测度论与概率

English | PDF | 1996 | 217 Pages | ISBN : 0817638849 | 12.1 MB

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.

测度论与积分从概率学的角度向本科生介绍。第一章展示了为什么在概率问题的表述中需要引入测度论，解释了如果没有勒贝格理论的存在（不然的话），要应对大数定律中的悖论是迫不得已的事情。然后测度论的方法引出了一些有趣的结论，并涵盖了R[superscript n]上的勒贝格度量（使用度量空间方法）、博雷尔-康提利尔引理、直线上的测度论（勒贝格积分）。第三章进一步阐述了抽象傅里叶分析、傅里叶级数和傅里叶积分，它们在概率学中有许多美妙的应用：泊松定理关于随机游走、卡茨对塞格定理的证明以及中心极限定理。在这本简短的书中，很多应用到了概率的问题都被浓缩到练习题中了。“……这本书对它所考虑的主题是用户友好的，并且应该非常容易理解…教统计测度论课程的教师和学生会欣赏其中许多信息性的习题；对于大部分问题，都提供了有帮助的提示或解答线索。总而言之，这本教材应该是专业人士和学生的有用参考书。”—《美国统计协会杂志》。

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