Measure Theory 2 - graduate-measure- ¢  NOTES ON MEASURE THEORY M. Papadimitrakis...

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Transcript of Measure Theory 2 - graduate-measure- ¢  NOTES ON MEASURE THEORY M. Papadimitrakis...

  • 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 Borel σ-algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

    1.4 Algebras and monotone classes. . . . . . . . . . . . . . . . . . . . 13

    1.5 Restriction of a σ-algebra. . . . . . . . . . . . . . . . . . . . . . . 16

    1.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

    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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

    3 Outer measures 33

    3.1 Outer measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

    3.2 Construction of outer measures. . . . . . . . . . . . . . . . . . . . 35

    3.3 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

    4 Lebesgue-measure in Rn 41

    4.1 Volume of intervals. . . . . . . . . . . . . . . . . . . . . . . . . . 41

    4.2 Lebesgue-measure in Rn. . . . . . . . . . . . . . . . . . . . . . . 43

    4.3 Lebesgue-measure and simple transformations. . . . . . . . . . . 46

    4.4 Cantor’s set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

    4.5 A non-Lebesgue-measurable set in R. . . . . . . . . . . . . . . . 52

    4.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

    5 Borel measures 59

    5.1 Lebesgue-Stieltjes-measures in R. . . . . . . . . . . . . . . . . . . 59

    5.2 Borel measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

    5.3 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

    3

  • 4 CONTENTS

    6 Measurable functions 71 6.1 Measurability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 6.2 Restriction and gluing. . . . . . . . . . . . . . . . . . . . . . . . . 71 6.3 Functions with arithmetical values. . . . . . . . . . . . . . . . . . 72 6.4 Composition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 6.5 Sums and products. . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.6 Absolute value and signum. . . . . . . . . . . . . . . . . . . . . . 76 6.7 Maximum and minimum. . . . . . . . . . . . . . . . . . . . . . . 77 6.8 Truncation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 6.9 Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.10 Simple functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.11 The role of null sets. . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.12 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

    7 Integrals 91 7.1 Integrals of non-negative simple functions. . . . . . . . . . . . . . 91 7.2 Integrals of non-negative functions. . . . . . . . . . . . . . . . . . 94 7.3 Integrals of complex valued functions. . . . . . . . . . . . . . . . 97 7.4 Integrals over subsets. . . . . . . . . . . . . . . . . . . . . . . . . 104 7.5 Point-mass distributions. . . . . . . . . . . . . . . . . . . . . . . . 107 7.6 Lebesgue-integral. . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.7 Lebesgue-Stieltjes-integral. . . . . . . . . . . . . . . . . . . . . . . 117 7.8 Reduction to integrals over R. . . . . . . . . . . . . . . . . . . . 121 7.9 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

    8 Product-measures 135 8.1 Product-σ-algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . 135 8.2 Product-measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 8.3 Multiple integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . 146 8.4 Surface-measure on Sn−1. . . . . . . . . . . . . . . . . . . . . . . 153 8.5 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

    9 Convergence of functions 169 9.1 a.e. convergence and uniformly a.e. convergence. . . . . . . . . . 169 9.2 Convergence in the mean. . . . . . . . . . . . . . . . . . . . . . . 170 9.3 Convergence in measure. . . . . . . . . . . . . . . . . . . . . . . . 173 9.4 Almost uniform convergence. . . . . . . . . . . . . . . . . . . . . 176 9.5 Relations between types of convergence. . . . . . . . . . . . . . . 178 9.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

    10 Signed measures and complex measures 187 10.1 Signed measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 10.2 The Hahn and Jordan decompositions, I. . . . . . . . . . . . . . . 189 10.3 The Hahn and Jordan decompositions, II. . . . . . . . . . . . . . 195 10.4 Complex measures. . . . . . . . . . . . . . . . . . . . . . . . . . . 198 10.5 Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

  • CONTENTS 5

    10.6 Lebesgue decomposition, Radon-Nikodym derivative. . . . . . . . 204 10.7 Differentiation of indefinite integrals in Rn. . . . . . . . . . . . . 212 10.8 Differentiation of Borel measures in Rn. . . . . . . . . . . . . . . 218 10.9 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

    11 The classical Banach spaces 225 11.1 Normed spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 11.2 The spaces Lp(X,Σ, µ). . . . . . . . . . . . . . . . . . . . . . . . 232 11.3 The dual of Lp(X,Σ, µ). . . . . . . . . . . . . . . . . . . . . . . . 243 11.4 The space M(X,Σ). . . . . . . . . . . . . . . . . . . . . . . . . . 250 11.5 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

  • 6 CONTENTS

  • 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 ∈ Σ, (ii) if A ∈ Σ, then Ac ∈ Σ, where Ac = X \A, and (iii) if An ∈ Σ for all n ∈ N, then ∪+∞n=1An ∈ 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 . (a) Take any A ∈ Σ and consider the sets A1 = A and An = Ac for all n ≥ 2. Then X = A ∪Ac = ∪+∞n=1An ∈ Σ and also ∅ = Xc ∈ Σ. (b) Let A1, . . . , AN ∈ Σ. Consider An = AN for all n > N and get that ∪Nn=1An = ∪+∞n=1An ∈ Σ. (c) Let An ∈ Σ for all n. Then ∩+∞n=1An = (∪+∞n=1Acn)c ∈ Σ. (d) Let A1, . . . , AN ∈ Σ. Using the result of (b), we get that ∩Nn=1An = (∪Nn=1Acn)c ∈ Σ. (e) Finally, let A,B ∈ Σ. Using the result of (d), we get that A\B = A∩Bc ∈ Σ.

    Here are some simple examples.

    Examples. 1. The collection {∅, X} is a σ-algebra of subsets of X . 2. If E ⊆ X and E is non-empty and different from X , then the collection {∅, E,Ec, X} is a σ-algebra of subsets of X .

    7

  • 8 CHAPTER 1. σ-ALGEBRAS

    3. P(X), the collection of all subsets of X , is a σ-algebra of subsets of X . 4. LetX be an uncountable set. The collection {A ⊆ X |A is countable or Ac is countable} 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 ∈ N. If all An’s are countable, then ∪+∞n=1An is also countable. If at least one of the Acn’s, say A

    c n0

    , is countable, then (∪+∞n=1An)c ⊆ Acn0 is also countable. In any case, ∪+∞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 sequence {An}Nn=1 or an infinite sequence {An} in Σ. Then there exists a finite sequence {Bn}Nn=1 or, respectively, an infinite sequence {Bn} in Σ with the properties: (i) Bn ⊆ An for all n = 1, . . . , N or, respectively, all n ∈ N, (ii) ∪Nn=1Bn = ∪Nn=1An or, respectively, ∪+∞n=1Bn = ∪+∞n=1An and (iii) the Bn’s are pairwise disjoint.

    Proof: Trivial, by taking B1 = A1 and Bk = Ak \ (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 {Σi}i∈I be any collection of σ-algebras of subsets ofX , indexed by an arbitrary non-empty set I of indices, and consider the intersection Σ = ∩i∈IΣi. (i) Since ∅ ∈ Σi for all i ∈ I, we get ∅ ∈ Σ and, hence, Σ is non-empty. (ii) Let A ∈ Σ. Then A ∈ Σi for all i ∈ I and, since all Σi’s are σ-algebras, Ac ∈ Σi for all i ∈ I. Therefore Ac ∈ Σ. (iii) Let An ∈ Σ for all n ∈ N. Then An ∈ Σi for all i ∈ I