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Title: Introductory Real Analysis by A. N. Kolmogorov, S. V. Fomin ISBN: 0-486-61226-0 Publisher: Dover Pubns Pub. Date: 01 June, 1975 Format: Paperback Volumes: 1 List Price(USD): $15.95

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Average Customer Rating: 4.36 (22 reviews)

Rating: 5
Summary: Very readable introduction by two eminent mathematicians
Comment: Years ago I used this book as a supplementary text for a course in functional analysis and measure theory. When I learned that it was being republished by Dover I immediately bought my own copy. It is a thoroughly readable book with lots of examples to illustrate concepts. The chapters on measure theory and the Lebesgue integral were exceptional. And the chapters on linear functionals and operators also very good. On the downside the introductory chapter on definitions of concepts like open and closed sets and the treatment of compactness and the Heine-Borel theorem could have been presented more clearly (I preferred Dieudonne's presentation in Foundations of Modern Analysis). I strongly recommend this book as excellent value for money.

Rating: 5
Summary: Concise, Lucid, Thorough
Comment: Is there anywhere a more logical, concise and lucid presentation of real analysis than Kolmogorov and Fomin's Introductory Real Analysis?

The material proceeds in such a beautiful order that I found myself, in a matter of days, going from set theory to linear functionals. The chapters on metric spaces and topological spaces were particularly great, with excellent problems. Kolmogorov was not only a great mathematician but also a great teacher and expositor, like many of the other great Russian mathematicians like Gelfand, Khinchin.

It's hard to believe that such a slim volume could provide a solid first course in real analysis. But it's so compact and well-priced that it should be snapped up quickly.

Rating: 3
Summary: Are all Silverman's "translations" like this one?
Comment: First, let us be precise in reviewing this book. It is NOT a book by Kolmogorov/Fomin, but rather an edited version by Silverman. So, if you read the first lines in the Editor's Preface, it states, "The present course is a freely revised and restyled version of ... the Russian original". Further down it continues, "...As in the other volumes of this series, I have not hesitated to make a number of pedagogical and mathematical improvements that occurred to me...". Read it as a big red warning flag. Alas, I would have to agree with the reader from Rio de Janeiro. I've been working through this book to rehash my knowledge of measure theory and Lebesgue integration as a prerequisite for stochastic calculus. And I've encountered many results of "mathematical improvements" that occurred to the esteemed "translator". Things are fine when topics/theorems are not too sophisticated (I guess not much room for "improvements"). Not so when you work through some more subtle proofs. Most mistakes I discovered are relatively easy to rectify (and I'm ignoring typos). But the latest is rather egregious. The proof of theorem 1 from ch. 9 (p.344-345) (about the Hahn decomposition induced on X by a signed measure F) contains such a blatant error, I am very hard pressed to believe it comes from the original. That book survived generations of math students at Moscow State, and believe me, they would go through each letter of the proofs. Astounded by such an obvious nonsense, I grabed the only other reference book on the subject I had at hand, "Measure Theory" by Halmos. The equivalent there is theorem A, sec. 29 (p.121 of Springer-Verlag edition), which has a correct proof.
For those interested in details, Silverman's proof states that positive integers are strictly ordered: k1Unfortunately, I don't have the Russian original. Instead, I'm trying to get the other, hopefully real translation, "Elements of the Theory of Functions and Functional Analysis". BTW, this is the actual title of the original, not "Introductory Real Analysis". Which apparently is causing significant confussion amoung past and present readers. To give you a background info, the Russian original is (or has been, at least) used as a textbook for a third-year subject for (hard-core) math students. Meaning, in the preceding two years they would complete a pre-requisite four-semester calculus course. For example, criteria of convergence of series and their properties is an assumed knowledge in presentation of Lebesgue integral. So, I think most of the critique from earlier reivews is a bit misdirected. The original book is a great starting book into functional analyis/Lebesgue integration and differentiation, but proofs require solid understanding of fundamentals of calculus.
The best part about Kolmogorov's text is the clarity of conceptual structure of the presented subject a reader would gain, if he/she puts some effort. You would gain a thorough understanding, not just a knowledge of the subject. There is quite a difference between the two, and not that many authors succeed in delivering that.
But to gain that from Kolmogorov, I would suggest the other, "unimproved" but real, translation.

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