积分 测度与实分析

【实例截图】

【核心代码】

Contents
About the Author vi
Preface for Students xiii
Preface for Instructors xiv
Acknowledgments xviii
1 Riemann Integration 1
1A Review: Riemann Integral 2
Exercises 1A 7
1B Riemann Integral Is Not Good Enough 9
Exercises 1B 12
2 Measures 13
2A Outer Measure on R 14
Motivation and Definition of Outer Measure 14
Good Properties of Outer Measure 15
Outer Measure of Closed Bounded Interval 18
Outer Measure is Not Additive 21
Exercises 2A 23
2B Measurable Spaces and Functions 25
σ-Algebras 26
Borel Subsets of R 28
Inverse Images 29
Measurable Functions 31
Exercises 2B 38
2C Measures and Their Properties 41
Definition and Examples of Measures 41
Properties of Measures 42
Exercises 2C 45
Measure, Integration & Real Analysis, by Sheldon Axler
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viii Contents
2D Lebesgue Measure 47
Additivity of Outer Measure on Borel Sets 47
Lebesgue Measurable Sets 52
Cantor Set and Cantor Function 55
Exercises 2D 60
2E Convergence of Measurable Functions 62
Pointwise and Uniform Convergence 62
Egorov’s Theorem 63
Approximation by Simple Functions 65
Luzin’s Theorem 66
Lebesgue Measurable Functions 69
Exercises 2E 71
3 Integration 73
3A Integration with Respect to a Measure 74
Integration of Nonnegative Functions 74
Monotone Convergence Theorem 77
Integration of Real-Valued Functions 81
Exercises 3A 84
3B Limits of Integrals & Integrals of Limits 88
Bounded Convergence Theorem 88
Sets of Measure 0 in Integration Theorems 89
Dominated Convergence Theorem 90
Riemann Integrals and Lebesgue Integrals 93
Approximation by Nice Functions 95
Exercises 3B 99
4 Differentiation 101
4A Hardy–Littlewood Maximal Function 102
Markov’s Inequality 102
Vitali Covering Lemma 103
Hardy–Littlewood Maximal Inequality 104
Exercises 4A 106
4B Derivatives of Integrals 108
Lebesgue Differentiation Theorem 108
Derivatives 110
Density 112
Exercises 4B 115
Measure, Integration & Real Analysis, by Sheldon Axler
Contents ix
5 Product Measures 116
5A Products of Measure Spaces 117
Products of σ-Algebras 117
Monotone Class Theorem 120
Products of Measures 123
Exercises 5A 128
5B Iterated Integrals 129
Tonelli’s Theorem 129
Fubini’s Theorem 131
Area Under Graph 133
Exercises 5B 135
5C Lebesgue Integration on R n 136
Borel Subsets of R n 136
Lebesgue Measure on R n 139
Volume of Unit Ball in R n 140
Equality of Mixed Partial Derivatives Via Fubini’s Theorem 142
Exercises 5C 144
6 Banach Spaces 146
6A Metric Spaces 147
Open Sets, Closed Sets, and Continuity 147
Cauchy Sequences and Completeness 151
Exercises 6A 153
6B Vector Spaces 155
Integration of Complex-Valued Functions 155
Vector Spaces and Subspaces 159
Exercises 6B 162
6C Normed Vector Spaces 163
Norms and Complete Norms 163
Bounded Linear Maps 167
Exercises 6C 170
6D Linear Functionals 172
Bounded Linear Functionals 172
Discontinuous Linear Functionals 174
Hahn–Banach Theorem 177
Exercises 6D 181
Measure, Integration & Real Analysis, by Sheldon Axler
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6E Consequences of Baire’s Theorem 184
Baire’s Theorem 184
Open Mapping Theorem and Inverse Mapping Theorem 186
Closed Graph Theorem 188
Principle of Uniform Boundedness 189
Exercises 6E 190
7 L p Spaces 193
7A L p ( µ ) 194
Hölder’s Inequality 194
Minkowski’s Inequality 198
Exercises 7A 199
7B L p ( µ ) 202
Definition of L p ( µ ) 202
L p ( µ ) Is a Banach Space 204
Duality 206
Exercises 7B 208
8 Hilbert Spaces 211
8A Inner Product Spaces 212
Inner Products 212
Cauchy–Schwarz Inequality and Triangle Inequality 214
Exercises 8A 221
8B Orthogonality 224
Orthogonal Projections 224
Orthogonal Complements 229
Riesz Representation Theorem 233
Exercises 8B 234
8C Orthonormal Bases 237
Bessel’s Inequality 237
Parseval’s Identity 243
Gram–Schmidt Process and Existence of Orthonormal Bases 245
Riesz Representation Theorem, Revisited 250
Exercises 8C 251
Measure, Integration & Real Analysis, by Sheldon Axler
Contents xi
9 Real and Complex Measures 255
9A Total Variation 256
Properties of Real and Complex Measures 256
Total Variation Measure 259
The Banach Space of Measures 262
Exercises 9A 265
9B Decomposition Theorems 267
Hahn Decomposition Theorem 267
Jordan Decomposition Theorem 268
Lebesgue Decomposition Theorem 270
Radon–Nikodym Theorem 272
Dual Space of L p ( µ ) 275
Exercises 9B 278
10 Linear Maps on Hilbert Spaces 280
10A Adjoints and Invertibility 281
Adjoints of Linear Maps on Hilbert Spaces 281
Null Spaces and Ranges in Terms of Adjoints 285
Invertibility of Operators 286
Exercises 10A 292
10B Spectrum 294
Spectrum of an Operator 294
Self-adjoint Operators 299
Normal Operators 302
Isometries and Unitary Operators 305
Exercises 10B 309
10C Compact Operators 312
The Ideal of Compact Operators 312
Spectrum of Compact Operator and Fredholm Alternative 316
Exercises 10C 323
10D Spectral Theorem for Compact Operators 326
Orthonormal Bases Consisting of Eigenvectors 326
Singular Value Decomposition 332
Exercises 10D 336
Measure, Integration & Real Analysis, by Sheldon Axler
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11 Fourier Analysis 339
11A Fourier Series and Poisson Integral 340
Fourier Coefficients and Riemann–Lebesgue Lemma 340
Poisson Kernel 344
Solution to Dirichlet Problem on Disk 348
Fourier Series of Smooth Functions 350
Exercises 11A 352
11B Fourier Series and L p of Unit Circle 355
Orthonormal Basis for L 2 of Unit Circle 355
Convolution on Unit Circle 357
Exercises 11B 361
11C Fourier Transform 363
Fourier Transform on L 1 ( R ) 363
Convolution on R 368
Poisson Kernel on Upper Half-Plane 370
Fourier Inversion Formula 374
Extending Fourier Transform to L 2 ( R ) 375
Exercises 11C 377
12 Probability Measures 380
Probability Spaces 381
Independent Events and Independent Random Variables 383
Variance and Standard Deviation 388
Conditional Probability and Bayes’ Theorem 390
Distribution and Density Functions of Random Variables 392
Weak Law of Large Numbers 396
Exercises 12 398
Photo Credits 400
Bibliography 402
Notation Index 403
Index 406
Colophon: Notes on Typesetting 411
Measure, Integration & Real Analysis, by Sheldon Axler
Preface for Students
You are about to immerse yourself in serious mathematics, with an emphasis on
attaining a deep understanding of the definitions, theorems, and proofs related to
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 should indeed do the verification, which will usually require
some writing on your part. When steps are left out, you need to supply the missing
pieces. You should ponder and internalize each definition. For each theorem, you
should seek examples to show why each hypothesis is necessary.
Working on the exercises should be your main mode of learning after you have
read a section. Discussions and joint work with other students may be especially
effective. Active learning promotes long-term understanding much better than passive
learning. Thus you will benefit considerably from struggling with an exercise and
eventually coming up with a solution, perhaps working with other students. Finding
and 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 theorems
are in blue boxes, in both print and electronic versions. Each theorem has an informal
descriptive name. The electronic version of this manuscript has links in blue.
Pleasecheckthewebsitebelow(ortheSpringerwebsite)foradditionalinformation
about the book. These websites link to the electronic version of this book, which is
free to the world because this book has been published under Springer’s Open Access
program. Your suggestions for improvements and corrections for a future edition are
most welcome (send to the email address below).
The prerequisite for using this book includes a good understanding of elementary
undergraduate real analysis. You can download from the website below or from the
Springer website the document titled Supplement for Measure, Integration & Real
Analysis. That supplement can serve as a review of the elementary undergraduate real
analysis used in this book.
Best wishes for success and enjoyment in learning measure, integration, and real
analysis!
Sheldon Axler
Mathematics Department
San Francisco State University
San Francisco, CA 94132, USA
website: http://measure.axler.net
e-mail: measure@axler.net
Twitter: @AxlerLinear
Measure, Integration & Real Analysis, by Sheldon Axler
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Preface for Instructors