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
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