## 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.* Clear, readable style * Solutions to many problems presented in text * Solutions manual for instructors * Material new to the second edition on ergodic theory, Brownian motion, and convergence theorems used in statistics * No knowledge of general topology required, just basic analysis and metric spaces * Efficient organization |

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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 |

### Common terms and phrases

a-field absolutely continuous arbitrary assume Borel measurable function Borel sets bounded Brownian motion characteristic function complex-valued continuity points converges a.e. converges weakly countably additive cr-field defined definition denoted density disjoint sets distribution function dominated convergence theorem entropy ergodic example exists f(co finite measure finitely additive Fubini's theorem function F given h(co hence Hilbert space hypothesis implies increasing sequence independent random variables inequality infinitely divisible large numbers Lebesgue measure Lebesgue-Stieltjes measure Lemma Let F lim sup liminf limit linear operator martingale measurable rectangles measure space measure-preserving transformation nonnegative normed linear space obtain orthonormal pointwise positive integer probability measure probability space Problem Proof properties prove real numbers real-valued result follows right-continuous Section set function signed measure simple functions submartingale subspace supermartingale uniformly integrable vector space