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42 of 42 people found the following review helpful:
5.0 out of 5 stars
The best introduction to probability and measure, February 8, 2005
This review is from: Probability & Measure Theory, Second Edition (Hardcover)
The book very nicely develops the basics of measure theory from a probability perspective (e.g. includes Caratheodory extension theorem, Lebesgue-Stieltjes measures, weak convergence and Kolmogorov extension theorem). It then gives a brief introduction to functional analysis and proceeds to probability theory, martingales and concludes with brownian motion and stochastic integration.
All standard results are given and the book is self-contained. It is a concise, yet readable introduction to this area (less concise then Rudin, Williams but more than Billingsly). An excellent feature of this book is that full solutions to some of the exercises are provided at the end. This makes this book ideal for self-study. The only prerequisite for this book is elementary real analysis (say chapters 1-7 of Rudin's principles of mathematical analysis).
There are other excellent books on measure theory (Rudin, Royden), but if you are interested in measure theory from a probabilistic view this is the book to choose.
As far as a probability textbook, it is clearer and more readable than Billingsly, Chung, Williams and Durrett.
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34 of 34 people found the following review helpful:
5.0 out of 5 stars
Exceptionally Clear, July 9, 2002
By A Customer
This review is from: Probability & Measure Theory, Second Edition (Hardcover)
I first used this text in the earlier version, which comprises the first half of the book, in a one-year course in Hilbert Spaces and Lebesgue Measure theory when in the first year of grad school. The material is presented in a clearly written manner and the exposition is some of the clearest mathematical writing I've seen in a subject which is replete with textbooks. Anyone who wants to be inaugurated into the "mysteries" of measure theory and the fine points of the rigorous theory of stochastic processes and the Ito integral, will do himself or herself a favor by using this text. If it is not assigned to your class and you have the extra cash, order it anyway. It is also well-suited for self-study.
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20 of 20 people found the following review helpful:
5.0 out of 5 stars
Excellent and rigurous, February 28, 2007
This review is from: Probability & Measure Theory, Second Edition (Hardcover)
This book deal with the whole picture of probability. One learns the very first roots of rigurous probability. And when I say rigurous I am not regarding it as "engineers rigour = nothing" but as "mathematicians rigour". The book is self-contained, the exposition is clear and is organized in the mathematic classical fashion: definition, lema, proof, theorem, proof.
That rigour, when it comes to probability beyond "number of successful cases / total number of cases", can only be achieved when the theory is developed in the most general background: measure theory. This gives general tools (theorems) which are applied to measures in general, a particular case of which is probability. Measure theory and general abstract Lebesgue integration go together, so the book defines and deepens in Lebesgue theory: integration, convergence theorems, Fubini's theorem, etc.
Once you feel confident and capable of deal with almost anything regarding Lebesgue integration the books moves on relations between integrals and measures: the Radon-Nikodym theorem which is perhaps one of the most important results of the book and whose proof is outstanding. It provides the reader with the tools to tackle Lebesgue almost everywhere differentiation theorem and absolutely continuous measures and functions.
Finally, before starting with probability as special case, there is a functional analysis chapter which gives proof of the three most important theorems of functional analisys in Hilbert and Banach spaces.
From chapter 4 on, everything about probability is covered. From basic distributions to martingales, ergodicity or central limit theoroem. But instead of making up ad-hoc theorems, theorems proved for measures in first chapters renders the proofs in this stage simply colloralies.
Once you read the book you will feel confident about anything touching probability, measure theory and Lebesgue integration and equipped with the most fundamental tools of functional analysis which are used widespread.
I couldn't recommend the book more.
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