose μis an outer measure on Ω and then the restriction of μto A denoted byμ ∣ A(B)=μ(A∩B) for Show that μ ∣ A is an outer measure on Ω and every μ-measurable set is also μ ∣ A –measurable.

Properties of Measurable sets p.1 Suppose μmeasures Ω. (1) If A is μ-measurable, then so is Ω\A=A c (2) If A 1, A 2 are μ-measurable, then so is A 1 ∪ A 2

Properties of Measurable sets p.2 Remark : By induction the union of finitely many μ-measurable sets is μ-measurable. This fact together with (1) implies that the intersection of finitely many μ-measurable set is μ-measurable.

Properties of Measurable sets p.3 is a disjointed sequence of(3) If then μ-measurable sets in Ω and

is a disjointed sequence of(4) If is μ-measurable. μ-measurable sets in Ω, then

Properties of Measurable sets p.4 is a sequence of(5) If μ-measurable sets in Ω, then so are

Properties of Measurable sets p.5 sets in Ω, then (Ω, Σ, μ) is a measure (6) Let Σ be the family of all μ-measurable space.

Exercise 1.4 p.1 (i) If is an increasing μ-measurable sets in Ω, then

Exercise 1.4 p.2 (ii) If is a decreasing μ-measurable sets in Ω with μ(A 1 )<+∞, then

Regular A measure μ on Ωis called regular if for each there is a μ-measurable set such that μ(A)=μ(B)

Exercise 1.5 If is a sequence of sets in Ω and μis a regular measure on Ω, then

Theorem The family Σ μ of μ-measurable subsets of Ω is σ- algebra and μ=μ ︳ Σμ is σ- additive. i.e. (Ω, Σ μ, μ) is a measure space.

Premeasure

prem easure If Ω is a nonempty set, G a class of subsets of Ω containing ψ, and τ: G→[0,+∞] satisfy τ(ψ)=0. τis called a premeasure.

Outer M easure constructed from τ For a premeasure τ, Define μ:2 Ω →[0,+∞] by Then μ measures Ω and is called the outer measure constructed fromτby Method I.

Example 2.1 The Lebesgue measure on R n Let G be the class of all oriented rectangles in R n with ψ adjoined and let τ(I)=the volumn of I if I is an oriented rectangle τ(ψ)=0 the measure on R n constructed fromτ by Method I is called the Lebesgue measure on R n.

Exercise 2.1 For ε>0, Let G ε be the class of all open oriented rectangles in R n with diameter <εand τ ε (I)=volume of I for Show that the measure on R n constructed from τ ε by Method I is the Lebesgue measure.

Exercise 2.2 p.1 Let μ be the Lebesgue measure on R n (i) Show that μ(I)=volume I if I is an open oriented rectangle.

Exercise 2.2 p.2 (ii)Show that every open oriented rectangle is μ-measurable and hence so is every open set in R n ( μ-measurable set is called Lebesgue measurable set in this case.)

Exercise 2.2 p.3 (iii) If A and B are subsets of R n and dist(A,B)>0, then μ(A ∪ B)=μ(A)+μ(B)

Metric spaces

Metric Space Let M be a nonempty set and ρ:MXM→[0, ∞) satisfies ρis called a metric on M (M,ρ)is called a metric space.

Example 1 for Metric Space Let M=R n and let

Example 2 for Metric Space Let M=R n and let

Example 3 for Metric Space Let M=C[a,b] (- ∞<a<b<∞ ) and let

Example 4 for Metric Space Let M=C(K), where K is a compace set in R n and let Unless statement otherwise, C(K) will denote the metric space with the metric so defined.

Example 5 for Metric Space Let M= L p (Ω, Σ, μ)and let

Converge Let (M, ρ) be a metric space. A sequence is said to converge to if for any ε>0, there is such that ρ(x n,x 0 )<ε whenever Since x 0 is uniquely determined, x 0 is denoted by lim n →∞ x n If lim n →∞ x n exists, then we say that { x n } converges in M.

Example 3.1 for converge converges if and only if converges uniformly for

Cauchy sequence A sequence is called a Cauchy sequence if for any ε>0 there is such that

Exercise 3.1 Show that if converges, then is a Cauchy sequence.

Complete Metric space A metric space M is called complete if every Cauchy sequence in M converges in M

Examples for Complete (1) Let be compact, then is complete. (2) L p (Ω, Σ, μ) is complete.

Normed vector space p.1 Let K=R or C and let E be a vector space over K. Suppose that for each there is a nonnegative number associated with it so that

Normed vector space p.2 Then E is called a normed vector space (n.v.s) with norm

Normed vector space p.3 Then ρ is a metric on E and is called the metric associated with norm Let E be a n.v.s and Unless stated otherwise, for a n.v.s., we always consider this metric.

Banach space Both C(K) with K a compact subset of R n and L p (Ω, Σ, μ) are Banach spaces. A normed vector space is called a Banach space if it is a complete metric space

Continuous mapping Let M 1 and M 2 be metric spaces with metrics ρ 1 and ρ 2 respectively. A mapping T: M 1 →M 2 is continuous at if for any ε >0, there is δ>0 such that

Open and Closed set in a metric space A set G in a metric space is called an open set if there is δ>0 such that The complete of an open set is called a closed set.

Exercise 3.2 p.1 Let M 1 and M 2 be metric spaces with metrics ρ 1 and ρ 2 respectively and let T: M 1 →M 2 (1) Show that T is continuous at if and only if for any open set contains an open subset which contains x 0.

Exercise 3.2 p.2 (2) Show that T is continuous on M if and only if for any open set is an open subset of M 1.

Exercise 3.3 Let O be the family of all open subsets of a metric space. Show that

4.Carathéodory measure

Carathédory M easure If Ω is a metric space, then a measureμ is called Carathédory measure if whenever

Example 4.1 The Lebesgue measure on R n is a Carathédory measure.

Lemma 4.1 be an increasing sequence of subsets of Ω and for each n

Theorem 4.1 If μ is Carathédory measure onΩ, then every closed subset of Ωis μ- measurable.

Borel sets B(Ω) is the smallest σ-algebra of subsets Elements of B(Ω) are called Borel sets of Ω. of Ω that contains all closed subsets of Ω

Corollary 4.1 If μ is Carathédory measure onΩ, then all Borel subsets of Ωareμ- measurable.

Lebesgue M easure p.1 Ω = R, I: open finite interval of R Define L(A) then L is a Carathédory measure. L is called the Lebesgue measure.

Lebesgue M easure p.2 (R, Σ L,L) Similar construction on R n with I replaced by n-dimensional intervals L n is a Carathédory measure. L n is called the Lebesgue measure on R n.

Regularity of Measure

Regular measure A measure μ on Ωis called regular if for each there is a μ-measurable set such that μ(A)=μ(B)

Borel regular measure A measure μ on Ωis called Borel if every Borel set is μ-measurable. It is called Borel regular if it is Borel and for every such that μ(A)=μ(B) there is a Borel set

Radon measure A measure μ on Ωis called Radon measure if it is Borel regular and μ(K)<∞ for each compact set K. We already known that Carathéodory measure is Borel.

Theorem 6.1 Let μ be a Borel regular on a metric space Ω and suppose is μ-measurable andμ(A)<∞ Then μ ︱ A is a Radon measure.

Measure Theoretical Approximation of Sets in R n

Lemma 7.1 p.1 Let μ be a Borel measure on R n and B is a Borel set (i)If μ(B)<∞, then for each ε >0 there is a closed set such that

Lemma 7.1 p.2 (ii) If μ is Radon measure, then for each ε >0 there is an open set such that

Theorem 7.1 Approximation by open and compact sets Let μbe a Radon measure on R n, then (1) For (2) If A is μ-measurable on R n, then

Exercise 7.1 p.1 (i)Show that the Lebesgue measure on R n is a Radon measure. (ii) Letshow that there is a G δ set such that L n (H)=L n (A),where L n denotes the Lebesgue measure on R n.

Exercise 7.1 p.2 (iii) Let be Lebesgue measure show that there is a F σ set with L n (M)=L n (A).

Exercise 7.1 p.3 (iv) Let be Lebesgue measurable show that f is equivalent to a Borel measurable function.

Theorem Let μ=L n be the Lebesgue measure on R n, then (1) For (2) If A is Lebesgue measurable on R n, then

(A, Σ ︳ A, μ) (Ω, Σ, μ): measure space (A, Σ ︳ A, μ ) is a measure space

L p (Ω) L p (Ω, Σ,μ)= L p (Ω) Σ: the family of Lebesgue measurable subsets of Ω μ: the Lebesgue measure

C c (Ω) then f is continuous and C c (Ω) is the space of all continuous functions with compact surport in Ω i.e. if is a compact set in Ω

Lemma such that Let B be a measurable subset of Ωwith =L p (B)<∞, then for any ε>0, there is

Corollary such that Let be a simple function on Ω, then for any ε>0

Theorem C c (Ω) is dence in L p (Ω), 1 ≦ p<∞

Chapter IV L p space

IV 1 Some result for integration which one must know

Theorem (Beppo-Levi) Let {f n } be a increasing sequence of integrable functions such that f n ↗ f. Then f is integrable and (Ω, Σ, μ) is a measure space

Lebesque Dominated Convergence Theorem If f n,n=1,2,…, and f are measurable functions and f n →f a.e. Suppose that a.e. with g being an integrable function.Then (Ω, Σ, μ) is a measure space

Fatous Lemma Let {f n } be a sequence of extended real-valued measurable functions which is bounded from below by an integrable function. Then (Ω, Σ, μ) is a measure space

Theorem IV.3 (Desity Theorem) C c (Ω) is dense in L p (Ω), 1 ≦ p<∞

Theorem IV.4(Tonelli) Suppose that and that Then

Theorem IV.5(Fubini) p.1 Suppose that and that then for a.e.

Theorem IV.5(Fubini) p.2 and that Similarly, for a.e.

Theorem IV.5(Fubini) p.3 Furthermore, we have

IV 2 Definition and elementary properties of the space L p

Exercise 5.1 Show that