NOTES ON MEASURE THEORY M. Papadimitrakis Department of Mathematics University of Crete Autumn of 2004 2 Contents 1 σ-algebras 7 1.1 σ-algebras. 7 1.2 Generated σ-algebras. 8 1.3 Algebras and monotone classes. 9 1.4 Restriction of a σ-algebra. 11 1.5 Borel σ-algebras. 12 1.6 Exercises. 17 2 Measures 21 2.1 General measures. 21 2.2 Point-mass distributions. 23 2.3 Complete measures. 25 2.4 Restriction of a measure. 27 2.5 Uniqueness of measures. 28 2.6 Exercises. 28 3 Outer measures 33 3.1 Outer measures. 33 3.2 Construction of outer measures. 35 3.3 Exercises. 36 4 Lebesgue measure on Rn 39 4.1 Volume of intervals. 39 4.2 Lebesgue measure in Rn. ...................... 41 4.3 Lebesgue measure and simple transformations. 45 4.4 Cantor set. 49 4.5 A non-Lebesgue set in R. ...................... 50 4.6 Exercises. 52 5 Borel measures 57 5.1 Lebesgue-Stieltjes measures in R................... 57 5.2 Borel measures. 62 5.3 Metric outer measures. 67 5.4 Hausdorff measure. 68 3 5.5 Exercises. 70 6 Measurable functions 73 6.1 Measurability. 73 6.2 Restriction and gluing. 73 6.3 Functions with arithmetical values. 74 6.4 Composition. 76 6.5 Sums and products. 76 6.6 Absolute value and signum. 78 6.7 Maximum and minimum. 79 6.8 Truncation. 80 6.9 Limits. 81 6.10 Simple functions. 83 6.11 The role of null sets. 86 6.12 Exercises. 87 7 Integrals 93 7.1 Integrals of non-negative simple functions. 93 7.2 Integrals of non-negative functions. 96 7.3 Integrals of complex valued functions. 99 7.4 Integrals over subsets. 106 7.5 Point-mass distributions. 109 7.6 Lebesgue integral. 112 7.7 Lebesgue-Stieltjes integral. 119 7.8 Reduction to integrals over R. 123 7.9 Exercises. 125 8 Product measures 137 8.1 Product σ-algebra. 137 8.2 Product measure. 139 8.3 Multiple integrals. 146 8.4 Surface measure on Sn−1. 153 8.5 Exercises. 161 9 Convergence of functions 167 9.1 a.e. convergence and uniformly a.e. convergence. 167 9.2 Convergence in the mean. 168 9.3 Convergence in measure. 171 9.4 Almost uniform convergence. 174 9.5 Relations between types of convergence. 176 9.6 Exercises. 180 10 Signed measures and complex measures 183 10.1 Signed measures. 183 10.2 The Hahn and Jordan decompositions, I. 185 10.3 The Hahn and Jordan decompositions, II. 191 4 10.4 Complex measures. 195 10.5 Integration. 197 10.6 Lebesgue decomposition, Radon-Nikodym derivative. 200 10.7 Differentiation of indefinite integrals in Rn. 208 10.8 Differentiation of Borel measures in Rn. 215 10.9 Exercises. 217 11 The classical Banach spaces 221 11.1 Normed spaces. 221 11.2 The spaces Lp(X; Σ; µ). 229 11.3 The dual of Lp(X; Σ; µ). 240 11.4 The space M(X; Σ). 247 11.5 The space C0(X) and its dual. 249 11.6 Exercises. 259 5 6 Chapter 1 σ-algebras 1.1 σ-algebras. Definition 1.1 Let X be a non-empty set and Σ a collection of subsets of X. We call Σ a σ-algebra of subsets of X if it is non-empty, closed under complements and closed under countable unions. This means: (i) there exists at least one A ⊆ X so that A 2 Σ, (ii) if A 2 Σ, then Ac 2 Σ, where Ac = X n A, and +1 (iii) if An 2 Σ for all n 2 N, then [n=1An 2 Σ. The pair (X; Σ) of a non-empty set X and a σ-algebra Σ of subsets of X is called a measurable space. Proposition 1.1 Every σ-algebra of subsets of X contains at least the sets ; and X, it is closed under finite unions, under countable intersections, under finite intersections and under set-theoretic differences. Proof: Let Σ be any σ-algebra of subsets of X. c (a) Take any A 2 Σ and consider the sets A1 = A and An = A for all n ≥ 2. c +1 c Then X = A [ A = [n=1An 2 Σ and also ; = X 2 Σ. (b) Let A1;:::;AN 2 Σ. Consider An = AN for all n > N and get that N +1 [n=1An = [n=1An 2 Σ. +1 +1 c c (c) Let An 2 Σ for all n. Then \n=1An = ([n=1An) 2 Σ. N (d) Let A1;:::;AN 2 Σ. Using the result of (b), we get that \n=1An = N c c ([n=1An) 2 Σ. (e) Finally, let A; B 2 Σ. Using the result of (d), we get that AnB = A\Bc 2 Σ. Here are some simple examples. Examples 1. The collection f;;Xg is a σ-algebra of subsets of X. 2. If E ⊆ X is non-empty and different from X, then the collection f;; E; Ec;Xg is a σ-algebra of subsets of X. 7 3. P(X), the collection of all subsets of X, is a σ-algebra of subsets of X. 4. Let X be uncountable. The fA ⊆ X j A is countable or Ac is countableg is a σ-algebra of subsets of X. Firstly, ; is countable and, hence, the collection is non-empty. If A is in the collection, then, considering cases, we see that Ac is also in the collection. Finally, let An be in the collection for all n 2 N. If all +1 c An's are countable, then [n=1An is also countable. If at least one of the An's, say Ac , is countable, then ([+1 A )c ⊆ Ac is also countable. In any case, n0 n=1 n n0 +1 [n=1An belongs to the collection. The following result is useful. Proposition 1.2 Let Σ be a σ-algebra of subsets of X and consider a finite N sequence fAngn=1 or an infinite sequence fAng in Σ. Then there exists a finite N sequence fBngn=1 or, respectively, an infinite sequence fBng in Σ with the properties: (i) Bn ⊆ An for all n = 1;:::;N or, respectively, all n 2 N. N N +1 +1 (ii) [n=1Bn = [n=1An or, respectively, [n=1Bn = [n=1An. (iii) the Bn's are pairwise disjoint. Proof: Trivial, by taking B1 = A1 and Bk = Ak n (A1 [···[ Ak−1) for all k = 2;:::;N or, respectively, all k = 2; 3;::: . 1.2 Generated σ-algebras. Proposition 1.3 The intersection of any σ-algebras of subsets of the same X is a σ-algebra of subsets of X. Proof: Let fΣigi2I be any collection of σ-algebras of subsets of X, indexed by an arbitrary non-empty set I of indices, and consider the intersection Σ = \i2I Σi. Since ; 2 Σi for all i 2 I, we get ; 2 Σ and, hence, Σ is non-empty. Let A 2 Σ. Then A 2 Σi for all i 2 I and, since all Σi's are σ-algebras, c c A 2 Σi for all i 2 I. Therefore A 2 Σ. Let An 2 Σ for all n 2 N. Then An 2 Σi for all i 2 I and all n 2 N +1 and, since all Σi's are σ-algebras, we get [n=1An 2 Σi for all i 2 I. Thus, +1 [n=1An 2 Σ. Definition 1.2 Let X be a non-empty set and E be an arbitrary collection of subsets of X. The intersection of all σ-algebras of subsets of X which include E is called the σ-algebra generated by E and it is denoted by Σ(E). Namely Σ(E) = \fΣ j Σ is a σ-algebra of subsets of X and E ⊆ Σg : Note that there is at least one σ-algebra of subsets of X which includes E and this is P(X). Note also that the term σ-algebra used in the name of Σ(E) is justified by its definition and by Proposition 1.3. Proposition 1.4 Let E be any collection of subsets of the non-empty X. Then Σ(E) is the smallest σ-algebra of subsets of X which includes E. Namely, if Σ is any σ-algebra of subsets of X such that E ⊆ Σ, then Σ(E) ⊆ Σ. 8 Proof: If Σ is any σ-algebra of subsets of X such that E ⊆ Σ, then Σ is one of the σ-algebras whose intersection is denoted Σ(E). Therefore Σ(E) ⊆ Σ. Looking back at two of the examples of σ-algebras, we easily get the following examples. Examples. 1. Let E ⊆ X and E be non-empty and different from X and consider E = fEg. Then Σ(E) = f;; E; Ec;Xg. To see this just observe that f;; E; Ec;Xg is a σ-algebra of subsets of X which contains E and that there can be no smaller σ-algebra of subsets of X containing E, since such a σ-algebra must necessarily contain ;;X and Ec besides E. 2. Let X be an uncountable set and consider E = fA ⊆ X j A is countableg. Then Σ(E) = fA ⊆ XjA is countable or Ac is countableg. The argument is the same as before. fA ⊆ XjA is countable or Ac is countableg is a σ-algebra of subsets of X which contains all countable subsets of X and there is no smaller σ-algebra of subsets of X containing all countable subsets of X, since any such σ-algebra must contain all the complements of countable subsets of X.