MEASURE THEORY Oscar Blasco Contents 1 Abstract Measure 5 1.1 Basic notions on sets . 5 1.2 Basic notions on set functions. 9 1.3 Outer measures. 11 1.4 Extension of measures. 15 1.5 Borel-Stieltjes measures on R................... 17 1.6 Measurable and non-measurable sets. 23 1.7 Exercises . 26 2 Measurable and Integrable functions 29 2.1 Measurable functions . 29 2.2 Some types of convergence. 35 2.3 Integrable functions. 38 2.4 Exercises . 50 3 The product measure and Fubini’s theorem 57 3.1 The product measure . 57 3.2 Fubini theorem . 63 3.3 Applications . 64 3.4 Exercises . 68 4 The Radon-Nikodym Theorem 73 4.1 Complex and real measures. 73 4.2 The theorem and its proof. 80 4.3 Applications . 84 4.4 Exercises . 88 3 Chapter 1 Abstract Measure 1.1 Basic notions on sets Definition 1.1.1 A non empty family of subsets of X, say A ⊂ P(X), is called an algebra if (i) ∅ ∈ A, (ii) If A ∈ A then X \ A ∈ A, (iii) If A, B ∈ A then A ∪ B ∈ A. Definition 1.1.2 A non empty family of subsets of X, say Σ ⊂ P(X), is called a σ-algebra if (i) ∅ ∈ Σ, (ii) If A ∈ Σ then X \ A ∈ Σ, (iii) If An ∈ Σ for all n ∈ N then ∪nAn ∈ Σ. Definition 1.1.3 A non empty family of subsets of X, say M ⊂ P(X), is called a monotone class if for all monotone sequence of sets An ∈ M, increasing An ⊂ An+1 (respect. decreasing An+1 ⊂ An), then ∪nAn ∈ M (respect. ∩nAn ∈ M). Definition 1.1.4 A non empty family of subsets of X, say R ⊂ P(X), is called a ring if (i) ∅ ∈ R, (ii) If A, B ∈ R then A \ B ∈ R, (iii) If A, B ∈ R then A ∪ B ∈ R. 5 6 Chapter 1. Abstract measure Remark 1.1.1 Let Σ ⊂ P(X). Σ is a σ-algebra if and only if Σ is a mono- tone class and an algebra. Let A ⊂ P(X). A is an algebra if and only if A is a ring containing X. Example 1.1.1 (1) The trivial σ-algebras are P(X) and Σ = {∅,X}. (2) Let M = {An : n ∈ N} ∪ X where A1 = ∅, An ⊂ An+1 for all n ∈ N and ∪nAn = X. This is a monotone class but not necessarily a ring. (3) Let X = [0, 1) and A = {finite unions of intervals [a, b), 0 ≤ a ≤ b ≤ 1} is an algebra but not σ-algebra. If the intervals in the previous family are assumed to have 0 ≤ a ≤ b < 1 then it is a ring but not an algebra. (4) Let X be non empty and numerable. R = {A ⊂ X : card(A) < ∞} is a ring but not algebra. Definition 1.1.5 Let An ∈ P(X) for n ∈ N. The upper limit (respect. lower limit) of the sequence is defined by ∞ ∞ lim sup An = ∩n=1 ∪k=n Ak (respect. ∞ ∞ lim inf An = ∪n=1 ∩k=n Ak.) A sequence is said to have limit if lim sup An = lim inf An. Such a set is called lim An. Remark 1.1.2 Any monotone sequence has a limit. If An is increasing (respect. decreasing) then lim An = ∪nAn (respect. lim An = ∩nAn). Proposition 1.1.6 Let R be a ring. If A, B ∈ R then A M B ∈ R and A ∩ B ∈ R. Proof: Note that A M B = ((A ∪ B) \ A) ∪ ((A ∪ B) \ B) and A ∩ B = A ∪ B \ (A M B). Proposition 1.1.7 Let Σ be a σ-algebra. If An ∈ Σ for all n ∈ N then ∩nAn ∈ Σ, lim sup An ∈ Σ and lim inf An ∈ Σ Proof: Write ∩nAn = X \ ∪n(X \ An) and apply the properties of an σ-algebra. 1.1. Basic notions on sets 7 Definition 1.1.8 A pair (X, Σ) given by a non-empty set X and a σ-algebra over X is called a measurable space. We shall give several methods of constructing measurable spaces. The proofs are rather straightforward and left to the reader. Definition 1.1.9 (Induced σ-algebra) Let (X, Σ) be a measurable space and let Y ⊂ X. Then ΣY = {A ∩ Y : A ∈ Σ} is a σ-algebra over Y . If Y ∈ Σ then ΣY = {A ⊂ Y, A ∈ Σ}. Definition 1.1.10 (Image of a σ-algebra ) Let (X, Σ) be a measurable space and let f : X → Y be a function. We define f(Σ) = {B ⊂ Y : f −1(B) ∈ Σ}. Then (Y, f(Σ)) is a measurable space. Definition 1.1.11 (σ-algebra generated by a family) Let F ⊂ P(X). We denote σ(F) the smallest σ-algebra containing F, which is called the σ-algebra generated by F. It is elementary to see that σ(F) = ∩{Σ:Σ σ-algebra , F ⊂ Σ}. Remark 1.1.3 (1) If F1 ⊂ F2 ⊂ σ(F1) then σ(F1) = σ(F2). (2) F1 ⊂ σ(F2) and F2 ⊂ σ(F1) if and only if σ(F1) = σ(F2). Definition 1.1.12 (Borel σ-algebra ) Let (X, τ) be a topological space and let G be the collection of open sets for the topology τ. The σ-algebra σ(G) is called the Borel σ-algebra and denoted B(X). The elements in B(X) are called Borel sets. Remark 1.1.4 Closed sets, Gδ sets (numerable intersection of open sets) or Fσ sets (numerable union of closed sets) are examples of Borel sets. Using Remark 1.1.3 one easily sees the following facts: If F denotes the collection of closed sets in τ then B(X) = σ(F). If (X, d) is a separable metric space (or a metric space where any open set is a numerable union of balls) and E = {B(x, r): x ∈ X, r > 0} then B(X) = σ(E). Proposition 1.1.13 Let X = R and let us consider the following collections E1 = {(a, b): a ≤ b}, E2 = {(a, b]: a ≤ b}, E3 = {[a, b): a ≤ b} and E4 = {[a, b]: a ≤ b}. Then B(R) = σ(Ei) for i = 1, 2, 3, 4. 8 Chapter 1. Abstract measure Proof: Notice that any open set is a numerable union of open intervals. Hence Remark 1.1.3 shows that B(R) = σ(E1). 1 1 Let us observe that (a, b] = ∩n(a, b + n ), [a, b) = ∪n(a − n , b) and [a, b] = 1 1 ∩n(a − n , b + n ) to get the other cases. n Proposition 1.1.14 Let n ∈ N and X = R and consider E = {(a1, b1] × n (a2, b2] × ...(an, bn]: ai ≤ bi, i = 1, ..., n}. Then B(R ) = σ(E). Proof: We sketch the proof for the case n = 2. Consider J0 = {(n, n + 1] × (m, m + 1] : n, m ∈ Z} and Jn the collection of intervals resulting of dividing each square of the previous family into four of the same area. 2 Now for each x ∈ R there exists a unique sequence of intervals Ik(x) ∈ Jk such that x ∈ Ik(x) for all k ∈ N, Ik+1(x) ⊂ Ik(x) and Area(Ik(x)) → 0 as k → ∞. Given an open set G we can consider the family F = {J ∈ ∪kJk : J ⊂ G}. Of course G = ∪J∈FJ and with a little effort it can be seen that only a numerable number of sets is needed. Since F ⊂ E then the proof is finished. Definition 1.1.15 Let E ⊂ P(X). M(E) stands for the smallest monotone class containing E, which is called the monotone class generated by E. Theorem 1.1.16 (The monotone class theorem) Let A be an algebra over X. Then σ(A) = M(A). Proof: It suffices to see that M(A) is σ-algebra . Since M(A) is a monotone class we have only to show that it is an algebra. Clearly ∅ ∈ A ⊂ M(A). Now given A ∈ M(A), to see that X \A ∈ M(A) let us define Σ = {A ∈ M(A): X \ A ∈ M(A)} and show that Σ = M(A). For such a purpose we shall see that Σ is a monotone class and contains A. Indeed, if An ∈ Σ is a monotone sequence then lim An ∈ Σ. Clearly if A ∈ A then X \ A ∈ A ⊂ M(A). Therefore A ⊂ Σ. Given now A, B ∈ M(A), we need to show that A ∪ B ∈ M(A). Let us define ΓA = {B ∈ M(A): A ∪ B ∈ M(A)}. Note that ΓA is a monotone class (since (Bn)n monotone sequence in ΓA has limit in ΓA). Now let us deal first with the case A ∈ A. In this case ΓA contains A (since B ∈ A implies A ∪ B ∈ A ⊂ M(A)). Therefore ΓA = M(A) for all A ∈ A. 1.2. Basic notions on set functions. 9 Now let us consider A ∈ M(A) \ A. To see that A ⊂ ΓA observe that B ∈ A belongs to ΓA since we have that B ∈ ΓA if and only if A ∈ ΓB and this was shown to be true in the previous case. 1.2 Basic notions on set functions. Definition 1.2.1 Let A be an algebra over X, a set function µ : A → [0, ∞] is called additive (or finitely additive) if µ(A ∪ B) = µ(A) + µ(B) for all A, B ∈ A such that A ∩ B = ∅. P∞ µ is called a measure over A if µ(∅) = 0 and µ(∪nAn) = n=1 µ(An) for any sequence (An) of pairwise disjoint sets in A such that ∪nAn ∈ A.