Probability and Measure TheoryProbability and Measure Theory, Second Edition, is a text for a graduate-level course in probability that includes essential background topics in analysis. It provides extensive coverage of conditional probability and expectation, strong laws of large numbers, martingale theory, the central limit theorem, ergodic theory, and Brownian motion.
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Contents
Fundamentals of Measure and Integration Theory | 1 |
Further Results in Measure and Integration Theory | 60 |
Introduction to Functional Analysis | 127 |
Basic Concepts of Probability | 166 |
Conditional Probability and Expectation | 201 |
Strong Laws of Large Numbers and Martingale Theory | 235 |
The Central Limit Theorem | 290 |
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Common terms and phrases
absolutely continuous arbitrary assume B₁ B₂ Borel measurable Borel measurable function Borel sets bounded Brownian motion characteristic function conditional convergence theorem converges a.e. countably additive defined definition denoted density dF(x distribution function dominated convergence theorem ergodic example exists finite measure Fn(x function F given h₂ hence Hilbert space hypothesis implies independent random variables inequality intervals large numbers Lebesgue measure Lebesgue-Stieltjes measure Lemma Let f Let X1 lim inf lim sup Markov martingale measurable function measurable rectangles measure space measure-preserving transformation nonnegative o-field o-finite obtain p(Xo P₁ pointwise probability measure probability space Problem PROOF prove real numbers real-valued result follows right-continuous right-semiclosed set function simple functions submartingale subset subspace supermartingale uniformly integrable vector space w₁ X₁ Xn+1 Y₁