So I decide to self-study the real analysis (measure theory, Banach space, etc.). Surprisingly, I found that Rudin-RCA is quite readable; it is less terse than his PMA. Although the required text for my introductory analysis course was PMA, I mostly studied from Hairer/Wanner's Analysis by Its History (I did not like PMA at all). Although I said readable, I do not know if I actually understand whole materials as I am middle of first chapter, and I already have topology background from Singer/Thorpe and Engelking. I actually like Rudin-RCA, but I am not sure if I am taking great risk as many experience people seem to not liking Rudin for learning...

Is Rudin-RCA suitable for a first introduction to the real analysis? Is it outdated? What should I know if I decide to study Rudin-RCA.

I am not planning to read the chapters in complex analysis as I am reading Barry Simon's excellent books in the complex analysis.


closed as primarily opinion-based by Aloizio Macedo, JMP, user91500, iadvd, Claude Leibovici Aug 23 '16 at 8:49

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ This is a bit hard to answer since it really depends on your maturity. What have you studied so far? Edit: No, the Green book is not at all an introductory text! $\endgroup$ – Faraad Armwood Aug 23 '16 at 5:21
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    $\begingroup$ Certainly if you are finding it readable then there is no reason not to read it. However, I think it would be beneficial to supplement your reading with other books. For example, Rudin's "construction" of Lebesgue measure via the Riesz representation theorm is slick, but surely it would be beneficial to see how almost everyone else does it. See, e.g. Stein's Real Analysis or Jones' Lebesgue Integration on Euclidean Space for more conventional treatments. Both are great books. $\endgroup$ – Bungo Aug 23 '16 at 6:02
  • $\begingroup$ @FaraadArmwood I studied Hairer/Wanner and Singer/Thorpe, and I am currently reading Engelking's General Topology along with Spanier's Algebraic Topology. $\endgroup$ – MathWanderer Aug 23 '16 at 13:41

No, No, and No. True, all the statements and proofs are squeaky clean, but the exposition, IMHO, is completely unsuitable for educational purposes.

Specifically for measure theory, I would recommend Vulikh's Brief course in the theory of functions of a real variable. For Banach spaces, Kolmogorov and Fomin's Introductory Real Analysis.

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    $\begingroup$ I agree totally! I personally like the old Royden and Halmos (specifically the integration section...it's perfect!). $\endgroup$ – Faraad Armwood Aug 23 '16 at 5:30
  • $\begingroup$ Halmos was a great expositor. However, his book on measure theory may be a bit too advanced for a beginner. Another nuance: Vulikh's book is the only one on measure theory that I know that has concise proofs of limits of measures of nested sets. He attains his concision by using $\sigma$-rings instead of $\sigma$-algebras, which also greatly helps him in his coverage of the Lebesgue measure on the Euclidean spaces. $\endgroup$ – avs Aug 23 '16 at 7:14
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    $\begingroup$ I apologize, but I am not longer up to date in the analysis world. I just remember studying for my prelim in my first year of graduate school and that section on integration was extremely simple and easy to understand. Up to that point I had only had 2 courses in analysis, so not quite a beginner but far from an expert :). $\endgroup$ – Faraad Armwood Aug 23 '16 at 7:17
  • $\begingroup$ For complex analysis, I would recommend none other than H. Cartan, and the problem book by Luntz, Volkovyskii and Aramanovich. $\endgroup$ – avs Oct 19 '16 at 17:19

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