Probability and Measure / Edition 4

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Intertwines measure theory and modern probability: probability problems generate an interest in measure theory and measure theory is then developed and applied to probability. Illustrates the connections probability theory has with applied mathematics on the one hand and with pure mathematics on the other.
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Editorial Reviews

A text that offers advanced students, scientists, and engineers an integrated introduction to measure theory and probability. The coverage extends to probability, measure, integration, random variables and expected values, convergence of distributions, derivatives and conditional probability, and stochastic processes. This edition features an improved version of Brownian motion and the replacement of queuing theory with ergodic theory. Annotation c. Book News, Inc., Portland, OR (
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Product Details

  • ISBN-13: 9781118122372
  • Publisher: Wiley, John & Sons, Incorporated
  • Publication date: 2/28/2012
  • Series: Wiley Series in Probability and Statistics Series , #894
  • Edition description: Anniversary Edition
  • Edition number: 4
  • Pages: 656
  • Product dimensions: 7.10 (w) x 10.10 (h) x 1.50 (d)

Meet the Author

PATRICK BILLINGSLEY is Professor of Statistics and Mathematics at the University of Chicago. He is the coauthor (with Watson et al.) of Statistics for Management and Economics; (with D. L. Huntsberger) of Elements of Statistical Inference; and the author of Convergence of Probability Measures (Wiley-Interscience), among other works. Dr. Billingsley has also edited the Annals of Probability for the Institute of Mathematical Statistics. He received his PhD in mathematics from Princeton University.
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Table of Contents

Ch. 1 Probability 1
1 Borel's Normal Number Theorem 1
2 Probability Measures 17
3 Existence and Extension 36
4 Denumerable Probabilities 51
5 Simple Random Variables 67
6 The Law of Large Numbers 85
7 Gambling Systems 92
8 Markov Chains 111
9 Large Deviations and the Law of the Iterated Logarithm 145
Ch. 2 Measure 158
10 General Measures 158
11 Outer Measure 165
12 Measures in Euclidean Space 171
13 Measurable Functions and Mappings 182
14 Distribution Functions 187
Ch. 3 Integration 199
15 The Integral 199
16 Properties of the Integral 206
17 The Integral with Respect to Lebesgue Measure 221
18 Product Measure and Fubini's Theorem 231
19 The L[superscript p] Spaces 241
Ch. 4 Random Variables and Expected Values 254
20 Random Variables and Distributions 254
21 Expected Values 273
22 Sums of Independent Random Variables 282
23 The Poisson Process 297
24 The Ergodic Theorem 310
Ch. 5 Convergence of Distributions 327
25 Weak Convergence 327
26 Characteristic Functions 342
27 The Central Limit Theorem 357
28 Infinitely Divisible Distributions 371
29 Limit Theorems in R[superscript k] 378
30 The Method of Moments 388
Ch. 6 Derivatives and Conditional Probability 400
31 Derivatives on the Line 400
32 The Radon-Nikodym Theorem 419
33 Conditional Probability 427
34 Conditional Expectation 445
35 Martingales 458
Ch. 7 Stochastic Processes 482
36 Kolmogorov's Existence Theorem 482
37 Brownian Motion 498
38 Nondenumerable Probabilities 526
Appendix 536
Notes on the Problems 552
Bibliography 581
List of Symbols 585
Index 587
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