MEASURE and INTEGRATION Problems with Solutions

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Mô tả: MEASURE and INTEGRATIONProblems with Solutions MEASURE and INTEGRATIONProblems with SolutionsAnh Quang Le, Ph.D.October 8, 20131NOTATIONSA(X): The σ-algebra of subsets of X.(X, A(X), µ) : The measure space on X.B(X): The σ-algebra of Borel sets in a topological space X.ML: The σ-algebra of Lebesgue measurable sets in R.(R, ML, µL): The Lebesgue measure space on R.µL: The Lebesgue measure on R.µ∗L: The Lebesgue outer measure on R.1Eor χE: The characteristic function of the set E.www.MATHVN.com - Anh Quang Le, PhDwww.MathVn.com - Math Vietnam2www.MATHVN.com - Anh Quang Le, PhDwww.MathVn.com - Math VietnamContentsContents 11 Measure on a σ-Algebra of Sets 52 Lebesgue Measure on R 213 Measurable Functions 334 Convergence a.e. and Convergence in Measure 455 Integration of Bounded Functions on Sets of Finite Measure 536 Integration of Nonnegative Functions 637 Integration of Measurable Functions 758 Signed Measures and Radon-Nikodym Theorem 979 Differentiation and Integration 10910 LpSpaces 12111 Integration on Product Measure Space 14112 Some More Real Analysis Problems 1513www.MATHVN.com - Anh Quang Le, PhDwww.MathVn.com - Math Vietnam4 CONTENTSwww.MATHVN.com - Anh Quang Le, PhDwww.MathVn.com - Math VietnamChapter 1Measure on a σ-Algebra of Sets1. Limits of sequences of setsDefinition 1 Let (An)n∈Nbe a sequence of subsets of a set X.(a) We say that (An) is increasing if An⊂ An+1for all n ∈ N, and decreasing if An⊃ An+1forall n ∈ N.(b) For an increasing sequence (An), we definelimn→∞An:=∞n=1An.For a decreasing sequence (An), we definelimn→∞An:=∞n=1An.Definition 2 For any sequence (An) of subsets of a set X, we definelim infn→∞An:=n∈Nk≥nAklim supn→∞An:=n∈Nk≥nAk.Proposition 1 Let (An) be a sequence of subsets of a set X. Then(i) lim infn→∞An= {x ∈ X : x ∈ Anfor all but finitely many n ∈ N}.(ii) lim supn→∞An= {x ∈ X : x ∈ Anfor infinitely many n ∈ N}.(iii) lim infn→∞An⊂ lim supn→∞An.2. σ-algebra of sets5www.MATHVN.com - Anh Quang Le, PhDwww.MathVn.com - Math Vietnam6 CHAPTER 1. MEASURE ON A σ-ALGEBRA OF SETSDefinition 3 (σ-algebra)Let X be an arbitrary set. A collection A of subsets of X is called an algebra if it satisfies thefollowing conditions:1. X ∈ A.2. A ∈ A ⇒ Ac∈ A.3. A, B ∈ A ⇒ A ∪ B ∈ A.An algebra A of a set X is called a σ-algebra if it satisfies the additional condition:4. An∈ A, ∀n ∈ N ⇒n∈NAn∈ n ∈ N.Definition 4 (Borel σ-algebra)Let (X, O) be a topological space. We call the Borel σ-algebra B(X) the smallest σ-algebra of Xcontaining O.It is evident that open sets and closed sets in X are Borel sets.3. Measure on a σ-algebraDefinition 5 (Measure)Let A be a σ-algebra of subsets of X. A set function µ defined on A is called a measure if itsatisfies the following conditions:1. µ(E) ∈ [0, ∞] for every E ∈ A.2. µ(∅) = 0.3. (En)n∈N⊂ A, disjoint ⇒ µn∈NEn=n∈Nµ(En).Notice that if E ∈ A such that µ(E) = 0, then E is called a null set. If any subset E0of a null setE is also a null set, then the measure space (X, A, µ) is called complete.Proposition 2 (Properties of a measure)A measure µ on a σ-algebra A of subsets of X has the following properties:(1) Finite additivity: (E1, E2, , En) ⊂ A, disjoint =⇒ µ (nk=1Ek) =nk=1µ(Ek).(2) Monotonicity: E1, E2∈ A, E1⊂ E2=⇒ µ(E1) ≤ m(E2).(3) E1, E2∈ A, E1⊂ E2, µ(E1) < ∞ =⇒ µ(E2\ E1) = µ(E2) − µ(E1).(4) Countable subadditivity: (En) ⊂ A =⇒ µn∈NEn≤n∈Nµ(En).Definition 6 (Finite, σ-finite measure)Let (X, A, µ) be a measure space.1. µ is called finite if µ(X) < ∞.2. µ is called σ-finite if there exists a sequence (En) of subsets of X such thatX =n∈NEnand µ(En) < ∞, ∀n ∈ N.www.MATHVN.com - Anh Quang Le, PhDwww.MathVn.com - Math Vietnam74. Outer measuresDefinition 7 (Outer measure)Let X be a set. A set function µ∗defined on the σ-algebra P(X) of all subsets of X is called anouter measure on X if it satisfies the following conditions:(i) µ∗(E) ∈ [0, ∞] for every E ∈ P(X).(ii) µ∗(∅) = 0.(iii) E, F ∈ P(X), E ⊂ F ⇒ µ∗(E) ≤ µ∗(F ).(iv) countable subadditivity:(En)n∈N⊂ P(X), µ∗n∈NEn≤n∈Nµ∗(En).Definition 8 (Caratheodory condition)We say that E ∈ P(X) is µ∗-measurable if it satisfies the Caratheodory condition:µ∗(A) = µ∗(A ∩ E) + µ∗(A ∩ Ec) for every A ∈ P(X).We write M(µ∗) for the collection of all µ∗-measurable E ∈ P(X). Then M(µ∗) is a σ-algebra.Proposition 3 (Properties of µ∗)(a) If E1, E2∈ M(µ∗), then E1∪ E2∈ M(µ∗).(b) µ∗is additive on M(µ∗), that is,E1, E2∈ M(µ∗), E1∩ E2= ∅ =⇒ µ∗(E1∪ E2) = µ∗(E1) + µ∗(E2).∗ ∗ ∗∗www.MATHVN.com - Anh Quang Le, PhDwww.MathVn.com - Math Vietnam8 CHAPTER 1. MEASURE ON A σ-ALGEBRA OF SETSProblem 1Let A be a collection of subsets of a set X with the following properties:1. X ∈ A.2. A, B ∈ A ⇒ A \ B ∈ A.Show that A is an algebra.Solution(i) X ∈ A.(ii) A ∈ A ⇒ Ac= X \ A ∈ A (by 2).(iii) A, B ∈ A ⇒ A ∩B = A \ Bc∈ A since Bc∈ A (by (ii)).Since Ac, Bc∈ A, (A ∪B)c= Ac∩ Bc∈ A. Thus, A ∪B ∈ A. Problem 2(a) Show that if (An)n∈Nis an increasing sequence of algebras of subsets of a setX, thenn∈NAnis an algebra of subsets of X.(b) Show by example that even if Anin (a) is a σ-algebra for every n ∈ N, theunion still may not be a σ-algebra.Solution(a) Let A =n∈NAn. We show that A is an algebra.(i) Since X ∈ An, ∀n ∈ N, so X ∈ A.(ii) Let A ∈ A. Then A ∈ Anfor some n. And so Ac∈ An( since Anis analgebra). Thus, Ac∈ A.(iii) Suppose A, B ∈ A. We shall show A ∪B ∈ A.Since {An} is increasing, i.e., A1⊂ A2⊂ and A, B ∈n∈NAn, there issome n0∈ N such that A, B ∈ A0. Thus, A ∪B ∈ A0. Hence, A ∪B ∈ A.(b) Let X = N, An= the family of all subsets of {1, 2, , n} and their complements.Clearly, Anis a σ-algebra and A1⊂ A2⊂ However,n∈NAnis the family of allfinite and co-finite subsets of N, which is not a σ-algebra. www.MATHVN.com - Anh Quang Le, PhDwww.MathVn.com - Math Vietnam9Problem 3Let X be an arbitrary infinite set. We say that a subset A of X is co-finite if itscomplement Acis a finite subset of X. Let A consists of all the finite and theco-finite subsets of a set X.(a) Show that A is an algebra of subsets of X.(b) Show that A is a σ-algebra if and only if X is a finite set.Solution(a)(i) X ∈ A since X is co-finite.(ii) Let A ∈ A. If A is finite then Acis co-finite, so Ac∈ A. If A co-finite then Acis finite, so Ac∈ A. In both cases,A ∈ A ⇒ Ac∈ A.(iii) Let A, B ∈ A. We shall show A ∪B ∈ A.If A and B are finite, then A ∪ B is finite, so A ∪ B ∈ A. Otherwise, assumethat A is co-finite, then A ∪B is co-finite, so A ∪B ∈ A. In both cases,A, B ∈ A ⇒ A ∪B ∈ A.(b) If X is finite then A = P(X), which is a σ-algebra.To show the reserve, i.e., if A is a σ -algebra then X is finite, we assume that Xis infinite. So we can find an infinite sequence (a1, a2, ) of distinct elements of Xsuch that X \ {a1, a2, } is infinite. Let An= {an}. Then An∈ A for any n ∈ N,whilen∈NAnis neither finite nor co-finite. Son∈NAn/∈ A. Thus, A is not aσ-algebra: a contradiction! Note:For an arbitrary collection C of subsets of a set X, we write σ(C) for the smallestσ-algebra of subsets of X containing C and call it the σ-algebra generated by C.Problem 4Let C be an arbitrary collection of subsets of a set X. Show that for a givenA ∈ σ(C), there exists a countable sub-collection CAof C depending on A suchthat A ∈ σ(CA). (We say that every member of σ(C) is countable generated).www.MATHVN.com - Anh Quang Le, PhDwww.MathVn.com - Math Vietnam . function of the set E.www .MATHVN. com - Anh Quang Le, PhDwww .MathVn. com - Math Vietnam2www .MATHVN. com - Anh Quang Le, PhDwww .MathVn. com - Math VietnamContentsContents. Problems 1513www .MATHVN. com - Anh Quang Le, PhDwww .MathVn. com - Math Vietnam4 CONTENTSwww .MATHVN. com - Anh Quang Le, PhDwww .MathVn. com - Math VietnamChapter

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