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What are the best textbooks to introduce measure theory and Lebesgue integration to undergraduate math majors? Many students in such a class will go on to graduate school and be required to take a similar course in their first year. As such, is it better to focus on $\mathbb{R}^n$ or more abstract spaces?

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    $\begingroup$ Is such a question acceptable on this site? There is some precedent with the question asking for a Calculus textbook that uses infinitesimals. And on the topic of tags, I'm not sure which tags is more appropriate: "reference request" or "textbooks". Maybe they are both suitable. Also, should this question be tagged "real-analysis"? Plenty of questions have already been tagged "calculus", so it seems fitting. Also, should the "college" tag be used? Though it certainly applies, it doesn't seem particularly important for this question. $\endgroup$ – Gamma Function Mar 15 '14 at 3:29
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    $\begingroup$ Is using Henstock–Kurzweil integral an option? It's much easier to introduce (especially for people knowing Riemann integral), and you can get Lebesgue integral and measure basically for free. The "classical" approach has one drawback: you have to introduce quite a lot of notions to get even to the definition of the integral. (There are a few good introductory books on H–K integral.) That said, I learned Lebesgue integration from Rudin's "Real and Complex Analysis". It was great, since it it is very concise, so I got to the fun part very quickly.;) $\endgroup$ – mbork Apr 6 '14 at 22:05
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Frank Jones' Lebesgue Integration in Euclidean Space is possibly the best math book I have ever read. It goes over the Lebesgue integral in small steps over many sections before moving on to applications and properties.

It is appropriate for advanced undergrads and the exercises are excellent (I went through the book over the summer after my class ended and did every exercise because they helped me learn so well).

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First, I highly recommend David Bressoud's book A Radical Approach to Lebesgue's Theory of Integration for this. It's highly motivated and historically grounded, written with undergraduates as the intended audience, and has a clear, engaging expository style and plenty of good exercises.

Second, In my opinion the case of $\mathbb{R}^n$ makes sense as the focus for the situation you're describing; in fact, it might make sense to focus mostly on $\mathbb{R}$. The abstract measure spaces they'll encounter in grad school will be both more comprehensible and more engaging after they've had some experience with the most important-to-them special case. (I'm thinking back on how much more sense the definition of a functor made to me after I had some experience with the fundamental group functor.) That said, it's a matter of your taste as a teacher. If you are really excited about some abstract measure spaces, then that can make sense too.

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I really like A Primer of Lebesgue Integration by H.S. Bear. The book is short and very readable, and it introduces Lebesgue integration on the real line in a very understandable way.

In general, I think that it is much better to introduce measure theory and Lebesgue integration in the specific context of the real line and $\mathbb{R}^n$, perhaps moving on to general measure spaces after this is done. As with many things in mathematics, the most general case is very hard to understand if you haven't yet seen a motivating example.

I also really like Bressoud's A Radical Approach to Lebesgue's Theory of Integration, which includes a generous helping of motivation and historical context. Personally, I prefer to use Bear's (shorter) book as the textbook, and to use Bressoud's book to help prepare my lectures.

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I think Terry Tao's analysis, measure theory and real analysis books (numbers 4,10, and 11 on this list) are excellent.

With special emphasis on his Analysis 1-2 books, they can be used in an excellent way to teach advanced undergraduates.

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Royden's Real Analysis is a great book on Lebesgue measure. It is very accessible. I haven't seen the 4th edition yet, but the 3rd edition starts out by giving a short overview of set theory and goes over the contruction of the real numbers. Later in the book there is discussion of abstract measures.

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