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Probability and Measure Theory, Second Edition, is a text for a graduate-level course in probability that includes essential background topics in analysis. It provides extensive coverage of conditional probability and expectation, strong laws of large numbers, martingale theory, the central limit theorem, ergodic theory, and Brownian motion.
- Clear, readable style
- Solutions to many problems presented in text
- Solutions manual for instructors
- Material new to the second edition on ergodic theory, Brownian motion, and convergence theorems used in statistics
- No knowledge of general topology required, just basic analysis and metric spaces
- Efficient organization
- Sales Rank: #996054 in Books
- Published on: 1999-12-20
- Original language: English
- Number of items: 1
- Dimensions: 9.02" h x 1.13" w x 5.98" l, 2.11 pounds
- Binding: Hardcover
- 516 pages
Review
"There are numerous probability texts on the market, which makes choosing one difficult. If you are a financial professional who knows basic probability theory, but wants to take the next step in sophistication, this is the essential text. It introduces basic measure theory and functional analysis, and then delves into probability. The writing is clear and highly accessible. The choice of topics is perfect for financial engineers or financial risk managers: martingales, the inversion theorem, the central limit theorem, Brownian motion and stochastic integrals. I can't praise this book enough. It is exceptional!" --http://www.contingencyanalysis.com
About the Author
Robert B. Ash as written about, taught, or studied virtually every area of mathematics. His books include Information Theory, Topics in Stochastic Processes, The Calculus Tutoring Book, Introduction to Discrete Mathematics, and A Primer of Mathematics.
Most helpful customer reviews
70 of 71 people found the following review helpful.
The best introduction to probability and measure
By Kindle Customer
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.
50 of 51 people found the following review helpful.
Exceptionally Clear
By A Customer
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.
39 of 40 people found the following review helpful.
Excellent and rigurous
By Fernando Sanz Gamiz
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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