Unfortunately, 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.
Rating: 
Summary: Excellent intro to real analysis
Review: I find this a great introduction to real analysis. Contrary to what one reviewer has suggested, I think the book is fairly rigorous. It is true that some details are omitted, but they can always be filled up by the reader. In fact, this is the one of the most fun parts of reading the book!To give a concrete example: One reviewer has suggested that the theorem "Every infinite set has a countable subset" is proved without stating that the axiom of choice is required. This is certainly a serious lapse of rigour, BUT, in a later page, the author explains the axiom of choice (and several equivalent assertions) and also touches upon the fact that there are some very deep set theoretic questions, not yet fully resolved, concerning this axiom. He goes on to say "The axiom of choice will be assumed in this book. In fact, without it, we will be severely hampered for making various set-theoretic constructions". It is evident that the above theorem is one such construction.
This book emphasizes an intuitive approach to the subject, something which in my opinion is neglected by far too many books. Rigour is necessary but never sufficient to acheive proficiency in math!
Rating:
Summary: Concise, Lucid, Thorough
Review: 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: 
Summary: Strong "introduction"
Review: Overall, this book is a very strong "introduction" (I use the word grudgingly, see below) to real analysis. Topics range from the basics of set theory through metrics, operators, and Lebesgue measures and integrals. Particularly well done are the section on linear maps and operators, which include excellent generalizations as well as the usual concrete examples. The book usually includes a large number of examples and exercises on each topic which aid in the understanding of the material (though in a few instances, most notably the introduction to measure, it would have been more helpful to have examples as the theory was being developed instead of spending 20 pages getting through the theorems and only then giving a few examples).
The main problem for this book, however, is that it is located at an awkward level in terms of its assumptions of what students have seen before. Most of the material covered is that of a first analysis course, and the book is probably usually used as such. The authors, however, sometimes make assumptions that students have had exposure to some of the concepts before, claiming that "the reader has probably already encountered the familiar Heine-Borel theorem", for example. One particularly annoying case was when the authors gave as an example that the set of polynomials with rational coefficients is dense in the set of continuous functions, and left it at that. Are we supposed to have encountered Weierstrass's theorem before we take our first analysis course?
Rating:
Summary: Strong "introduction"
Review: Overall, this book is a very strong "introduction" (I use the word grudgingly, see below) to real analysis. Topics range from the basics of set theory through metrics, operators, and Lebesgue measures and integrals. Particularly well done are the section on linear maps and operators, which include excellent generalizations as well as the usual concrete examples. The book usually includes a large number of examples and exercises on each topic which aid in the understanding of the material (though in a few instances, most notably the introduction to measure, it would have been more helpful to have examples as the theory was being developed instead of spending 20 pages getting through the theorems and only then giving a few examples).
The main problem for this book, however, is that it is located at an awkward level in terms of its assumptions of what students have seen before. Most of the material covered is that of a first analysis course, and the book is probably usually used as such. The authors, however, sometimes make assumptions that students have had exposure to some of the concepts before, claiming that "the reader has probably already encountered the familiar Heine-Borel theorem", for example. One particularly annoying case was when the authors gave as an example that the set of polynomials with rational coefficients is dense in the set of continuous functions, and left it at that. Are we supposed to have encountered Weierstrass's theorem before we take our first analysis course?
Rating:
Summary: Highly Motivated
Review: This is a most beautiful exposition of Analysis going upto graduate level! The wonderful thing about the book is the examples, examples, examples! Every definition and many of the theorems are followed by concrete examples, many of them closely related to familiar notions such as the real line or R^n. He begins measure theory by constrution of measures on plane sets, then proceeds to generalize, one example of the conrcrete approach in the book. Kolmogorov also provided us with the axiomatics of Functional Analysis in 3 clear chapters.
I heartily recommend this book as a stop-over before ,say, a study of Rudin's Real and Complex whose expostion (especially of measure theory) is as abstract as it is beautiful.
And then of course, enough cannot be said about the price.....!
Rating:
Summary: Really nice book for learning analysis on one's own
Review: This is one of the most-thumbed books on my shelves (a little less so than "medium Rudin"). I look here for facts about metric spaces, topological spaces, and the other "basic" topics that make up the beginning of the book.
Because it's Russian, it's probably only suitable for advanced undergraduates and above. Even though the price has gone up by half since I bought it ten years ago, I'd still recommend it!
Rating: 
Summary: Are all Silverman's "translations" like this one?
Review: This textbook has several major virtues: it is dirt cheap, it is concise, and it touches on many advanced topics. Unfortunately, it has equally major flaws.
Many of the "proofs," especially in the first few chapters, are simply vague outlines of proofs. New notation is introduced without formal definition, terminology is used sloppily (sometimes even inaccurately), and explanations are invariably terse.
Before reading each chapter, I found it was necessary to first consult a more down-to-earth text. Sometimes I got the impression that the authors were more interested in showing off their brilliance than teaching me about analysis.
If you want to learn analysis, I would recommend first working through Rudin's Principles of Mathematical Analysis, then using this book as a source of challenging problems and interesting remarks.
Rating:
Summary: Very readable introduction by two eminent mathematicians
Review: 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.
