Springer - Measure Theory And Probability Theory: Springer - Measure Theory And Probability Theory.pdf

Springer - Measure Theory And Probability Theory

Krishna B. Athreya Soumendra N. Lahiri
Department of Mathematics and Department of Statistics
Department of Statistics Iowa State University
Iowa State University Ames, IA 50011
Ames, IA 50011 snlahiri@iastate.edu
kba@iastate.edu
Editorial Board
George Casella Stephen Fienberg Ingram Olkin
Department of Statistics Department of Statistics Department of Statistics
University of Florida Carnegie Mellon University Stanford University
Gainesville, FL 32611-8545 Pittsburgh, PA 15213-3890 Stanford, CA 94305
USA USA USA
Library of Congress Control Number: 2006922767
ISBN-10: 0-387-32903-X e-ISBN: 0-387-35434-4
ISBN-13: 978-0387-32903-1
Printed on acid-free paper.
©2006 Springer Science+Business Media, LLC
All rights reserved. This work may not be translated or copied in whole or in part without the
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Printed in the United States of America. (MVY)
9 8

This book arose out of two graduate courses that the authors have taught
during the past several years; the first one being on measure theory followed
by the second one on advanced probability theory.
The traditional approach to a first course in measure theory, such as in
Royden (1988), is to teach the Lebesgue measure on the real line, then the
differentation theorems of Lebesgue, Lp-spaces on R, and do general measure
at the end of the course with one main application to the construction
of product measures. This approach does have the pedagogic advantage
of seeing one concrete case first before going to the general one. But this
also has the disadvantage in making many students’ perspective on measure
theory somewhat narrow. It leads them to think only in terms of the
Lebesgue measure on the real line and to believe that measure theory is
intimately tied to the topology of the real line. As students of statistics,
probability, physics, engineering, economics, and biology know very well,
there are mass distributions that are typically nonuniform, and hence it is
useful to gain a general perspective.
This book attempts to provide that general perspective right from the
beginning. The opening chapter gives an informal introduction to measure
and integration theory. It shows that the notions of σ-algebra of sets and
countable additivity of a set function are dictated by certain very natural
approximation procedures from practical applications and that they
are not just some abstract ideas. Next, the general extension theorem of
Carathedory is presented in Chapter 1. As immediate examples, the construction
of the large class of Lebesgue-Stieltjes measures on the real line
and Euclidean spaces is discussed, as are measures on finite and countable
viii Preface
spaces. Concrete examples such as the classical Lebesgue measure and various
probability distributions on the real line are provided. This is further
developed in Chapter 6 leading to the construction of measures on sequence
spaces (i.e., sequences of random variables) via Kolmogorov’s consistency
theorem.
After providing a fairly comprehensive treatment of measure and integration
theory in the first part (Introduction and Chapters 1–5), the focus
moves onto probability theory in the second part (Chapters 6–13). The feature
that distinguishes probability theory from measure theory, namely, the
notion of independence and dependence of random variables (i.e., measureable
functions) is carefully developed first. Then the laws of large numbers
are taken up. This is followed by convergence in distribution and the central
limit theorems. Next the notion of conditional expectation and probability
is developed, followed by discrete parameter martingales. Although the development
of these topics is based on a rigorous measure theoretic foundation,
the heuristic and intuitive backgrounds of the results are emphasized
throughout. Along the way, some applications of the results from probability
theory to proving classical results in analysis are given. These include,
for example, the density of normal numbers on (0,1) and the Wierstrass
approximation theorem. These are intended to emphasize the benefits of
studying both areas in a rigorous and combined fashion. The approach
to conditional expectation is via the mean square approximation of the
“unknown” given the “known” and then a careful approximation for the
L1-case. This is a natural and intuitive approach and is preferred over the
“black box” approach based on the Radon-Nikodym theorem.
The final part of the book provides a basic outline of a number of special
topics. These include Markov chains including Markov chain Monte Carlo
(MCMC), Poisson processes, Brownian motion, bootstrap theory, mixing
processes, and branching processes. The first two parts can be used for a
two-semester sequence, and the last part could serve as a starting point for
a seminar course on special topics.
This book presents the basic material on measure and integration theory
and probability theory in a self-contained and step-by-step manner. It is
hoped that students will find it accessible, informative, and useful and also
that they will be motivated to master the details by carefully working out
the text material as well as the large number of exercises. The authors hope
that the presentation here is found to be clear and comprehensive without
being intimidating.
Here is a quick summary of the various chapters of the book. After giving
an informal introduction to the ideas of measure and integration theory,
the construction of measures starting with set functions on a small class of
sets is taken up in Chapter 1 where the Caratheodory extension theorem is
proved and then applied to construct Lebesgue-Stieltjes measures. Integration
theory is taken up in Chapter 2 where all the basic convergence theorems
including the MCT, Fatou, DCT, BCT, Egorov’s, and Scheffe’s are
Preface ix
proved. Included here are also the notion of uniform integrability and the
classical approximation theorem of Lusin and its use in Lp-approximation
by smooth functions. The third chapter presents basic inequalities for Lpspaces,
the Riesz-Fischer theorem, and elementary theory of Banach and
Hilbert spaces. Chapter 4 deals with Radon-Nikodym theory via the Riesz
representation on L2-spaces and its application to differentiation theorems
on the real line as well as to signed measures. Chapter 5 deals with product
measures and the Fubini-Tonelli theorems. Two constructions of the
product measure are presented: one using the extension theorem and another
via iterated integrals. This is followed by a discussion on convolutions,
Laplace transforms, Fourier series, and Fourier transforms. Kolmogorov’s
consistency theorem for the construction of stochastic processes is taken up
in Chapter 6 followed by the notion of independence in Chapter 7. The laws
of large numbers are presented in a unified manner in Chapter 8 where the
classical Kolmogorov’s strong law as well as Etemadi’s strong law are presented
followed by Marcinkiewicz-Zygmund laws. There are also sections
on renewal theory and ergodic theorems. The notion of weak convergence of
probability measures on R is taken up in Chapter 9, and Chapter 10 introduces
characteristic functions (Fourier transform of probability measures),
the inversion formula, and the Levy-Cramer continuity theorem. Chapter
11 is devoted to the central limit theorem and its extensions to stable and
infinitely divisible laws. Chapter 12 discusses conditional expectation and
probability where an L2-approach followed by an approximation to L1 is
presented. Discrete time martingales are introduced in Chapter 13 where
the basic inequalities as well as convergence results are developed. Some
applications to random walks are indicated as well. Chapter 14 discusses
discrete time Markov chains with a discrete state space first. This is followed
by discrete time Markov chains with general state spaces where the
regeneration approach for Harris chains is carefully explained and is used
to derive the basic limit theorems via the iid cycles approach. There are
also discussions of Feller Markov chains on Polish spaces and Markov chain
Monte Carlo methods. An elementary treatment of Brownian motion is
presented in Chapter 15 along with a treatment of continuous time jump
Markov chains. Chapters 16–18 provide brief outlines respectively of the
bootstrap theory, mixing processes, and branching processes. There is an
Appendix that reviews basic material on elementary set theory, real and
complex numbers, and metric spaces.
Here are some suggestions on how to use the book.
1. For a one-semester course on real analysis (i.e., measure end integration
theory), material up to Chapter 5 and the Appendix should
provide adequate coverage with Chapter 6 being optional.
2. A one-semester course on advanced probability theory for those with
the necessary measure theory background could be based on Chapters
6–13 with a selection of topics from Chapters 14–18.
x Preface
3. A one-semester course on combined treatment of measure theory and
probability theory could be built around Chapters 1, 2, Sections 3.1–
3.2 of Chapter 3, all of Chapter 4 (Section 4.2 optional), Sections
5.1 and 5.2 of Chapter 5, Chapters 6, 7, and Sections 8.1, 8.2, 8.3
(Sections 8.5 and 8.6 optional) of Chapter 8. Such a course could
be followed by another that includes some coverage of Chapters 9–
12 before moving on to other areas such as mathematical statistics
or martingales and financial mathematics. This will be particularly
useful for graduate programs in statistics.
4. A one-semester course on an introduction to stochastic processes or
a seminar on special topics could be based on Chapters 14–18.
A word on the numbering system used in the book. Statements of results
(i.e., Theorems, Corollaries, Lemmas, and Propositions) are numbered consecutively
within each section, in the format a.b.c, where a is the chapter
number, b is the section number, and c is the counter. Definitions, Examples,
and Remarks are numbered individually within each section, also of
the form a.b.c, as above. Sections are referred to as a.b where a is the chapter
number and b is the section number. Equation numbers appear on the
right, in the form (b.c), where b is the section number and c is the equation
number. Equations in a given chapter a are referred to as (b.c) within the
chapter but as (a.b.c) outside chapter a. Problems are listed at the end of
each chapter in the form a.c, where a is the chapter number and c is the
problem number.
In the writing of this book, material from existing books such as Apostol
(1974), Billingsley (1995), Chow and Teicher (2001), Chung (1974), Durrett
(2004), Royden (1988), and Rudin (1976, 1987) has been freely used.
The authors owe a great debt to these books. The authors have used this
material for courses taught over several years and have benefited greatly
from suggestions for improvement from students and colleagues at Iowa
State University, Cornell University, the Indian Institute of Science, and
the Indian Statistical Institute. We are grateful to them.
Our special thanks go to Dean Issacson, Ken Koehler, and Justin Peters
at Iowa State University for their administrative support of this long
project. Krishna Athreya would also like to thank Cornell University for
its support.
We are most indebted to Sharon Shepard who typed and retyped several
times this book, patiently putting up with our never-ending “final” versions.
Without her patient and generous help, this book could not have been
written. We are also grateful to Denise Riker who typed portions of an
earlier version of this book.
John Kimmel of Springer got the book reviewed at various stages. The
referee reports were very helpful and encouraging. Our grateful thanks to
both John Kimmel and the referees.

Preface vii
Measures and Integration: An Informal Introduction 1
1 Measures 9
1.1 Classes of sets . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 The extension theorems and Lebesgue-Stieltjes measures . . 19
1.3.1 Caratheodory extension of measures . . . . . . . . . 19
1.3.2 Lebesgue-Stieltjes measures on R . . . . . . . . . . . 25
1.3.3 Lebesgue-Stieltjes measures on R2 . . . . . . . . . . 27
1.3.4 More on extension of measures . . . . . . . . . . . . 28
1.4 Completeness of measures . . . . . . . . . . . . . . . . . . . 30
1.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 Integration 39
2.1 Measurable transformations . . . . . . . . . . . . . . . . . . 39
2.2 Induced measures, distribution functions . . . . . . . . . . . 44
2.2.1 Generalizations to higher dimensions . . . . . . . . . 47
2.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4 Riemann and Lebesgue integrals . . . . . . . . . . . . . . . 59
2.5 More on convergence . . . . . . . . . . . . . . . . . . . . . . 61
2.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
xiv Contents
3 Lp-Spaces 83
3.1 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.2 Lp-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.2.1 Basic properties . . . . . . . . . . . . . . . . . . . . 89
3.2.2 Dual spaces . . . . . . . . . . . . . . . . . . . . . . . 93
3.3 Banach and Hilbert spaces . . . . . . . . . . . . . . . . . . . 94
3.3.1 Banach spaces . . . . . . . . . . . . . . . . . . . . . 94
3.3.2 Linear transformations . . . . . . . . . . . . . . . . . 96
3.3.3 Dual spaces . . . . . . . . . . . . . . . . . . . . . . . 97
3.3.4 Hilbert space . . . . . . . . . . . . . . . . . . . . . . 98
3.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4 Differentiation 113
4.1 The Lebesgue-Radon-Nikodymtheorem . . . . . . . . . . . 113
4.2 Signed measures . . . . . . . . . . . . . . . . . . . . . . . . 119
4.3 Functions of bounded variation . . . . . . . . . . . . . . . . 125
4.4 Absolutely continuous functions on R . . . . . . . . . . . . 128
4.5 Singular distributions . . . . . . . . . . . . . . . . . . . . . 133
4.5.1 Decomposition of a cdf . . . . . . . . . . . . . . . . . 133
4.5.2 Cantor ternary set . . . . . . . . . . . . . . . . . . . 134
4.5.3 Cantor ternary function . . . . . . . . . . . . . . . . 136
4.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5 Product Measures, Convolutions, and Transforms 147
5.1 Product spaces and product measures . . . . . . . . . . . . 147
5.2 Fubini-Tonelli theorems . . . . . . . . . . . . . . . . . . . . 152
5.3 Extensions to products of higher orders . . . . . . . . . . . 157
5.4 Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.4.1 Convolution of measures on


R, B(R)

. . . . . . . . 160
5.4.2 Convolution of sequences . . . . . . . . . . . . . . . 162
5.4.3 Convolution of functions in L1(R) . . . . . . . . . . 162
5.4.4 Convolution of functions and measures . . . . . . . . 164
5.5 Generating functions and Laplace transforms . . . . . . . . 164
5.6 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5.7 Fourier transforms on R . . . . . . . . . . . . . . . . . . . . 173
5.8 Plancherel transform . . . . . . . . . . . . . . . . . . . . . . 178
5.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6 Probability Spaces 189
6.1 Kolmogorov’s probability model . . . . . . . . . . . . . . . . 189
6.2 Randomvariables and randomvectors . . . . . . . . . . . . 191
6.3 Kolmogorov’s consistency theorem . . . . . . . . . . . . . . 199
6.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
7 Independence 219
Contents xv
7.1 Independent events and randomvariables . . . . . . . . . . 219
7.2 Borel-Cantelli lemmas, tail σ-algebras, and Kolmogorov’s
zero-one law . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
7.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
8 Laws of Large Numbers 237
8.1 Weak laws of large numbers . . . . . . . . . . . . . . . . . . 237
8.2 Strong laws of large numbers . . . . . . . . . . . . . . . . . 240
8.3 Series of independent randomvariables . . . . . . . . . . . . 249
8.4 Kolmogorov and Marcinkiewz-Zygmund SLLNs . . . . . . . 254
8.5 Renewal theory . . . . . . . . . . . . . . . . . . . . . . . . . 260
8.5.1 Definitions and basic properties . . . . . . . . . . . . 260
8.5.2 Wald’s equation . . . . . . . . . . . . . . . . . . . . 262
8.5.3 The renewal theorems . . . . . . . . . . . . . . . . . 264
8.5.4 Renewal equations . . . . . . . . . . . . . . . . . . . 266
8.5.5 Applications . . . . . . . . . . . . . . . . . . . . . . 268
8.6 Ergodic theorems . . . . . . . . . . . . . . . . . . . . . . . . 271
8.6.1 Basic definitions and examples . . . . . . . . . . . . 271
8.6.2 Birkhoff’s ergodic theorem. . . . . . . . . . . . . . . 274
8.7 Law of the iterated logarithm . . . . . . . . . . . . . . . . . 278
8.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
9 Convergence in Distribution 287
9.1 Definitions and basic properties . . . . . . . . . . . . . . . . 287
9.2 Vague convergence, Helly-Bray theorems, and tightness . . 291
9.3 Weak convergence on metric spaces . . . . . . . . . . . . . . 299
9.4 Skorohod’s theorem and the continuous mapping theorem . 303
9.5 Themethod of moments and themoment problem . . . . . 306
9.5.1 Convergence of moments . . . . . . . . . . . . . . . . 306
9.5.2 Themethod of moments . . . . . . . . . . . . . . . . 307
9.5.3 Themoment problem . . . . . . . . . . . . . . . . . 307
9.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
10 Characteristic Functions 317
10.1 Definition and examples . . . . . . . . . . . . . . . . . . . . 317
10.2 Inversion formulas . . . . . . . . . . . . . . . . . . . . . . . 323
10.3 Levy-Cramer continuity theorem . . . . . . . . . . . . . . . 327
10.4 Extension to Rk . . . . . . . . . . . . . . . . . . . . . . . . 332
10.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
11 Central Limit Theorems 343
11.1 Lindeberg-Feller theorems . . . . . . . . . . . . . . . . . . . 343
11.2 Stable distributions . . . . . . . . . . . . . . . . . . . . . . . 352
11.3 Infinitely divisible distributions . . . . . . . . . . . . . . . . 358
11.4 Refinements and extensions of the CLT . . . . . . . . . . . 361
xvi Contents
11.4.1 The Berry-Esseen theorem . . . . . . . . . . . . . . . 361
11.4.2 Edgeworth expansions . . . . . . . . . . . . . . . . . 364
11.4.3 Large deviations . . . . . . . . . . . . . . . . . . . . 368
11.4.4 The functional central limit theorem . . . . . . . . . 372
11.4.5 Empirical process and Brownian bridge . . . . . . . 374
11.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
12 Conditional Expectation and Conditional Probability 383
12.1 Conditional expectation: Definitions and examples . . . . . 383
12.2 Convergence theorems . . . . . . . . . . . . . . . . . . . . . 389
12.3 Conditional probability . . . . . . . . . . . . . . . . . . . . 392
12.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
13 Discrete Parameter Martingales 399
13.1 Definitions and examples . . . . . . . . . . . . . . . . . . . . 399
13.2 Stopping times and optional stopping theorems . . . . . . . 405
13.3 Martingale convergence theorems . . . . . . . . . . . . . . . 417
13.4 Applications of martingalemethods . . . . . . . . . . . . . 424
13.4.1 Supercritical branching processes . . . . . . . . . . . 424
13.4.2 Investment sequences . . . . . . . . . . . . . . . . . 425
13.4.3 A conditional Borel-Cantelli lemma . . . . . . . . . . 425
13.4.4 Decomposition of probability measures . . . . . . . . 427
13.4.5 Kakutani’s theorem . . . . . . . . . . . . . . . . . . 429
13.4.6 de Finetti’s theorem . . . . . . . . . . . . . . . . . . 430
13.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
14 Markov Chains and MCMC 439
14.1 Markov chains: Countable state space . . . . . . . . . . . . 439
14.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . 439
14.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . 440
14.1.3 Existence of aMarkov chain . . . . . . . . . . . . . . 442
14.1.4 Limit theory . . . . . . . . . . . . . . . . . . . . . . 443
14.2 Markov chains on a general state space . . . . . . . . . . . . 457
14.2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . 457
14.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . 458
14.2.3 Chapman-Kolmogorov equations . . . . . . . . . . . 461
14.2.4 Harris irreducibility, recurrence, and minorization . . 462
14.2.5 Theminorization theorem . . . . . . . . . . . . . . . 464
14.2.6 The fundamental regeneration theorem . . . . . . . 465
14.2.7 Limit theory for regenerative sequences . . . . . . . 467
14.2.8 Limit theory of Harris recurrent Markov chains . . . 469
14.2.9 Markov chains on metric spaces . . . . . . . . . . . . 473
14.3 Markov chainMonte Carlo (MCMC) . . . . . . . . . . . . . 477
14.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 477
14.3.2 Metropolis-Hastings algorithm . . . . . . . . . . . . 478
Contents xvii
14.3.3 The Gibbs sampler . . . . . . . . . . . . . . . . . . . 480
14.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
15 Stochastic Processes 487
15.1 Continuous timeMarkov chains . . . . . . . . . . . . . . . . 487
15.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . 487
15.1.2 Kolmogorov’s differential equations . . . . . . . . . . 488
15.1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . 489
15.1.4 Limit theorems . . . . . . . . . . . . . . . . . . . . . 491
15.2 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . 493
15.2.1 Construction of SBM. . . . . . . . . . . . . . . . . . 493
15.2.2 Basic properties of SBM . . . . . . . . . . . . . . . . 495
15.2.3 Some related processes . . . . . . . . . . . . . . . . . 498
15.2.4 Some limit theorems . . . . . . . . . . . . . . . . . . 498
15.2.5 Some sample path properties of SBM . . . . . . . . 499
15.2.6 Brownian motion and martingales . . . . . . . . . . 501
15.2.7 Some applications . . . . . . . . . . . . . . . . . . . 502
15.2.8 The Black-Scholes formula for stock price option . . 503
15.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504
16 Limit Theorems for Dependent Processes 509
16.1 A central limit theoremfor martingales . . . . . . . . . . . 509
16.2 Mixing sequences . . . . . . . . . . . . . . . . . . . . . . . . 513
16.2.1 Mixing coefficients . . . . . . . . . . . . . . . . . . . 514
16.2.2 Coupling and covariance inequalities . . . . . . . . . 516
16.3 Central limit theorems for mixing sequences . . . . . . . . . 519
16.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529
17 The Bootstrap 533
17.1 The bootstrap method for independent variables . . . . . . 533
17.1.1 A description of the bootstrap method . . . . . . . . 533
17.1.2 Validity of the bootstrap: Samplemean . . . . . . . 535
17.1.3 Second order correctness of the bootstrap . . . . . . 536
17.1.4 Bootstrap for lattice distributions . . . . . . . . . . 537
17.1.5 Bootstrap for heavy tailed randomvariables . . . . . 540
17.2 Inadequacy of resampling single values under dependence . 545
17.3 Block bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . 547
17.4 Properties of theMBB . . . . . . . . . . . . . . . . . . . . . 548
17.4.1 Consistency ofMBB variance estimators . . . . . . . 549
17.4.2 Consistency ofMBB cdf estimators . . . . . . . . . . 552
17.4.3 Second order properties of theMBB . . . . . . . . . 554
17.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556
18 Branching Processes 561
18.1 Bienyeme-Galton-Watson branching process . . . . . . . . . 562
xviii Contents
18.2 BGW process: Multitype case . . . . . . . . . . . . . . . . . 564
18.3 Continuous time branching processes . . . . . . . . . . . . . 566
18.4 Embedding of Urn schemes in continuous time branching
processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568
18.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569
A Advanced Calculus: A Review 573
A.1 Elementary set theory . . . . . . . . . . . . . . . . . . . . . 573
A.1.1 Set operations . . . . . . . . . . . . . . . . . . . . . 574
A.1.2 The principle of induction . . . . . . . . . . . . . . . 577
A.1.3 Equivalence relations . . . . . . . . . . . . . . . . . . 577
A.2 Real numbers, continuity, differentiability, and integration . 578
A.2.1 Real numbers . . . . . . . . . . . . . . . . . . . . . . 578
A.2.2 Sequences, series, limits, limsup, liminf . . . . . . . . 580
A.2.3 Continuity and differentiability . . . . . . . . . . . . 582
A.2.4 Riemann integration . . . . . . . . . . . . . . . . . . 584
A.3 Complex numbers, exponential and trigonometric functions 586
A.4 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 590
A.4.1 Basic definitions . . . . . . . . . . . . . . . . . . . . 590
A.4.2 Continuous functions . . . . . . . . . . . . . . . . . . 592
A.4.3 Compactness . . . . . . . . . . . . . . . . . . . . . . 592
A.4.4 Sequences of functions and uniform convergence . . 593
A.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594
B List of Abbreviations and Symbols 599
B.1 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . 599
B.2 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600
References 603
Author Index 610
Subject Index 612

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:30bb :30bb :30bb :30bb 谢谢！！！！！！

好資料
真是感激
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maybe useful for me:29bb :29bb

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