Real Analysis and Probability / Edition 2

Real Analysis and Probability / Edition 2

by R. M. Dudley, Dudley R. M.
     
 

ISBN-10: 052180972X

ISBN-13: 9780521809726

Pub. Date: 10/28/2002

Publisher: Cambridge University Press

This is a reissue of textbook covering graduate level courses in probability theory and real analysis, each conceived as a one-semester course. Dudley (Massachusetts Institute of Technology), in an effort to make the text self-contained, has added a treatment of the Stone- Weierstrass theorem. Annotation c. Book News, Inc., Portland, OR  See more details below

Overview

This is a reissue of textbook covering graduate level courses in probability theory and real analysis, each conceived as a one-semester course. Dudley (Massachusetts Institute of Technology), in an effort to make the text self-contained, has added a treatment of the Stone- Weierstrass theorem. Annotation c. Book News, Inc., Portland, OR

Product Details

ISBN-13:
9780521809726
Publisher:
Cambridge University Press
Publication date:
10/28/2002
Series:
Cambridge Studies in Advanced Mathematics Series, #74
Edition description:
REV
Pages:
568
Product dimensions:
5.98(w) x 9.02(h) x 1.38(d)

Table of Contents

Preface to the Cambridge Edition
1Foundations; Set Theory1
1.1Definitions for Set Theory and the Real Number System1
1.2Relations and Orderings9
1.3Transfinite Induction and Recursion12
1.4Cardinality16
1.5The Axiom of Choice and Its Equivalents18
2General Topology24
2.1Topologies, Metrics, and Continuity24
2.2Compactness and Product Topologies34
2.3Complete and Compact Metric Spaces44
2.4Some Metrics for Function Spaces48
2.5Completion and Completeness of Metric Spaces58
2.6Extension of Continuous Functions63
2.7Uniformities and Uniform Spaces67
2.8Compactification71
3Measures85
3.1Introduction to Measures85
3.2Semirings and Rings94
3.3Completion of Measures101
3.4Lebesgue Measure and Nonmeasurable Sets105
3.5Atomic and Nonatomic Measures109
4Integration114
4.1Simple Functions114
4.2Measurability123
4.3Convergence Theorems for Integrals130
4.4Product Measures134
4.5Daniell-Stone Integrals142
5L[superscript p] Spaces; Introduction to Functional Analysis152
5.1Inequalities for Integrals152
5.2Norms and Completeness of LP158
5.3Hilbert Spaces160
5.4Orthonormal Sets and Bases165
5.5Linear Forms on Hilbert Spaces, Inclusions of IP Spaces, and Relations Between Two Measures173
5.6Signed Measures178
6Convex Sets and Duality of Normed Spaces188
6.1Lipschitz, Continuous, and Bounded Functionals188
6.2Convex Sets and Their Separation195
6.3Convex Functions203
6.4Duality of L[superscript p] Spaces208
6.5Uniform Boundedness and Closed Graphs211
6.6The Brunn-Minkowski Inequality215
7Measure, Topology, and Differentiation222
7.1Baire and Borel [sigma]-Algebras and Regularity of Measures222
7.2Lebesgue's Differentiation Theorems228
7.3The Regularity Extension235
7.4The Dual of C(K) and Fourier Series239
7.5Almost Uniform Convergence and Lusin's Theorem243
8Introduction to Probability Theory250
8.1Basic Definitions251
8.2Infinite Products of Probability Spaces255
8.3Laws of Large Numbers260
8.4Ergodic Theorems267
9Convergence of Laws and Central Limit Theorems282
9.1Distribution Functions and Densities282
9.2Convergence of Random Variables287
9.3Convergence of Laws291
9.4Characteristic Functions298
9.5Uniqueness of Characteristic Functions and a Central Limit Theorem303
9.6Triangular Arrays and Lindeberg's Theorem315
9.7Sums of Independent Real Random Variables320
9.8The Levy Continuity Theorem; Infinitely Divisible and Stable Laws325
10Conditional Expectations and Martingales336
10.1Conditional Expectations336
10.2Regular Conditional Probabilities and Jensen's Inequality341
10.3Martingales353
10.4Optional Stopping and Uniform Integrability358
10.5Convergence of Martingales and Submartingales364
10.6Reversed Martingales and Submartingales370
10.7Subadditive and Superadditive Ergodic Theorems374
11Convergence of Laws on Separable Metric Spaces385
11.1Laws and Their Convergence385
11.2Lipschitz Functions390
11.3Metrics for Convergence of Laws393
11.4Convergence of Empirical Measures399
11.5Tightness and Uniform Tightness402
11.6Strassen's Theorem: Nearby Variables with Nearby Laws406
11.7A Uniformity for Laws and Almost Surely Converging Realizations of Converging Laws413
11.8Kantorovich-Rubinstein Theorems420
11.9U-Statistics426
12Stochastic Processes439
12.1Existence of Processes and Brownian Motion439
12.2The Strong Markov Property of Brownian Motion450
12.3Reflection Principles, The Brownian Bridge, and Laws of Suprema459
12.4Laws of Brownian Motion at Markov Times: Skorohod Imbedding469
12.5Laws of the Iterated Logarithm476
13Measurability: Borel Isomorphism and Analytic Sets487
13.1Borel Isomorphism487
13.2Analytic Sets493
App. AAxiomatic Set Theory503
App. BComplex Numbers, Vector Spaces, and Taylor's Theorem with Remainder521
App. CThe Problem of Measure526
App. DRearranging Sums of Nonnegative Terms528
App. EPathologies of Compact Nonmetric Spaces530
Author Index541
Subject Index546
Notation Index554

Read More

Customer Reviews

Average Review:

Write a Review

and post it to your social network

     

Most Helpful Customer Reviews

See all customer reviews >