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I am currently planning to get a book on Real Analysis for self studying before diving into my 4th year real analysis course. The standard textbook for my 4th year course is Stein's Measure, but I do not like much about abstract measure introduced near the end. Perhaps because I am currently taking 3rd year real analysis course in the level of Pugh with some other additional materials.

Anyway, I am considering one of the followings: Folland - Real Analysis, Bruckner, Bruckner, Thomson - Real Analysis, Yeh - Real Analysis, Kantorovitz - Introduction to Modern Analysis (and maybe Cohn - Measure Theory)

(Note: Royden is omitted because I am waiting for 2nd printing and waiting so that I can get it cheap from some website (like abebooks), so 12 pages of erratas are all fixed)

Which book do you think is most suitable for self-study? (My 4th year course is cross-listed, meaning it is equivalent to first year graduate real analysis course)

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It depends a lot in which country you are in - here in Brazil we have a couple of good books, but of course they are in portuguese so I guess it won't help you. – Marra Mar 27 '12 at 1:19
I live in Canada, and I try to find good book to study to. – chhan92 Mar 30 '12 at 2:05

If you want an interesting alternative that goes deep into why things work out as they do in real analysis, especially things like sound and convincing explanations of Lebesgue Measure, then take a look at Terence Tao's 2-volume Analysis textbook.

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+1 for a really good and underrated analysis text. But to be honest,Tao gives a much more detailed presentation of Lebesgue and abstract measure theory in his sequel, An Introduction to Measure Theory,published by the AMS and fully described at Indeed-the 4 textbooks together can serve as the basis for a remarkable 3 year sequence in real analysis;from third year undergraduate all the way up to the graduate qualifying exam in analysis for second year grad students. – Mathemagician1234 Mar 27 '12 at 4:41
Tao is indeed excellent and underrated. Hmm. It is funny that we are all always taking about books rather than teachers. I've become very skeptical of University level mathematics courses where most professors cannot adequately teach the material, or else, have no real way to allow students to "check-in" to see what they have learned. The mechanisms for learning are deficient not only at the elementary and secondary school levels but at the Uni level as well. So, are all really good mathematicians then simply self-taught? If so, why not have universities offer 30 semester hours of self-study? – Kurt Lovelace Mar 27 '12 at 18:50
I like Tao's Book, I will definitely check this one out along my other possible lists. – chhan92 Mar 30 '12 at 2:13
@KurtLovelace They don't get paid to teach,hence the culture at most universities-especially research based ones-is "those that can't,teach." Which is not only elitist and utterly self serving,but utterly false,as many good teachers at prominent universities prove. This shallowly egoist environment is one of the reasons US university scientific training is towards the bottom of the pile compared with other nations. It's sad because it really costs nothing to encourage teaching and it's simply a choice based on callousness. – Mathemagician1234 Nov 30 '14 at 18:10

One of the very best books on analysis, which also contains so much more then just measure and integration theory,is available very cheap from Dover Books: General Theory Of Functions And Integration by Angus Taylor. You can probably get a used copy for 2 bucks or less and it contains everything you ever wanted to know about not only measure and integration theory, but point set topology on Euclidean spaces. It also has some of the best exercises I've ever seen and all come with fantastic hints. This is my favorite book on analysis and I think you'll find it immensely helpful for not only integration theory, but a whole lot more.I don't know if it's available where you live,but you can probably get it incredibly cheap online. I strongly encourage you to find a copy-I think you'll find it immensely helpful. I think this book should be mandatory summer reading before a graduate analysis course based on Big Rudin- I think students would find Rudin far less difficult after spending a few weeks working through Taylor's classic.

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How about Walter Rudin's Real and Complex Analysis and Theory of Functions by Edward Charles Titchmarsh. These two books have rather different styles, both cover measure theory. The first is rather abstract and the later is written with the old school hard analysis style. I do highly recommend the later, the first one is a best seller.

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I recommend Klambauer's Real Analysis from Dover especially for its clear presentation of the Lebesgue measure and integral. It uses Royden's strategy of working on the real line first, generalizing later.

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