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Real Analysis and Probability

av R. M. Dudley

MedlemmarRecensionerPopularitetGenomsnittligt betygDiskussioner
371505,829 (3.25)Ingen/inga
This classic textbook offers a clear exposition of modern probability theory and of the interplay between the properties of metric spaces and probability measures. The first half of the book gives an exposition of real analysis: basic set theory, general topology, measure theory, integration, an introduction to functional analysis in Banach and Hilbert spaces, convex sets and functions and measure on topological spaces. The second half introduces probability based on measure theory, including laws of large numbers, ergodic theorems, the central limit theorem, conditional expectations and martingale's convergence. A chapter on stochastic processes introduces Brownian motion and the Brownian bridge. The edition has been made even more self-contained than before; it now includes a foundation of the real number system and the Stone-Weierstrass theorem on uniform approximation in algebras of functions. Several other sections have been revised and improved, and the comprehensive historical notes have been further amplified. A number of new exercises have been added, together with hints for solution.… (mer)

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DISCLAIMER: I only read most of the non-starred sections and did only few of the exercises.

The first half of the book covers real analysis (i.e., topological and metric spaces, measure and integration theory); the second half then introduces basic concepts from probability (mainly the central limit theorem, conditional expectations, and brownian motion) using a lot of the abstract topological stuff from the first part. The advantage of this approach is that many theorems from probability can be stated in a more general form. The disadvantage is that the results become harder to understand. I suspect the author has a tendency towards useless over-generalization of results.

The book is written carefully with much attention to detail, but is often also very terse; I often had to spend 10 minutes to understand a single line. The reason may either be that the author has problems to put himself in the shoes of a person how has not worked on the subject for his entire life, or he wants to "encourage the reader to think"; in both cases it is just odd and a big waste of time, especially since often just a few more words or equations could have reduced the reading time vastly. Another thing that is very annoying, is that the author often changes symbols when there is no need to do so. For example, in a Definition he may call a set "X" only to call the same thing "S" a quarter page later in the theorem that makes use of the Definition.

While the book may be formally self contained, it seems quite inappropriate as a first introduction into the matter. For example, the author often simply seems to have forgotten to state some simple results. Also, I often had problems to look up a certain result and I believe that the reason for this is in mainly this tendency to over-generalize. I conclude from this that while the book contains a huge amount of knowledge one will not be able to make use of that knowledge three month after reading the book.

The examples are purely academic but they often elucidate critical issues of a previous theorem and I found them very helpful. A lot of exercises are provided. The few of them at which I looked seemed to strike a good balance between easy ones and hard ones. ( )
  Tobias.Bruell | Sep 21, 2014 |
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Wikipedia på engelska (5)

This classic textbook offers a clear exposition of modern probability theory and of the interplay between the properties of metric spaces and probability measures. The first half of the book gives an exposition of real analysis: basic set theory, general topology, measure theory, integration, an introduction to functional analysis in Banach and Hilbert spaces, convex sets and functions and measure on topological spaces. The second half introduces probability based on measure theory, including laws of large numbers, ergodic theorems, the central limit theorem, conditional expectations and martingale's convergence. A chapter on stochastic processes introduces Brownian motion and the Brownian bridge. The edition has been made even more self-contained than before; it now includes a foundation of the real number system and the Stone-Weierstrass theorem on uniform approximation in algebras of functions. Several other sections have been revised and improved, and the comprehensive historical notes have been further amplified. A number of new exercises have been added, together with hints for solution.

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