# Measure Theory and Probability

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Measure theory and integration are presented to undergraduates from the perspective of probability theory. In fact, discrete probability theory is taught at many institutions as a freshman course. The early chapters, going under the rubric of the law of large numbers, show why measure theory is needed for the formulation of problems in probability, and explain 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 Rn (metric space approach), the Borel-Cantelli lemmas, straight measure theory (the Lebesgue integral). In this concise text, a number of applications to probability are packed into the exercises. Contents: Chapter1 Measure Theory Introduction Randomness Measure Theory Measure Theoretic Modeling Chapter 2 Integration Measurable functions The Lebesgue Integral Further Properties of the Integral; Convergence Theorems Lebesgue integration versus Riemann Integration Fubini Theorem Random Variables, Expectations Values, and Independence The Law of Large Numbers The Discrete Dirichlet Problem Chapter 3 Fourier Analysis L 1 Theory L 2 Theory The Geometry of Hilbert Space Fourier Series The Fourier Integral Some Applications of Fourier Series to Probability Theory An Application of Probability The Central Limit Theorem Appendix A, Metric Spaces Appendix B, On L p Matters References IndexIn this concise text, a number of applications to probability are packed into the exercises.

 Title : Measure Theory and Probability Author : Malcolm Adams, Victor Guillemin Publisher : Springer Science & Business Media - 1996