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I am a Phd student in Computer Science and I want to focus on Machine Learning, especially on statistical methods. My problem is, I always keep hitting the wall when it comes to studying underlying theories in more detail, since these mostly include Measure Theory, Lebesgue Measures and Lebesgue Integration, due to their probabilistic nature.

My problem is, I was mostly educated as a "coder" (Bs and Msc in Computer Science as well), not as a mathematician. Unfortunately I am still relying on my sloppy background which I obtained in the superficial Calculus courses I had taken years ago. Clearly, I am in need of building at least a working knowledge about Real Analysis, which covers topics like Lebesgue Theory, Multivariable Calculus, rigorous definitions of limits, derivation, integration etc. Since I am little bit late to study all of these, I am not able to spend my little available time with books having hundreds and hundreds of pages, it wouldn't be possible.

So, I am in need of an advice on a good and brief (as possible) Real Analysis book, which has the beginners as its target audience, as possible, so I can study it chapter by chapter from start to finish and can build myself a working knowledge of Lebesgue Integration, Measure Theory,etc. I read somewhere that Rudin's book is good but is too much detailed. By the way, I am also open to any advice on how to study Real Analysis in an efficient way, by myself.

Thanks in advance.

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  • $\begingroup$ You might want to refer to the answers to this related MSE question. $\endgroup$ – Jose Arnaldo Bebita-Dris Sep 5 '14 at 13:15
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    $\begingroup$ I recommend Royden and Fitzpatrick. However, for the purposes you've described, I think you are best off not working straight through the whole thing. Instead I think you are better off getting some intuition from the first section and then jumping to the section on abstract measure theory. Note that, in the abstract section, they occasionally omit proofs which were included in the concrete section. These results are usually word-for-word modifications of the concrete case, so it is easy enough to just look back at the concrete proof to understand the abstract proof. $\endgroup$ – Ian Sep 5 '14 at 13:52
  • $\begingroup$ How about Rudin's book? I heard that it builds up from the very basic definitons about real numbers, covers all basic topics about limits, derivatives, Riemann Integration up to the Lebesgue Theory. Is it an appropriate source for self study? $\endgroup$ – Ufuk Can Bicici Sep 5 '14 at 14:25
  • $\begingroup$ Rudin is a very difficult book. I don't know that I would recommend it to someone with little math background for self study. Rudin is very good for single variable analysis. For multivariable and lebesgue theory it is probably not the best. $\endgroup$ – Seth Sep 5 '14 at 14:43
  • $\begingroup$ Most of the books I examined either jumps right into the advanced topics of Measure Theory or they limit themselves with Riemann Integration theory without touching Lebesgue theory. I see that among the ones I looked to, Rudin's one is the only book which iteratively expands from very basics like basic topology and limits to Lebesgue theory. I do not require (or able) to master the content with all its details so maybe a book like Rudin's, only more easier, may be the one for me. $\endgroup$ – Ufuk Can Bicici Sep 5 '14 at 15:43
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I'm afraid Real Analysis just isn't a "beginner" topic--unless you define "beginner" as someone who has completed advanced calculus. You need more background than the superficial introductory calculus courses, particularly if multi-variable calculus is a problem for you.

If you've completed Calculus 1-3 at most universities (including multi-variable calculus), you can get a taste of measure theory at the very end of Rudin, Principles of Mathematical Analysis (known as baby Rudin). If you've worked through that book cover to cover, you can get a solid understanding of measure theory with just part 1 of Royden and Fitzpatrick, Real Analysis.

There are no shortcuts if you want to have a solid understanding of the material. That's just the sequence. Otherwise, I'd say use the same books, but just skim and "believe" each theorem, but at least try to understand what each theorem is saying. If the highest level math that you've taken is intro calculus, I have my doubts that baby Rudin will be accessible, but that is at least theoretically possible.

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