MEASURE THEORY - University of papadim/measure_   NOTES ON MEASURE THEORY M. Papadimitrakis

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Transcript of MEASURE THEORY - University of papadim/measure_   NOTES ON MEASURE THEORY M. Papadimitrakis

  • NOTES ON

    MEASURE THEORY

    M. PapadimitrakisDepartment of MathematicsUniversity of Crete

    Autumn of 2004

  • 2

  • Contents

    1 -algebras 71.1 -algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Generated -algebras. . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Algebras and monotone classes. . . . . . . . . . . . . . . . . . . . 91.4 Restriction of a -algebra. . . . . . . . . . . . . . . . . . . . . . . 111.5 Borel -algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

    2 Measures 212.1 General measures. . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Point-mass distributions. . . . . . . . . . . . . . . . . . . . . . . . 232.3 Complete measures. . . . . . . . . . . . . . . . . . . . . . . . . . 252.4 Restriction of a measure. . . . . . . . . . . . . . . . . . . . . . . . 272.5 Uniqueness of measures. . . . . . . . . . . . . . . . . . . . . . . . 282.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

    3 Outer measures 333.1 Outer measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Construction of outer measures. . . . . . . . . . . . . . . . . . . . 353.3 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

    4 Lebesgue measure on Rn 394.1 Volume of intervals. . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Lebesgue measure in Rn. . . . . . . . . . . . . . . . . . . . . . . 414.3 Lebesgue measure and simple transformations. . . . . . . . . . . 454.4 Cantor set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.5 A non-Lebesgue set in R. . . . . . . . . . . . . . . . . . . . . . . 504.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

    5 Borel measures 575.1 Lebesgue-Stieltjes measures in R. . . . . . . . . . . . . . . . . . . 575.2 Borel measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.3 Metric outer measures. . . . . . . . . . . . . . . . . . . . . . . . . 675.4 Hausdorff measure. . . . . . . . . . . . . . . . . . . . . . . . . . . 68

    3

  • 5.5 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

    6 Measurable functions 736.1 Measurability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.2 Restriction and gluing. . . . . . . . . . . . . . . . . . . . . . . . . 736.3 Functions with arithmetical values. . . . . . . . . . . . . . . . . . 746.4 Composition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.5 Sums and products. . . . . . . . . . . . . . . . . . . . . . . . . . 766.6 Absolute value and signum. . . . . . . . . . . . . . . . . . . . . . 786.7 Maximum and minimum. . . . . . . . . . . . . . . . . . . . . . . 796.8 Truncation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.9 Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.10 Simple functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.11 The role of null sets. . . . . . . . . . . . . . . . . . . . . . . . . . 866.12 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

    7 Integrals 937.1 Integrals of non-negative simple functions. . . . . . . . . . . . . . 937.2 Integrals of non-negative functions. . . . . . . . . . . . . . . . . . 967.3 Integrals of complex valued functions. . . . . . . . . . . . . . . . 997.4 Integrals over subsets. . . . . . . . . . . . . . . . . . . . . . . . . 1067.5 Point-mass distributions. . . . . . . . . . . . . . . . . . . . . . . . 1097.6 Lebesgue integral. . . . . . . . . . . . . . . . . . . . . . . . . . . 1127.7 Lebesgue-Stieltjes integral. . . . . . . . . . . . . . . . . . . . . . . 1197.8 Reduction to integrals over R. . . . . . . . . . . . . . . . . . . . 1237.9 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

    8 Product measures 1378.1 Product -algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . 1378.2 Product measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1398.3 Multiple integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . 1468.4 Surface measure on Sn1. . . . . . . . . . . . . . . . . . . . . . . 1538.5 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

    9 Convergence of functions 1679.1 a.e. convergence and uniformly a.e. convergence. . . . . . . . . . 1679.2 Convergence in the mean. . . . . . . . . . . . . . . . . . . . . . . 1689.3 Convergence in measure. . . . . . . . . . . . . . . . . . . . . . . . 1719.4 Almost uniform convergence. . . . . . . . . . . . . . . . . . . . . 1749.5 Relations between types of convergence. . . . . . . . . . . . . . . 1769.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

    10 Signed measures and complex measures 18310.1 Signed measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18310.2 The Hahn and Jordan decompositions, I. . . . . . . . . . . . . . . 18510.3 The Hahn and Jordan decompositions, II. . . . . . . . . . . . . . 191

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  • 10.4 Complex measures. . . . . . . . . . . . . . . . . . . . . . . . . . . 19510.5 Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19710.6 Lebesgue decomposition, Radon-Nikodym derivative. . . . . . . . 20010.7 Differentiation of indefinite integrals in Rn. . . . . . . . . . . . . 20810.8 Differentiation of Borel measures in Rn. . . . . . . . . . . . . . . 21510.9 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

    11 The classical Banach spaces 22111.1 Normed spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22111.2 The spaces Lp(X,, ). . . . . . . . . . . . . . . . . . . . . . . . 22911.3 The dual of Lp(X,, ). . . . . . . . . . . . . . . . . . . . . . . . 24011.4 The space M(X,). . . . . . . . . . . . . . . . . . . . . . . . . . 24711.5 The space C0(X) and its dual. . . . . . . . . . . . . . . . . . . . 24911.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

    5

  • 6

  • Chapter 1

    -algebras

    1.1 -algebras.

    Definition 1.1 Let X be a non-empty set and a collection of subsets of X.We call a -algebra of subsets of X if it is non-empty, closed undercomplements and closed under countable unions. This means:(i) there exists at least one A X so that A ,(ii) if A , then Ac , where Ac = X \A, and(iii) if An for all n N, then +n=1An .

    The pair (X,) of a non-empty set X and a -algebra of subsets of X iscalled a measurable space.

    Proposition 1.1 Every -algebra of subsets of X contains at least the sets and X, it is closed under finite unions, under countable intersections, underfinite intersections and under set-theoretic differences.

    Proof: Let be any -algebra of subsets of X.(a) Take any A and consider the sets A1 = A and An = Ac for all n 2.Then X = A Ac = +n=1An and also = Xc .(b) Let A1, . . . , AN . Consider An = AN for all n > N and get thatNn=1An = +n=1An .(c) Let An for all n. Then +n=1An = (

    +n=1A

    cn)c .

    (d) Let A1, . . . , AN . Using the result of (b), we get that Nn=1An =(Nn=1Acn)c .(e) Finally, let A,B . Using the result of (d), we get that A\B = ABc .

    Here are some simple examples.

    Examples1. The collection {, X} is a -algebra of subsets of X.2. If E X is non-empty and different from X, then the collection {, E,Ec, X}is a -algebra of subsets of X.

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  • 3. P(X), the collection of all subsets of X, is a -algebra of subsets of X.4. Let X be uncountable. The {A X |A is countable or Ac is countable} isa -algebra of subsets of X. Firstly, is countable and, hence, the collection isnon-empty. If A is in the collection, then, considering cases, we see that Ac isalso in the collection. Finally, let An be in the collection for all n N. If allAns are countable, then +n=1An is also countable. If at least one of the Acns,say Acn0 , 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 be a -algebra of subsets of X and consider a finitesequence {An}Nn=1 or an infinite sequence {An} in . Then there exists a finitesequence {Bn}Nn=1 or, respectively, an infinite sequence {Bn} in with theproperties:(i) Bn An for all n = 1, . . . , N or, respectively, all n N.(ii) Nn=1Bn = Nn=1An or, respectively, +n=1Bn =

    +n=1An.

    (iii) the Bns are pairwise disjoint.

    Proof: Trivial, by taking B1 = A1 and Bk = Ak \ (A1 Ak1) for allk = 2, . . . , N or, respectively, all k = 2, 3, . . . .

    1.2 Generated -algebras.

    Proposition 1.3 The intersection of any -algebras of subsets of the same Xis a -algebra of subsets of X.

    Proof: Let {i}iI be any collection of -algebras of subsets of X, indexed by anarbitrary non-empty set I of indices, and consider the intersection = iIi.

    Since i for all i I, we get and, hence, is non-empty.Let A . Then A i for all i I and, since all is are -algebras,

    Ac i for all i I. Therefore Ac .Let An for all n N. Then An i for all i I and all n N

    and, since all is are -algebras, we get +n=1An i for all i I. Thus,+n=1An .

    Definition 1.2 Let X be a non-empty set and E be an arbitrary collection ofsubsets of X. The intersection of all -algebras of subsets of X which includeE is called the -algebra generated by E and it is denoted by (E). Namely

    (E) = { | is a -algebra of subsets of X and E } .

    Note that there is at least one -algebra of subsets of X which includes E andthis is P(X). Note also that the term -algebra used in the name of (E) isjustified by its definition and by Proposition 1.3.

    Proposition 1.4 Let E be any collection of subsets of the non-empt