[内容简介]This Anniversary Edition of Probability and Measure offers advanced students, scientists, and engineers an integrated introduction to measure theory and probability. Retaining intact the unique approach of the Third Edition, this text interweaves material on probability and measure, so that probability problems generate an interest in measure theory, which is then developed and applied to probability. Probability and Measure provides thorough coverage of probability, measure, integration, random variables and expected values, convergence of distributions, derivatives and conditional probability, stochastic processes, Brownian motion, and ergodic theory. The Anniversary Edition features a new, pedagogically sound interior design with an emphasis on open space.Like the previous editions, this Anniversary Edition will be well received by students of mathematics, statistics, economics, and a wide variety of disciplines that require a solid understanding of probability theory.
[目次]Chapter1 PROBABILITY 1
1. BOREL’S NORMAL NUMBER THEOREM, 1
2. PROBABILITY MEASURES, 18
3. EXISTENCE AND EXTENSION, 39
4. DENUMERABLE PROBABILITIES, 53
5. SIMPLE RANDOM VARIABLES, 72
6. THE LAW OF LARGE NUMBERS, 90
7. GAMBLING SYSTEMS, 98
8. MARKOV CHAINS, 117
9. LARGE DEVIATIONS AND THE LAW OF THE ITERATED LOGARITHM, 154
Chapter2 MEASURE 167
10. GENERAL MEASURES, 167
11. OUTER MEASURE, 174
12. MEASURES IN EUCLIDEAN SPACE, 181
13. MEASURABLE FUNCTIONS AND MAPPINGS, 192
14. DISTRIBUTION FUNCTIONS, 198
Chapter3 INTEGRATION 211
15. THE INTEGRAL, 211
16. PROPERTIES OF THE INTEGRAL, 218
17. THE INTEGRAL WITH RESPECT TO LEBESGUE MEASURE, 234
18. PRODUCT MEASURE AND FUBINI’S THEOREM, 245
19. THE Lp SPACES*, 256
Chapter4 RANDOM VARIABLES AND EXPECTED VALUES 271
20. RANDOM VARIABLES AND DISTRIBUTIONS, 271
21. EXPECTED VALUES, 291
22. SUMS OF INDEPENDENT RANDOM VARIABLES, 300
23. THE POISSON PROCESS, 316
24. THE ERGODIC THEOREM*, 330
Chapter5 CONVERGENCE OF DISTRIBUTIONS 349
25. WEAK CONVERGENCE, 349
26. CHARACTERISTIC FUNCTIONS, 365
27. THE CENTRAL LIMIT THEOREM, 380
28. INFINITELY DIVISIBLE DISTRIBUTIONS*, 394
29. LIMIT THEOREMS IN Rk, 402
30. THE METHOD OF MOMENTS*, 412
Chapter6 DERIVATIVES AND CONDITIONAL PROBABILITY 425
31. DERIVATIVES ON THE LINE*, 425
32. THE RADON–NIKODYM THEOREM, 446
33. CONDITIONAL PROBABILITY, 454
34. CONDITIONAL EXPECTATION, 472
35. MARTINGALES, 487
Chapter7 STOCHASTIC PROCESSES 513
36. KOLMOGOROV'S EXISTENCE THEOREM, 513
37. BROWNIAN MOTION, 530