## Measure Theory and ProbabilityMeasure theory and integration are presented to undergraduates from the perspective of probability theory. The first chapter shows why measure theory is needed for the formulation of problems in probability, and explains why one would have been forced to invent Lebesgue theory (had it not already existed) to contend with the paradoxes of large numbers. The measure-theoretic approach then leads to interesting applications and a range of topics that include the construction of the Lebesgue measure on R [superscript n] (metric space approach), the Borel-Cantelli lemmas, straight measure theory (the Lebesgue integral). Chapter 3 expands on abstract Fourier analysis, Fourier series and the Fourier integral, which have some beautiful probabilistic applications: Polya's theorem on random walks, Kac's proof of the Szegö theorem and the central limit theorem. In this concise text, quite a few applications to probability are packed into the exercises. "...the text is user friendly to the topics it considers and should be very accessible...Instructors and students of statistical measure theoretic courses will appreciate the numerous informative exercises; helpful hints or solution outlines are given with many of the problems. All in all, the text should make a useful reference for professionals and students."—The Journal of the American Statistical Association |

### What people are saying - Write a review

I'm a CS major and I wanted to gain some background in measure theory in order to understand the theory of how distributions and densities are formed. This book looked like a good, concise introduction to the topic, but it is mired with errors in some of the important proofs. Also, in some definitions of mathematical concepts, the authors seem to reuse the same symbols in different ways, or at least they omit some clarifications that would have made the book more complete. I feel as though I'm guessing at what was meant in some places, and wondering if we're still talking about the same set that we were two pages ago or not. Some of these ambiguities are not trivial and alter the meaning of subsequent concepts considerably. I'm only in section 1.3 but since there are other measure texts out there, and I will likely pursue one of those instead.

### Contents

Measure Theory | 1 |

12 Randomness | 14 |

13 Measure Theory | 24 |

14 Measure Theoretic Modeling | 42 |

Integration | 53 |

22 The Lebesgue Integral | 60 |

23 Further Properties of the Integral Convergence Theorems | 72 |

24 Lebesgue Integration versus Riemann Integration | 82 |

32 ℐ˛Theory | 124 |

33 The Geometry of Hilbert Space | 130 |

34 Fourier Series | 137 |

35 The Fourier Integral | 145 |

36 Some Applications of Fourier Series to Probability Theory | 156 |

37 An Application of Probability Theory to Fourier Series | 164 |

38 The Central Limit Theorem | 170 |

Metric Spaces | 178 |

25 Fubini Theorem | 89 |

26 Random Variables Expectation Values and Independence | 102 |

27 The Law of Large Numbers | 110 |

28 The Discrete Dirichlet Problem | 115 |

Fourier Analysis | 118 |

On ℐp Matters | 183 |

A NonMeasurable Subset of the Interval 01 | 199 |

202 | |

203 | |