Probability & Measure Theory, Second Edition

Contents......Page 4
Preface......Page 8
Summary of Notation......Page 10
1.1 Introduction......Page 15
1.2 Fields, u-Fields, and Measures......Page 17
1.3 Extension of Measures......Page 26
1.4 Lebesgue-Stieltjes Measures and Distribution Functions......Page 36
1.5 Measurable Functions and Integration......Page 49
1.6 Basic Integration Theorems......Page 59
1.7 Comparison of Lebesgue and Riemann Integrals......Page 69
2.1 Introduction......Page 74
2.2 Radon-Nikodym Theorem and Related Results......Page 78
2.3 Applications to Real Analysis......Page 86
2.4 LP Spaces......Page 97
2.5 Convergence of Sequences of Measurable Functions......Page 110
2.6 Product Measures and Fubini' s Theorem......Page 115
2.7 Measures on Infinite Product Spaces......Page 127
2.8 Weak Convergence of Measures......Page 135
2.9 References......Page 139
3.1 Introduction......Page 141
3.2 Basic Properties of Hilbert Spaces......Page 144
3.3 Linear Operators on Normed Linear Spaces......Page 155
3.4 Basic Theorems of Functional Analysis......Page 166
3.5 References......Page 179
4.1 Introduction......Page 180
4.3 Independence......Page 181
4.4 Bernoulli Trials......Page 184
4.5 Conditional Probability......Page 185
4.6 Random Variables......Page 187
4.7 Random Vectors......Page 190
4.8 Independent Random Variables......Page 192
4.9 Some Examples from Basic Probability......Page 195
4.10 Expectation......Page 202
4.11 Infinite Sequences of Random Variables......Page 210
4.12 References......Page 214
5.1 Introduction......Page 215
5.2 Applications......Page 216
5.3 The General Concept of Conditional Probability and Expectation......Page 218
5.4 Conditional Expectation Given au-Field......Page 229
5.5 Properties of Conditional Expectation......Page 234
5.6 Regular Conditional Probabilities......Page 242
6.1 Introduction......Page 249
6.2 Convergence Theorems......Page 253
6.3 Martingales......Page 262
6.4 Martingale Convergence Theorems......Page 271
6.5 Uniform Integrability......Page 276
6.6 Uniform Integrability and Martingale Theory......Page 280
6.7 Optional Sampling Theorems......Page 284
6.8 Applications of Martingale Theory......Page 291
6.9 Applications to Markov Chains......Page 299
6.10 References......Page 302
7.1 Introduction......Page 304
7.2 The Fundamental Weak Compactness Theorem......Page 314
7.3 Convergence to a Normal Distribution......Page 321
7.4 Stable Distributions......Page 331
7.5 Infinitely Divisible Distributions......Page 334
7.6 Uniform Convergence in the Central Limit Theorem......Page 343
7.7 The Skorokhod Construction and Other Convergence Theorems......Page 346
7.8 The k-Dimensional Central Limit Theorem......Page 350
7.9 References......Page 358
8.1 Introduction......Page 359
8.2 Ergodicity and Mixing......Page 364
8.3 The Pointwise Ergodic Theorem......Page 370
8.4 Applications to Markov Chains......Page 382
8.5 The Shannon-McMillan Theorem......Page 388
8.6 Entropy of a Transformation......Page 400
8.7 Bernoulli Shifts......Page 408
8.8 References......Page 411
9.1 Stochastic Processes......Page 413
9.2 Brownian Motion......Page 415
9.3 Nowhere Differentiability and Quadratic Variation of Paths......Page 422
9.4 Law of the Iterated Logarithm......Page 424
9.5 The Markov Property......Page 428
9.6 Martingales......Page 434
9.7 Ito Integrals......Page 440
9.8 Ito's Differentiation Formula......Page 446
9.9 References......Page 451
1. The Symmetric Random Walk in R^2......Page 452
2. Semicontinuous Functions......Page 455
3. Completion of the Proof of Theorem 7.3.2......Page 457
4. Proof of the Convergence of Types Theorem 7.3.4......Page 461
5. The Multivariate Normal Distribution......Page 463
Bibliography......Page 468
Solutions to Problems......Page 470
Index......Page 526