概率與測(cè)度論（英文版·第2版）

1　Fundamentals of Measure and Integration Theory　1
1.1　Introduction　1
1.2　Fields, σ-Fields, and Measures　3
1.3　Extension of Measures　12
1.4　Lebesgue-Stieltjes Measures and Distribution Functions　22
1.5　Measurable Functions and Integration　35
1.6　Basic Integration Theorems　45
1.7　Comparison of Lebesgue and Riemann Integrals　55
2　Further Results in Measure and Integration Theory　60
2.1　Introduction　60
2.2　Radon-Nikodym Theorem and Related Results　64
2.3　Applications to Real Analysis　72
2.4　LP Spaces　83
2.5　Convergence of Sequences of Measurable Functions　96
2.6　Product Measures and Fubini's Theorem　101
2.7　Measures on Infinite Product Spaces　113
2.8　Weak Convergence of Measures　121
2.9　References　125
3　Introduction to Functional Analysis　127
3.1　Introduction　127
3.2　Basic Properties of Hilbert Spaces　130
3.3　Linear Operators on Normed Linear Spaces　141
3.4　Basic Theorems of Functional Analysis　152
3.5　References　165
4　Basic Concepts of Probability　166
4.1　Introduction　166
4.2　Discrete Probability Spaces　167
4.3　Independence　167
4.4　Bernoulli Trials　170
4.5　Conditional Probability　171
4.6　Random Variables　173
4.7　Random Vectors　176
4.8　Independent Random Variables　178
4.9　Some Examples from Basic Probability　181
4.10　Expectation　188
4.11　Infinite Sequences of Random Variables　196
4.12　References　200
5　Conditional Probability and Expectation　201
5.1　Introduction　201
5.2　Applications　202
5.3　The General Concept of Conditional Probability and Expectation　204
5.4　Conditional Expectation Given a σ-Fields　215
5.5　Properties of Conditional Expectation　220
5.6　Regular Conditional Probabilities　228
6　Strong Laws of Large Numbers and Martingale Theory　235
6.1　Introduction　235
6.2　Convergence Theorems　239
6.3　Martingales　248
6.4　Martingale Convergence Theorems　257
6.5　Uniform Integrability　262
6.6　Uniform Integrability and Martingale Theory　266
6.7　Optional Sampling Theorems　270
6.8　Applications of Martingale Theory　277
6.9　Applications to Markov Chains　285
6.10　References　288
7　The Central Limit Theorem　290
7.1　Introduction　290
7.2　The Fundamental Weak Compactness Theorem　300
7.3　Convergence to a Normal Distribution　307
7.4　Stable Distributions　317
7.5　Infinitely Divisible Distributions　320
7.6　Uniform Convergence in the Central Limit Theorem　329
7.7　The Skorokhod Construction and Other Convergence Theorems　332
7.8　The k-Dimensional Central Limit Theorem　336
7.9　References　344
8　Ergodic Theory　345
8.1　Introduction　345
8.2　Ergodicity and Mixing　350
8.3　The Pointwise Ergodic Theorem　356
8.4　Applications to Markov Chains　368
8.5　The Shannon-McMillan Theorem　374
8.6　Entropy of a Transformation　386
8.7　Bernoulli Shifts　394
8.8　References　397
9　Brownian Motion and Stochastic Integrals　399
9.1　Stochastic Processes　399
9.2　Brownian Motion　401
9.3　Nowhere Differentiability and Quadratic Variation of Paths 408
9.4　Law of the Iterated Logarithm　410
9.5　The Markov Property　414
9.6　Martingales　420
9.7　It? Integrals　426
9.8　It? s Differentiation Formula　432
9.9　References　437
Appendices　438
1. The Symmetric Random Walk in Rk　438
2. Semicontinuous Functions　441
3. Completion of the Proof of Theorem 7.3.2　443
4. Proof of the Convergence of Types Theorem 7.3.4　447
5. The Multivariate Normal Distribution　449
Bibliography　454
Solutions to Problems　456
Index　512