Measure, Integration & Real Analysis
Measure, Integration& Real Analysis6 April 2021Sheldon AxlerxivContentsZ f g dµ Zp f dµ 1/p Zp0 g dµ 1/p0 Sheldon AxlerThis book is licensed under a Creative CommonsAttribution-NonCommercial 4.0 International License.The print version of this book is available from Springer.Measure, Integration & Real Analysis, by Sheldon Axler
Dedicated toPaul Halmos, Don Sarason, and Allen Shields,the three mathematicians who mosthelped me become a mathematician.Measure, Integration & Real Analysis, by Sheldon Axler
About the AuthorSheldon Axler was valedictorian of his high school in Miami, Florida. He received hisAB from Princeton University with highest honors, followed by a PhD in Mathematicsfrom the University of California at Berkeley.As a postdoctoral Moore Instructor at MIT, Axler received a university-wideteaching award. He was then an assistant professor, associate professor, and professorat Michigan State University, where he received the first J. Sutherland Frame TeachingAward and the Distinguished Faculty Award.Axler received the Lester R. Ford Award for expository writing from the Mathematical Association of America in 1996. In addition to publishing numerous researchpapers, he is the author of six mathematics textbooks, ranging from freshman tograduate level. His book Linear Algebra Done Right has been adopted as a textbookat over 300 universities and colleges.Axler has served as Editor-in-Chief of the Mathematical Intelligencer and Associate Editor of the American Mathematical Monthly. He has been a member ofthe Council of the American Mathematical Society and a member of the Board ofTrustees of the Mathematical Sciences Research Institute. He has also served on theeditorial board of Springer’s series Undergraduate Texts in Mathematics, GraduateTexts in Mathematics, Universitext, and Springer Monographs in Mathematics.He has been honored by appointments as a Fellow of the American MathematicalSociety and as a Senior Fellow of the California Council on Science and Technology.Axler joined San Francisco State University as Chair of the Mathematics Department in 1997. In 2002, he became Dean of the College of Science & Engineering atSan Francisco State University. After serving as Dean for thirteen years, he returnedto a regular faculty appointment as a professor in the Mathematics Department.Cover figure: Hölder’s Inequality, which is proved in Section 7A.viMeasure, Integration & Real Analysis, by Sheldon Axler
ContentsAbout the Author viPreface for Students xiiiPreface for Instructors xivAcknowledgments xviii1 Riemann Integration 11A Review: Riemann Integral2Exercises 1A 71B Riemann Integral Is Not Good Enough9Exercises 1B 122 Measures132A Outer Measure on R14Motivation and Definition of Outer Measure 14Good Properties of Outer Measure 15Outer Measure of Closed Bounded Interval 18Outer Measure is Not Additive 21Exercises 2A 232B Measurable Spaces and Functions25σ-Algebras 26Borel Subsets of R 28Inverse Images 29Measurable Functions 31Exercises 2B 382C Measures and Their Properties41Definition and Examples of Measures 41Properties of Measures 42Exercises 2C 45viiMeasure, Integration & Real Analysis, by Sheldon Axler
viiiContents2D Lebesgue Measure47Additivity of Outer Measure on Borel Sets 47Lebesgue Measurable Sets 52Cantor Set and Cantor Function 55Exercises 2D 60622E Convergence of Measurable FunctionsPointwise and Uniform Convergence 62Egorov’s Theorem 63Approximation by Simple Functions 65Luzin’s Theorem 66Lebesgue Measurable Functions 69Exercises 2E 713 Integration73743A Integration with Respect to a MeasureIntegration of Nonnegative Functions 74Monotone Convergence Theorem 77Integration of Real-Valued Functions 81Exercises 3A 843B Limits of Integrals & Integrals of Limits88Bounded Convergence Theorem 88Sets of Measure 0 in Integration Theorems 89Dominated Convergence Theorem 90Riemann Integrals and Lebesgue Integrals 93Approximation by Nice Functions 95Exercises 3B 994 Differentiation1014A Hardy–Littlewood Maximal Function102Markov’s Inequality 102Vitali Covering Lemma 103Hardy–Littlewood Maximal Inequality 104Exercises 4A 1064B Derivatives of Integrals108Lebesgue Differentiation Theorem 108Derivatives 110Density 112Exercises 4B 115Measure, Integration & Real Analysis, by Sheldon Axler
Contents5 Product Measures 1165A Products of Measure Spaces117Products of σ-Algebras 117Monotone Class Theorem 120Products of Measures 123Exercises 5A 1285B Iterated Integrals129Tonelli’s Theorem 129Fubini’s Theorem 131Area Under Graph 133Exercises 5B 1355C Lebesgue Integration on Rn136Borel Subsets of Rn 136Lebesgue Measure on Rn 139Volume of Unit Ball in Rn 140Equality of Mixed Partial Derivatives Via Fubini’s Theorem 142Exercises 5C 1446 Banach Spaces1466A Metric Spaces 147Open Sets, Closed Sets, and Continuity 147Cauchy Sequences and Completeness 151Exercises 6A 1536B Vector Spaces155Integration of Complex-Valued Functions 155Vector Spaces and Subspaces 159Exercises 6B 1626C Normed Vector Spaces163Norms and Complete Norms 163Bounded Linear Maps 167Exercises 6C 1706D Linear Functionals172Bounded Linear Functionals 172Discontinuous Linear Functionals 174Hahn–Banach Theorem 177Exercises 6D 181Measure, Integration & Real Analysis, by Sheldon Axlerix
xContents6E Consequences of Baire’s Theorem184Baire’s Theorem 184Open Mapping Theorem and Inverse Mapping Theorem 186Closed Graph Theorem 188Principle of Uniform Boundedness 189Exercises 6E 1907 L p Spaces1937A L p (µ)1947B L p (µ)202Hölder’s Inequality 194Minkowski’s Inequality 198Exercises 7A 199Definition of L p (µ) 202L p (µ) Is a Banach Space 204Duality 206Exercises 7B 2088 Hilbert Spaces 2118A Inner Product Spaces212Inner Products 212Cauchy–Schwarz Inequality and Triangle Inequality 214Exercises 8A 2218B Orthogonality224Orthogonal Projections 224Orthogonal Complements 229Riesz Representation Theorem 233Exercises 8B 2348C Orthonormal Bases237Bessel’s Inequality 237Parseval’s Identity 243Gram–Schmidt Process and Existence of Orthonormal Bases 245Riesz Representation Theorem, Revisited 250Exercises 8C 251Measure, Integration & Real Analysis, by Sheldon Axler
Contents9 Real and Complex Measures 2559A Total Variation256Properties of Real and Complex Measures 256Total Variation Measure 259The Banach Space of Measures 262Exercises 9A 2659B Decomposition Theorems267Hahn Decomposition Theorem 267Jordan Decomposition Theorem 268Lebesgue Decomposition Theorem 270Radon–Nikodym Theorem 272Dual Space of L p (µ) 275Exercises 9B 27810 Linear Maps on Hilbert Spaces28010A Adjoints and Invertibility281Adjoints of Linear Maps on Hilbert Spaces 281Null Spaces and Ranges in Terms of Adjoints 285Invertibility of Operators 286Exercises 10A 29210B Spectrum 294Spectrum of an Operator 294Self-adjoint Operators 299Normal Operators 302Isometries and Unitary Operators 305Exercises 10B 30910C Compact Operators312The Ideal of Compact Operators 312Spectrum of Compact Operator and Fredholm Alternative 316Exercises 10C 32310D Spectral Theorem for Compact Operators326Orthonormal Bases Consisting of Eigenvectors 326Singular Value Decomposition 332Exercises 10D 336Measure, Integration & Real Analysis, by Sheldon Axlerxi
xiiContents11 Fourier Analysis 33911A Fourier Series and Poisson Integral340Fourier Coefficients and Riemann–Lebesgue Lemma 340Poisson Kernel 344Solution to Dirichlet Problem on Disk 348Fourier Series of Smooth Functions 350Exercises 11A 35211B Fourier Series and L p of Unit Circle355L2Orthonormal Basis forof Unit Circle 355Convolution on Unit Circle 357Exercises 11B 36111C Fourier Transform363Fourier Transform on L1 (R) 363Convolution on R 368Poisson Kernel on Upper Half-Plane 370Fourier Inversion Formula 374Extending Fourier Transform to L2 (R)Exercises 11C 37737512 Probability Measures 380Probability Spaces 381Independent Events and Independent Random Variables 383Variance and Standard Deviation 388Conditional Probability and Bayes’ Theorem 390Distribution and Density Functions of Random Variables 392Weak Law of Large Numbers 396Exercises 12 398Photo Credits400Bibliography402Notation IndexIndex403406Colophon: Notes on Typesetting411Measure, Integration & Real Analysis, by Sheldon Axler
Preface for StudentsYou are about to immerse yourself in serious mathematics, with an emphasis onattaining a deep understanding of the definitions, theorems, and proofs related tomeasure, integration, and real analysis. This book aims to guide you to the wondersof this subject.You cannot read mathematics the way you read a novel. If you zip through a pagein less than an hour, you are probably going too fast. When you encounter the phraseas you should verify, you should indeed do the verification, which will usually requiresome writing on your part. When steps are left out, you need to supply the missingpieces. You should ponder and internalize each definition. For each theorem, youshould seek examples to show why each hypothesis is necessary.Working on the exercises should be your main mode of learning after you haveread a section. Discussions and joint work with other students may be especiallyeffective. Active learning promotes long-term understanding much better than passivelearning. Thus you will benefit considerably from struggling with an exercise andeventually coming up with a solution, perhaps working with other students. Findingand reading a solution on the internet will likely lead to little learning.As a visual aid, throughout this book definitions are in yellow boxes and theoremsare in blue boxes, in both print and electronic versions. Each theorem has an informaldescriptive name. The electronic version of this manuscript has links in blue.Please check the website below (or the Springer website) for additional informationabout the book. These websites link to the electronic version of this book, which isfree to the world because this book has been published under Springer’s Open Accessprogram. Your suggestions for improvements and corrections for a future edition aremost welcome (send to the email address below).The prerequisite for using this book includes a good understanding of elementaryundergraduate real analysis. You can download from the website below or from theSpringer website the document titled Supplement for Measure, Integration & RealAnalysis. That supplement can serve as a review of the elementary undergraduate realanalysis used in this book.Best wishes for success and enjoyment in learning measure, integration, and realanalysis!Sheldon AxlerMathematics DepartmentSan Francisco State UniversitySan Francisco, CA 94132, USAwebsite: http://measure.axler.nete-mail: measure@axler.netTwitter: @AxlerLinearxiiiMeasure, Integration & Real Analysis, by Sheldon Axler
Preface for InstructorsYou are about to teach a course, or possibly a two-semester sequence of courses, onmeasure, integration, and real analysis. In this textbook, I have tried to use a gentleapproach to serious mathematics, with an emphasis on students attaining a deepunderstanding. Thus new material often appears in a comfortable context insteadof the most general setting. For example, the Fourier transform in Chapter 11 isintroduced in the setting of R rather than Rn so that students can focus on the mainideas without the clutter of the extra bookkeeping needed for working in Rn .The basic prerequisite for your students to use this textbook is a good understanding of elementary undergraduate real analysis. Your students can download from thebook’s website (http://measure.axler.net) or from the Springer website the documenttitled Supplement for Measure, Integration & Real Analysis. That supplement canserve as a review of the elementary undergraduate real analysis used in this book.As a visual aid, throughout this book definitions are in yellow boxes and theoremsare in blue boxes, in both print and electronic versions. Each theorem has an informaldescriptive name. The electronic version of this manuscript has links in blue.Mathematics can be learned only by doing. Fortunately, real analysis has manygood homework exercises. When teaching this course, during each class I usuallyassign as homework several of the exercises, due the next class. I grade only oneexercise per homework set, but the students do not know ahead of time which one. Iencourage my students to work together on the homework or to come to me for help.However, I tell them that getting solutions from the internet is not allowed and wouldbe counterproductive for their learning goals.If you go at a leisurely pace, then covering Chapters 1–5 in the first semester maybe a good goal. If you go a bit faster, then covering Chapters 1–6 in the first semestermay be more appropriate. For a second-semester course, covering some subset ofChapters 6 through 12 should produce a good course. Most instructors will not havetime to cover all those chapters in a second semester; thus some choices need tobe made. The following chapter-by-chapter summary of the highlights of the bookshould help you decide what to cover and in what order: Chapter 1: This short chapter begins with a brief review of Riemann integration.Then a discussion of the deficiencies of the Riemann integral helps motivate theneed for a better theory of integration. Chapter 2: This chapter begins by defining outer measure on R as a naturalextension of the length function on intervals. After verifying some nice propertiesof outer measure, we see that it is not additive. This observation leads to restrictingour attention to the σ-algebra of Borel sets, defined as the smallest σ-algebra on Rcontaining all the open sets. This path leads us to measures.xivMeasure, Integration & Real Analysis, by Sheldon Axler
Preface for InstructorsxvAfter dealing with the properties of general measures, we come back to the settingof R, showing that outer measure restricted to the σ-algebra of Borel sets iscountably additive and thus is a measure. Then a subset of R is defined to beLebesgue measurable if it differs from a Borel set by a set of outer measure 0. Thisdefinition makes Lebesgue measurable sets seem more natural to students than theother competing equivalent definitions. The Cantor set and the Cantor functionthen stretch students’ intuition.Egorov’s Theorem, which states that pointwise convergence of a sequence ofmeasurable functions is close to uniform convergence, has multiple applications inlater chapters. Luzin’s Theorem, back in the context of R, sounds spectacular buthas no other uses in this book and thus can be skipped if you are pressed for time. Chapter 3: Integration with respect to a measure is defined in this chapter in anatural fashion first for nonnegative measurable functions, and then for real-valuedmeasurable functions. The Monotone Convergence Theorem and the DominatedConvergence Theorem are the big results in this chapter that allow us to interchangeintegrals and limits under appropriate conditions. Chapter 4: The highlight of this chapter is the Lebesgue Differentiation Theorem,which allows us to differentiate an integral. The main tool used to prove thisresult cleanly is the Hardy–Littlewood maximal inequality, which is interestingand important in its own right. This chapter also includes the Lebesgue DensityTheorem, showing that a Lebesgue measurable subset of R has density 1 at almostevery number in the set and density 0 at almost every number not in the set. Chapter 5: This chapter deals with product measures. The most important resultshere are Tonelli’s Theorem and Fubini’s Theorem, which allow us to evaluateintegrals with respect to product measures as iterated integrals and allow us tochange the order of integration under appropriate conditions. As an application ofproduct measures, we get Lebesgue measure on Rn from Lebesgue measure on R.To give students practice with using these concepts, this chapter finds a formula forthe volume of the unit ball in Rn . The chapter closes by using Fubini’s Theorem togive a simple proof that a mixed partial derivative with sufficient continuity doesnot depend upon the order of differentiation. Chapter 6: After a quick review of metric spaces and vector spaces, this chapterdefines normed vector spaces. The big result here is the Hahn–Banach Theoremabout extending bounded linear functionals from a subspace to the whole space.Then this chapter introduces Banach spaces. We see that completeness playsa major role in the key theorems: Open Mapping Theorem, Inverse MappingTheorem, Closed Graph Theorem, and Principle of Uniform Boundedness. Chapter 7: This chapter introduces the important class of Banach spaces L p (µ),where 1 p and µ is a measure, giving students additional opportunities touse results from earlier chapters about measure and integration theory. The crucialresults called Hölder’s inequality and Minkowski’s inequality are key tools here.0This chapter also shows that the dual of p is p for 1 p .Chapters 1 through 7 should be covered in order, before any of the later chapters.After Chapter 7, you can cover Chapter 8 or Chapter 12.Measure, Integration & Real Analysis, by Sheldon Axler
xviPreface for Instructors Chapter 8: This chapter focuses on Hilbert spaces, which play a central role inmodern mathematics. After proving the Cauchy–Schwarz inequality and the RieszRepresentation Theorem that describes the bounded linear functionals on a Hilbertspace, this chapter deals with orthonormal bases. Key results here include Bessel’sinequality, Parseval’s identity, and the Gram–Schmidt process. Chapter 9: Only positive measures have been discussed in the book up until thischapter. In this chapter, real and complex measures get consideration. These concepts lead to the Banach space of measures, with total variation as the norm. Keyresults that help describe real and complex measures are the Hahn DecompositionTheorem, the Jordan Decomposition Theorem, and the Lebesgue DecompositionTheorem. The Radon–Nikodym Theorem is proved using von Neumann’s slickHilbert space trick. Then the Radon–Nikodym Theorem is used to prove that the0dual of L p (µ) can be identified with L p (µ) for 1 p and µ a (positive)measure, completing a project that started in Chapter 7.The material in Chapter 9 is not used later in the book. Thus this chapter can beskipped or covered after one of the later chapters. Chapter 10: This chapter begins by discussing the adjoint of a bounded linearmap between Hilbert spaces. Then the rest of the chapter presents key resultsabout bounded linear operators from a Hilbert space to itself. The proof that eachbounded operator on a complex nonzero Hilbert space has a nonempty spectrumrequires a tiny bit of knowledge about analytic functions. Properties of specialclasses of operators (self-adjoint operators, normal operators, isometries, andunitary operators) are described.Then this chapter delves deeper into compact operators, proving the FredholmAlternative. The chapter concludes with two major results: the Spectral Theoremfor compact operators and the popular Singular Value Decomposition for compactoperators. Throughout this chapter, the Volterra operator is used as an example toillustrate the main results.Some instructors may prefer to cover Chapter 10 immediately after Chapter 8,because both chapters live in the context of Hilbert space. I chose the current orderto give students a breather between the two Hilbert space chapters, thinking thatbeing away from Hilbert space for a little while and then coming back to it mightstrengthen students’ understanding and provide some variety. However, coveringthe two Hilbert space chapters consecutively would also work fine. Chapter 11: Fourier analysis is a huge subject with a two-hundred year history.This chapter gives a gentle but modern introduction to Fourier series and theFourier transform.This chapter first develops results in the context of Fourier series, but then comesback later and develops parallel concepts in the cont
Exercises 6B 162 6CNormed Vector Spaces 163 Norms and Complete Norms 163 Bounded Linear Maps 167 Exercises 6C 170 6DLinear Functionals 172 Bounded Linear Functionals 172 Discontinuous Linear Functionals 174 Hahn–Banach Theorem 177 Exercises 6D 181 Measure, Integration & Real Analysis, by Sheldon Axler
5 Integration of Bounded Functions on Sets of Finite Measure 53 6 Integration of Nonnegative Functions 63 7 Integration of Measurable Functions 75 8 Signed Measures and Radon-Nikodym Theorem 97 9 Difierentiation and Integration 109 10 Lp Spaces 121 11 Integration on Product Measure Space 141
List of core vocabulary Cambridge Pre-U Mandarin Chinese 9778: List of core vocabulary. 5 3 Measure words 把 bǎ measure word, hold 杯 bēi a cup of / cup 本 běn measure word 遍 biàn number of times 层 céng layer; storey 次 cì number of times 段 duàn paragraph, section 队 duì team 封 fēng measure word 个 gè measure word 壶 hú measure word 件 jiàn measure word
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measure, integration, and real analysis. This book aims to guide you to the wonders of this subject. You cannot read mathematics the way you read a novel. If you zip through a page in less than an hour, you are probably going too fast. When you encounter the phrase as you should verify, you
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