Probability Measures on Metric Spaces by K. R. Parthasarathy (PDF)

Description

Probability Measures on Metric Spaces presents the general theory of probability measures in abstract metric spaces. This book deals with complete separable metric groups, locally impact abelian groups, Hilbert spaces, and the spaces of continuous functions.Organized into seven chapters, this book begins with an overview of isomorphism theorem, which states that two Borel subsets of complete separable metric spaces are isomorphic if and only if they have the same cardinality. This text then deals with properties such as tightness, regularity, and perfectness of measures defined on metric spaces. Other chapters consider the arithmetic of probability distributions in topological groups. This book discusses as well the proofs of the classical extension theorems and existence of conditional and regular conditional probabilities in standard Borel spaces. The final chapter deals with the compactness criteria for sets of probability measures and their applications to testing statistical hypotheses.This book is a valuable resource for statisticians.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I am a topologist and make use of compactifications all the time. It is well known that elements of compactifications can be identified with measures. This book has been an excellent resource for me to develop my measure theoretic understanding of completely regular spaces and their compactifications. The book has a great deal more than just what I needed. There is also a great introduction to some of the Ergodic theorems as well as a nice introduction to some general probability theory.I also think that this book would be a good companion to anyone studying measure theory for the first time. I do not think it could serve as a standalone text for such an endeavor as it is written with a different goal in mind(as the title suggests)-the book is self contained and the proofs are easy to read.

⭐Parthasarathy’s book gives proofs of the following results, and for each result the presentation is better than in most books: the Glivenko-Cantelli theorem, the portmanteau theorem, Prokhorov’s theorem, the Lévy–Khinchin formula, the Kolmogorov consistency theorem, the Kolmogorov continuity theorem, and Donsker’s theorem. I particularly like the chapter on Borel probability measures on C[0,1]: rather than speaking about continuous stochastic processes with index set [0,1], we can speak about Borel probability measures on C[0,1].

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