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Measure and Integral Jaroslav Lukeš Jan Malý matfyzpress PRAGUE 2005 All rights reserved, no part of this publication may be reproduced or transmitted in any form or by any means, electronic, mechanical, photocopying or otherwise, without the prior written permission of the publisher. © Jaroslav Lukeš, Jan Malý, 2005 © MATFYZPRESS by publishing house of the Faculty of Mathematics and Physics Charles University in Prague, 2005 ISBN 80-86732-68-1 ISBN 80-85863-06-5 (First edition) Motto: Everybody writes and nobody reads L.Feje´r Preface This text is based on lectures in measure and integration theory given by the authors during the past decade at Charles University, and on preliminary lecture notes published in Czech. It is impossible to thank individually all colleagues and students who assisted in the preparation of this manuscript, but we will just mention Michal Kubeˇcek who helped with the translation and TEX processing. Theauthors wish toexpresstheir deep gratitude toProfessor StylianosNegre- pontis, who was the chief coordinator of TEMPUS project JEP–1980. Without support from him and the Tempus programme the manuscript would never have appeared. The preparation of this manuscript was partially supported by the grant No. 201/93/2174 of the Czech Grant Agency and by the grant No. 354 of the Charles University. Prague, 1994 Jaroslav Lukeˇs and Jan Maly´ Preface to the second edition We have carried out only minor corrections. We wish to thank all who con- tributed by suggestions and comments. Prague, 2005 Jaroslav Lukeˇs and Jan Maly´ Contents List of Basic Notations and Frequently Used Symbols.............................. 1 A. Measures and Measurable Functions.............................................. 2 1. TheLebesgueMeasure 2. AbstractMeasures 3. MeasurableFunctions 4. ConstructionofMeasuresfromOuterMeasures 5. ClassesofSetsandSetFunctions 6. SignedandComplexMeasures B. The Abstract Lebesgue Integral.................................................. 27 7. IntegrationonR 8. TheAbstractLebesgueIntegral 9. IntegralsDependingonaParameter 10. TheLp Spaces 11. ProductMeasuresandtheFubiniTheorem 12. SequencesofMeasurableFunctions 13. TheRadon-Nikody´mTheoremandtheLebesgueDecomposition C. Radon Integral and Measure .................................................... 57 14. RadonIntegral 15. RadonMeasures 16. RieszRepresentationTheorem 17. SequencesofMeasures 18. Luzin’sTheorem 19. MeasuresonTopologicalGroups D. Integration on R .................................................................. 82 20. IntegralandDifferentiation 21. FunctionsofFiniteVariationandAbsolutelyContinuousFunctions 22. TheoremsonAlmostEverywhereDifferentiation 23. IndefiniteLebesgueIntegralandAbsoluteContinuity 24. RadonMeasuresonRandDistributionFunctions 25. Henstock–KurzweilIntegral E. Integration on Rn ................................................................ 103 26. LebesgueMeasureandIntegralonRn 27. CoveringTheorems 28. DifferentiationofMeasures 29. LebesgueDensityTheoremandApproximatelyContinuousFunctions 30. LipschitzFunctions 31. ApproximationTheorems 32. Distributions 33. FourierTransform F. Change of Variable and k-dimensional Measures ............................. 141 34. ChangeofVariableTheorem 35. TheDegreeofaMapping 36. HausdorffMeasures G. Surface and Curve Integrals..................................................... 163 37. IntegralCalculusinVectorAnalysis 38. IntegrationofDifferentialForms 39. IntegrationonManifolds H. Vector Integration ............................................................... 191 40. MeasurableFunctions 41. VectorMeasures 42. TheBochnerIntegral 43. TheDunfordandPettisIntegrals Appendix on Topology ............................................................. 203 References ........................................................................... 206 A Short Guide to the Notation..................................................... 217 Subject Index ....................................................................... 219 1 List of Basic Notations and Frequently Used Symbols In this manuscript we use the standard notation. In all what follows, N, Z, Q, R, C will denote the sets of all natural, integer, rational, real and complex numbers, respectively. Extended realnumbersetRconsists of Rtogether with twosymbols −∞and +∞equippedwiththeusualalgebraicstructureandtopology. Remindonlythat 0·±∞ and ±∞·0 are taken as 0. If X is a set, P(X) denotes the collection of all its subsets. Rn stands for the Euclidean n-dimensional space under the usual Euclidean norm |·| and the metric |x−y|, where x=[x1,...,xn]. Remember that when multiplied by a matrix (from the left), the vector x = [x1,...,xn] behaves like a “vertical vector”, i.e. like a matrix with one column ⎛ ⎞ x1 ⎝ ... ⎠. x n The horizontal notation is preferred for estetical and typographical reasons. The standard (or canonical) basis of the space Rn is denoted by {e1,...,en}, the vector e =[0,...,0,1,0,...,0] with 1 at the ith place. The inner product in i Rn is denoted by x·y. ByU(x,r)wedenotetheopenballinametricspace(P,ρ)ofradiusrroundthe x. The closed ball is denoted by B(x,r). Thus U(x,r) = {y ∈ P: ρ(x,y) < r}, B(x,r) = {y ∈ P: ρ(x,y) ≤ r}. For the diameter of a set we use the symbol ”diam” and for the distance of a pair of sets the symbol ”dist”. If nothing else is specified, a function on a set X is a mapping of X into R. If we want to emphasise that a function does not attain the values −∞ and +∞, we call it a real function. Insteadofthenotation{x∈X: f(x)>a}weoftenusetheabbreviatedversion {f >a}. Thesymbolc denotestheindicator function ofasetA⊂X, i.e.thefunction A (cid:6) 1 for x∈A, c (x)= A 0 for x∈X \A. We use f ⇒f to denote the uniform convergence of a sequence of functions. j 2 1. The Lebesgue Measure A. Measures and Measurable Functions 1. The Lebesgue Measure In the history, people wereengagedin the problem of measuring lenghts, areas and volumes. In mathematical formulation the task was, for a given set A, to determine its size (”measure”) λA. It was required that the volume of a cube or the area of a rectangle or a circle should agree with the well-known formulae. It was also clear by intuition that this measure should be positive and additive, i.e. it should satisfy the equality (cid:7) (cid:8) λ A = λA j j provided {A } is a finite disjoint collection of sets. For a succesful development j of the theory a further condition was imposed: The above equality was claimed to hold even for countable disjoint collections of sets. Moreover, the effort was paid to assign a measure to as many sets as possible. Now,wearegoingtoshowhowtoproceedontherealline. Thesameapproach will be used later in the Euclidean space Rn where the proofs will be given. 1.1. Outer Lebesgue Measure. For an arbitrary set A⊂R, define (cid:8)∞ (cid:7)∞ λ∗A:=inf{ (b −a ): (a ,b )⊃A}. i i i i i=1 i=1 The value λ∗A (which can also be +∞) is called the outer Lebesgue measure of a set A. 1.2. Properties of the Outer Lebesgue Measure. Onecanseeimmediately that λ∗A≤λ∗B if A⊂B and that the measure of a singleton is 0, and without mucheffortitbecomesclearthatλ∗I isthelengthofI incaseofI intervalofany type(seeExercise1.6). ThenitisrelativelyeasytoprovethattheouterLebesgue measure is translation invariant: If A ⊂ R and x ∈ R, then λ∗A = λ∗(x+A). Another important property is the σ-subadditivity: (cid:7)∞ (cid:8)∞ λ∗( A )≤ λ∗A . j j j=1 j=1 Inmathematicalterminology,theprefixσusuallyrelatestocountableunionsandδtocountable intersections. The question of whether λ∗ is an additive set function has a negative answer: There are disjoint sets A, B with λ∗(A∪B)<λ∗A+λ∗B (cf. 1.8), and we need to find a family of sets (as large as possible) on which the measureλ∗ isadditive. ThistaskwillbesolvedlaterinChapter4inamuchmore general case. Now we just briefly indicate one of its possible solutions in case of the Lebesgue measure. A. Measures and Measurable Functions 3 1.3. Lebesgue Measurable Sets. Let A be a subset of a bounded interval I. Defining the “inner measure” λ∗A=λ∗I−λ∗(I−A), it is natural to investigate the collection of sets for which λ∗A = λ∗A (cf. Exercise 1.7). This leads to the following definition. WesaythatasetA⊂Ris(Lebesgue)measurableifλ∗I =λ∗(A∩I)+λ∗(I\A) for every bounded interval I ⊂ R. The collection of all measurable sets on R will be denoted by M. Not every set is measurable as will be seen in 1.8. The set function M (cid:9)→ λ∗M, M ∈ M is denoted by λ and called the Lebesgue measure. Thus, on measurable sets, the set functions λ∗ and λ coincide but for nonmeasurable ones only λ∗ is defined. Another important property of the measure λ is contained in the following theorem which is now presented without proof. 1(cid:9).4. Theo(cid:10)rem. (a) If M1, M2,... are elements of M, then also M1 \M2, M and M are elements of M. If, in addition, the sets M are pairwise n n n disjoint, then (cid:11)(cid:7) (cid:12) (cid:8) λ M = λM . n n n n (b) Intervals of any type are in M. 1.5. Remark. The ingenuity of Lebesgue’s approach to the measure consists in considering thecountable coversofasetAwithintervals. Ifinthedefinitionofλ∗Aweconsideronlyfinite covers,wegetthenotionofso-called Jordan-Peano content. Inmodernanalysisthisnotionis farfrombeingasimportantastheLebesguemeasure. 1.6. Exercise. IfI⊂Risaninterval(ofanytype),showthatλ∗I isitslength. Hint. It is sufficient to consider the case I = [a,b]. Clearly λ∗[a,b] ≤ b−a (since [a,b] ⊂ ∞S (a−ε,b+ε)). Suppose (ai,bi)⊃[a,b]. Acompactnessargumentyieldstheexistenceofan i=1 Sn indexnsatisfying (ai,bi)⊃[a,b]. Usinginduction(withrespectton)itcanbeshownthat i=1 Pn b−a≤ (bi−ai). i=1 1.7. Exercise. ForeveryboundedsetA⊂R,define λ∗A:=λI−λ∗(I\A) whereI isaboundedintervalcontainingA. Showthat: (a)thevalueofλ∗AdoesnotdependonthechoiceofI; (b)aboundedsetA⊂Rismeasurableifandonlyifλ∗A=λ∗A; (c) a set M ⊂R is measurable if and only if its intersection with each bounded interval is measurable. In the next part of this chapter we introduce some significant sets on the real line. 1.8. A Nonmeasurable Set. Now we prove the existence of a nonmeasurable subset of R andconsequentlyprovethattheouterLebesguemeasurecannotbeadditive. Set x∼y if x−y is a rational number. It is easy to see that ∼ is an equivalence relation onR. ThereforeRsplitsintoanuncountablecollection ofpairwisedisjointclasses. AsetV belongstothiscollection ifandonlyifV =x+Qforsomex∈R. Bytheaxiomofchoice, 4 1. The Lebesgue Measure thereexistsasetE⊂(0,1)thatsharesexactlyonepointwitheachsetV ∈ . Weshowthat E isnotinM. Let{qn}beasequencecontainingallrationalnumbersfromtheinterval(−1,+1). Itisnot verydifficulttoshowthatthesetsEn:=qn+E arepairwisedisjointandthat [ (0,1)⊂ En⊂(−1,2). n S P Assuming that E ∈ M, then also En ∈ Mand Theorem 1.4 gives λ En = λEn. Distin- n n guishingtwocasesλE=0andλE>0weeasilyobtainthecontradiction. 1.9. Remarks. 1. The proof of the existence of a nonmeasurable set is not a constructive one (it uses the axiom of choice for an uncountable collection of sets). We return to the topic ofnonmeasurablesetsinNotes1.22. 2. By a simple argument, an even stronger proposition can be proved: Any measurable set M ⊂SR of a positive measure contains a nonmeasurable subset. It is sufficient to realize that M = M∩(E+q) where E is the nonmeasurable set from 1.8 and that any measurable q∈Q subsetofE isofzero(Lebesgue)measure. 3. VanVleck[1908]“constructed”asetE⊂[0,1]forwhichλ∗E=1andλ∗E=0. 1.10. Exercise. ShowthateverycountablesetS isofmeasurezero. ∞S Hint. Considercovers (rj−ε2−j,rj+ε2−j)where{rj}isasequenceofallelementsofthe j=1 setS. TheassertionalsofollowsfromTheorem1.4ifyourealizethatsingletonshavemeasure zero. 1.11. Examples of Sets of Measure Zero. (a) The set Q of all rational numbers is countable,thusbyExercise1.10ithasLebesguemeasurezero. (b) It can be seen from the hint to the exercise that for every k ∈ N there is an open set ∞T G such that Q⊂G and λ∗G ≤1/k. The set G has also Lebesgue measure zero, it is k k k k k=1 denseanduncountable(evenresidual). 1.12. Cantor Ternary Set. Consider the sequence { n} of finite collections of intervals definedinthefollowingway: 0={[0,1]}, 1={[0,13],[23,1]}. Ineachstepweconstruct n from n−1 asthecollectionofallclosedintervalswhicharetheleftorrightthirdofaninterval fromthecollection n−1(themiddlethirdsareomitted). Then nisacollectionof2ndisjoint closedintervals,eachofthemoflengtTh3−n. LetKndenotetheunionofthecollection n. The Cantor ternary set1 C isdefinedas Kn. ItisnotdifficulttoverifythatC consistsprecisely n P∞ of points of the form ai3−i where each ai is 0 or 2. Roughly speaking, in the Cantor set i=1 thereareexactlythosepointsoftheinterval[0,1]whoseternaryexpansionsdonotcontainthe digit1. TheCantorsethasthefollowingproperties: (a)C isacompactsetwithoutisolatedpoints; (b)C isanowheredense(andtotallydisconnected)set; (c)C isanuncountableset; (d)theLebesguemeasureofC iszero. 1.13 Discontinua of a Positive Measure. IfweconstructasetD⊂[0,1]liketheCantor setexceptthatwealwaysomitintervalsoflengthε3−n whereε∈(0,1)(notethattheircentres arenotthesameasthoseintheconstructionoftheCantorset),wegetaclosednowheredense set, for which λD = 1−ε. Sets having this property are called the discontinua of a positive measure. Another construction: If G is an open subset of the interval (0,1), containing all rationalpointsofthisintervalandλG=ε<1then[0,1]\Gisadiscontinuumofmeasure1−ε. 1sometimesalsocalledtheCantor discontinuum