# Measure Theory and Probability

Springer Science & Business Media, Jan 26, 1996 - Mathematics - 206 pages

Measure 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

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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 References 202 Index 203 Copyright