## An Introduction to Measure-theoretic ProbabilityThis book provides in a concise, yet detailed way, the bulk of the probabilistic tools that a student working toward an advanced degree in statistics, probability and other related areas, should be equipped with. The approach is classical, avoiding the use of mathematical tools not necessary for carrying out the discussions. All proofs are presented in full detail. * Excellent exposition marked by a clear, coherent and logical devleopment of the subject * Easy to understand, detailed discussion of material * Complete proofs |

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### Contents

1 | |

29 | |

CHAPTER 3 Some Modes of Convergence of Sequences of Random Variables and Their Relationships | 55 |

CHAPTER 4 The Integral of a Random Variable and Its Basic Properties | 71 |

Standard Convergence Theorems the Fubini Theorem | 89 |

Standard Moment and Probability Inequalities Convergence in the rth Mean and Its Implications | 119 |

CHAPTER 7 The HahnJordan Decomposition Theorem the Lebesgue Decomposition Theorem and the RadonNikodym Theorem | 147 |

CHAPTER 8 Distribution Functions and Their Basic Properties HellyBray Type Results | 167 |

CHAPTER 11 Topics from the Theory of Characteristic Functions | 235 |

The Centered Case | 289 |

The Central Limit Problem The Noncentered Case | 325 |

Topics from Sequences of Independent Random Variables | 345 |

Topics from Ergodic Theory | 383 |

Appendix | 421 |

Selected References | 431 |

433 | |

CHAPTER 9 Conditional Expectation and Conditional Probability and Related Properties and Results | 187 |

CHAPTER 10 Independence | 217 |

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### Common terms and phrases

0-ﬁnite measure A G A assume assumption B Q A B-measurable Borel ch.f Chapter 12 concept conditional expectation continuous convergence in distribution convergence in measure converges mutually Corollary countable deﬁned deﬁned on Q Deﬁnition denoted Dominated Convergence Theorem Exercise exists fact ﬁeld ﬁnite ﬁrst ﬁxed follows hence Hint holds ifand implies independent r.v.s inequality inﬁnite integrable intervals Large Numbers Lebesgue measure Lemma Let F Let the r.v.s lim inf lim sup measurable space measure-preserving monotone class Monotone Convergence Theorem nondecreasing nonnegative outer measure partition probability measure probability space Proof Let proof of Theorem Proposition r.v.s deﬁned r.v.s Xn random variables relation Remark respect result rth mean sequence set function simple r.v.s space Q stationary process subsets sufﬁces to show sufﬁcient conditions transformation true uniformly uniquely