Measure, Integration & Real Analysis

1 Measure, Integration & Real Analysis1 January 2022 Sheldon AxlerxivContents |f g|d ( |f|pd )1/p( |g|p d )1/p Sheldon AxlerThis book is licensed under a Creative CommonsAttribution-NonCommercial International print version of this book is available from , Integration & Real Analysis , by Sheldon AxlerDedicated toPaul Halmos, Don Sarason, and Allen Shields,the three mathematicians who mosthelped me become a , Integration & Real Analysis , by Sheldon AxlerAbout 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 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 received the Lester R.

2 Ford Award for expository writing from the Mathe-matical 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 bookLinear Algebra Done Righthas been adopted as a textbookat over 300 universities and has served as Editor-in-Chief of theMathematical Intelligencerand As-sociate Editor of theAmerican 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 has been honored by appointments as a Fellow of the American MathematicalSociety and as a Senior Fellow of the California Council on Science and joined San Francisco State University as Chair of the Mathematics Depart-ment in 1997.

3 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 figure: H lder s Inequality, which is proved in , Integration & Real Analysis , by Sheldon AxlerviContentsAbout the AuthorviPreface for StudentsxiiiPreface for InstructorsxivAcknowledgmentsxviii1 Riemann Integration11A Review: Riemann Integral2 Exercises 1A71B Riemann Integral Is Not Good Enough9 Exercises 1B122 Measures132A Outer Measure onR14 Motivation and Definition of Outer Measure14 Good Properties of Outer Measure15 Outer Measure of Closed Bounded Interval18 Outer Measure is Not Additive21 Exercises 2A232B Measurable Spaces and Functions25 -Algebras26 Borel Subsets ofR28 Inverse Images29 Measurable Functions31 Exercises 2B382C Measures and Their Properties41 Definition and Examples of Measures41 Properties of Measures42 Exercises 2C45 Measure, Integration & Real Analysis .

4 By Sheldon AxlerviiviiiContents2D Lebesgue Measure47 Additivity of Outer Measure on Borel Sets47 Lebesgue Measurable Sets52 Cantor Set and Cantor Function55 Exercises 2D602E Convergence of Measurable Functions62 Pointwise and Uniform Convergence62 Egorov s Theorem63 Approximation by Simple Functions65 Luzin s Theorem66 Lebesgue Measurable Functions69 Exercises 2E713 Integration733A Integration with Respect to a Measure74 Integration of Nonnegative Functions74 monotone Convergence Theorem77 Integration of Real-Valued Functions81 Exercises 3A843B Limits of Integrals & Integrals of Limits88 Bounded Convergence Theorem88 Sets of Measure0in Integration Theorems89 Dominated Convergence Theorem90 Riemann Integrals and Lebesgue Integrals93 Approximation by Nice Functions95 Exercises 3B994 Differentiation1014A Hardy Littlewood Maximal Function102 Markov s Inequality102 Vitali Covering Lemma103 Hardy Littlewood Maximal Inequality104 Exercises 4A1064B Derivatives of Integrals108 Lebesgue Differentiation Theorem108 Derivatives110 Density112 Exercises 4B115 Measure, Integration & Real Analysis , by Sheldon AxlerContentsix5 Product Measures1165A Products of Measure Spaces117 Products of -Algebras117 monotone Class Theorem120 Products of Measures123 Exercises 5A1285B Iterated Integrals129 Tonelli s Theorem129 Fubini s Theorem131 Area Under Graph133 Exercises 5B1355C Lebesgue Integration onRn136 Borel Subsets ofRn136 Lebesgue Measure onRn139 Volume of Unit Ball inRn140 Equality of Mixed Partial Derivatives Via Fubini s Theorem142 Exercises 5C1446 Banach Spaces1466A Metric Spaces147 Open Sets, Closed Sets.

5 And Continuity147 Cauchy Sequences and Completeness151 Exercises 6A1536B Vector Spaces155 Integration of Complex-Valued Functions155 Vector Spaces and Subspaces159 Exercises 6B1626C Normed Vector Spaces163 Norms and Complete Norms163 Bounded Linear Maps167 Exercises 6C1706D Linear Functionals172 Bounded Linear Functionals172 Discontinuous Linear Functionals174 Hahn Banach Theorem177 Exercises 6D181 Measure, Integration & Real Analysis , by Sheldon AxlerxContents6E Consequences of Baire s Theorem184 Baire s Theorem184 Open Mapping Theorem and Inverse Mapping Theorem186 Closed Graph Theorem188 Principle of Uniform Boundedness189 Exercises 6E1907 LpSpaces1937 ALp( )194H lder s Inequality194 Minkowski s Inequality198 Exercises 7A1997 BLp( )202 Definition ofLp( )202Lp( )Is a Banach Space204 Duality206 Exercises 7B2088 Hilbert Spaces2118A Inner Product Spaces212 Inner Products212 Cauchy Schwarz Inequality and Triangle Inequality214 Exercises 8A2218B Orthogonality224 Orthogonal Projections224 Orthogonal Complements229 Riesz Representation Theorem233 Exercises 8B2348C Orthonormal Bases237 Bessel s Inequality237 Parseval s Identity243 Gram Schmidt Process and Existence of Orthonormal Bases245 Riesz Representation Theorem, Revisited250 Exercises 8C251 Measure, Integration & Real Analysis , by Sheldon AxlerContentsxi9 Real and Complex Measures2559A Total Variation256 Properties of Real and Complex Measures256 Total Variation Measure259 The Banach Space of Measures262 Exercises 9A2659B Decomposition Theorems267 Hahn Decomposition Theorem267 Jordan Decomposition Theorem268 Lebesgue Decomposition Theorem270 Radon Nikodym Theorem272 Dual Space ofLp( )

6 275 Exercises 9B27810 Linear Maps on Hilbert Spaces28010A Adjoints and Invertibility281 Adjoints of Linear Maps on Hilbert Spaces281 Null Spaces and Ranges in Terms of Adjoints285 Invertibility of Operators286 Exercises 10A29210B Spectrum294 Spectrum of an Operator294 Self-adjoint Operators299 Normal Operators302 Isometries and Unitary Operators305 Exercises 10B30910C Compact Operators312 The Ideal of Compact Operators312 Spectrum of Compact Operator and Fredholm Alternative316 Exercises 10C32310D Spectral Theorem for Compact Operators326 Orthonormal Bases Consisting of Eigenvectors326 Singular Value Decomposition332 Exercises 10D336 Measure, Integration & Real Analysis , by Sheldon AxlerxiiContents11 Fourier Analysis33911A Fourier Series and Poisson Integral340 Fourier Coefficients and Riemann Lebesgue Lemma340 Poisson Kernel344 Solution to Dirichlet Problem on Disk348 Fourier Series of Smooth Functions350 Exercises 11A35211B Fourier Series andLpof Unit Circle355 Orthonormal Basis forL2of Unit Circle355 Convolution on Unit Circle357 Exercises 11B36111C Fourier Transform363 Fourier Transform onL1(R)363 Convolution onR368 Poisson Kernel on Upper Half-Plane370 Fourier Inversion Formula374 Extending Fourier Transform toL2(R)375 Exercises 11C37712 Probability Measures380 Probability Spaces381 Independent Events and Independent Random Variables383 Variance and Standard Deviation388 Conditional Probability and Bayes Theorem390 Distribution and Density Functions of Random Variables392 Weak Law of Large Numbers396 Exercises 12398 Photo Credits400 Bibliography402 Notation Index403 Index406 Colophon.

7 Notes on Typesetting411 Measure, Integration & Real Analysis , by Sheldon AxlerPreface 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 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 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.

8 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 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 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 .

9 You can download from the website below or from theSpringer website the document titledSupplement for Measure, Integration & RealAnalysis. That supplement can serve as a review of the elementary undergraduate realanalysis used in this wishes for success and enjoyment in learning measure, Integration , and realanalysis!Sheldon AxlerMathematics DepartmentSan Francisco State UniversitySan Francisco, CA 94132, USAwebsite: : @AxlerLinearMeasure, Integration & Real Analysis , by Sheldon AxlerxiiiPreface 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 Chapter11isintroduced in the setting ofRrather thanRnso that students can focus on the mainideas without the clutter of the extra bookkeeping needed for working basic prerequisite for your students to use this textbook is a good understand-ing of elementary undergraduate real Analysis .

10 Your students can download from thebook s website ( ) or from the Springer website the documenttitledSupplement for Measure, Integration & Real Analysis . That supplement canserve as a review of the elementary undergraduate real Analysis used in this 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 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 , I tell them that getting solutions from the internet is not allowed and wouldbe counterproductive for their learning you go at a leisurely pace, then covering Chapters1 5in the first semester maybe a good goal.