There are two books by Kolmogorov & Fomin that I am interested in purchasing, namely Introductory Real Analysis and Elements of the Theory of Functions and Functional Analysis. Now, this will be my third book on graduate analysis; in particular, I have studied from Bartle’s Elements of Integration (which develops measure theory in an axiomatic/general sort of way) and a delightful little book called Introduction to Lebesgue Measure and Fourier Series by authors Wilcox and Myers (which develops the idea of Lebesgue measurable sets of the closed interval $[0,1]$, Lebesgue outer measure, and the Lebesgue integral). Next year, I will be starting graduate school; and, it appears that I will be using Royden or Rudin’s Real/Complex Analysis (but most likely Royden). As such, I am looking for one more text to tie everything together. So, I suppose my questions are as follows:
(1) Which of the two Kolmogorov & Fomin texts is better suitable for self-study?
(2) Which of the two Kolmogorov & Fomin texts contains topics most closely in line with the Royden text?
Thank you! Your input will be greatly appreciated. Note: I am also open to other suggestions assuming the cost of the text is low (or Frank Jones’s Lebesgue Measure on Euclidean Spaces would be an option).
Best Answer
Kolmogorov and Fomin wrote only one book. First it was published as two volumes under the name "Elements of the Theory of Functions and Functional Analysis", in 1954. This is probably the second text you are referring to. The first text you are talking about is the "translation" by Silverman of the second Russian edition. I write "translation", because the translator states: "The present course is a freely revised and restyled version of ... the Russian original...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..." So this is not a book by Kolmogorov and Fomin per se, and they never titled their work "Introductory real analysis". After that there were a third and fourth editions of this classical book in Russian, with a lot of material added. If you can read Russian, I would suggest to pick the latest edition. If not, my advice would be to choose another introductory text in analysis.