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I would like to teach myself measure theory. Unfortunately most of the books that I've come across are very difficult and are quick to get into Lemmas and proofs. Can someone please recommend a layman's guide to measure theory? Something that reads a bit like this blog post, starts out very gently and places much emphasis on the intuition behind the subject and the many lemmas.

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Did you try Wheeden-Zygmund? Unfortunately, it's very expensive. –  t.b. Oct 10 '11 at 12:26
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A set of lecture notes titled Analysis, Measure and Probability by Marcus Pivato begins gently with motivations and pictures like the blog you mention. –  Dilip Sarwate Oct 10 '11 at 13:10
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Quite a large number of "lemmas" (to quote the question) needed to establish that Lebesgue measure satisfies various properties claimed for it temporarily blinded me to the fact that the definition of the Lebesgue integral is very simple. Here it is: The integral of a non-negative function is just the smallest number that is not to small to be the integral (and its $\infty$ if no number is too small). A number is too small if it's less than the integral of a simple function that is dominated by the function to be integrated. –  Michael Hardy Oct 10 '11 at 13:19
    
That's for non-negative functions. For other functions, one just looks at positive and negative parts separately. –  Michael Hardy Oct 10 '11 at 13:20
    
I am making this Community-Wiki. If you have any objections, let me know. –  robjohn Jan 6 '13 at 19:39

10 Answers 10

up vote 14 down vote accepted

Measures, Integrals and Martingales by René L. Schilling is a very gentle (mathematically rigorous, but that should be the case if you want to learn measure theory) introduction to measure theory. All the solutions to the exercises are available on the website of the author. Another advantage is that it is quite inexpensive.

However, I'd also suggest Measure and Integration Theory by Heinz Bauer. This is one of the best introductions to this subject I have ever seen (and my professor and some others seem to agree). One drawback is that it has a few typos but that keeps you sharp ;-). It is a translation of the author's original book in German where only the relevant topics are kept.

Here (TU Delft) they first used the first book which I mentioned and this year they use Bauer.

Both books are an excellent basis if you want to go in the direction of analysis or probability theory. Both fields require at least what is in these books.

A companion to Bauer's measure theory book if your goal is to learn probability theory is his probability theory book.

Another thing I would like to note is that you should have a reasonable knowledge of the foundations of real analysis before you embark on this. Measure theory is a "true" analytic topic and should not be treated like many calculus courses.

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After seeing your suggestion of the Schilling text a few weeks ago I ordered it; I have only worked through the first few chapters but I must say I really like this book. The solutions on the author's website are just as thorough and detailed as his writing. The appendices are also very useful. All-in-all, I'm finding this to be a very complete and useful resource. Thanks for the suggestion. –  ItsNotObvious Nov 7 '11 at 20:01
    
@3Sphere Glad to have helped :-). –  Jonas Teuwen Nov 7 '11 at 20:09

I suggest A Concise Introduction to the Theory of Integration by Daniel W. Stroock, which I found both a pleasure to read and straight to the point.

Edit: I was almost forgetting that it includes interesting exercises with hints/solutions, so it's good for self study.

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Stroock's book is ok,but nothing special. I think if you're going to spend that kind of money for a book about measure theory in Euclidean space,you might as well get Jones or an old beat up copy of Zygmund/Wheeden. –  Mathemagician1234 Oct 11 '11 at 4:37
    
I enjoyed his style of writing and his approach to the matter, so I felt justified in suggesting it. But of course other books are valid alternatives. –  Francesco Oct 11 '11 at 9:53

One of the very best books on analysis, which also contains so much more then just measure and integration theory,is also 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.

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I very much enjoyed these lecture notes.

It's written in a style that is suitable for self-study.

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I dislike books that spell everything out! That doesn't really promote "active" reading. –  Jonas Teuwen Oct 10 '11 at 18:21
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Yes it does! I read slowly. I read, think, then discover that what I thought was actually right and if it wasn't I get corrected. If I read an-everything-is-obvious-sort-of-book it's an almost complete waste of time because I have no way of checking whether I actually understood what I just read. –  Matt N. Oct 10 '11 at 18:27
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I think active reading should happen naturally, without the need to be promoted. I'm driven and curious. Reading something that spells out nothing is frustrating and tiring to me. –  Matt N. Oct 10 '11 at 18:31
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Okay, then I misunderstood what you mean with "spell everything out", I assumed that you mean that all the steps don't require considerable thought. –  Jonas Teuwen Oct 10 '11 at 18:31

My favourite book on measure theory is Cohn's. It has a manageable size and yet, it covers all the basics.

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I would recommend "Lebesgue Integration on Euclidean space" by Frank Jones. The analysis texts by Stein and Shakarchi are also very accessible.

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+1 For Jones,terrific book for undergraduates with a good real analysis course background!Unfortunately,it's quite expensive,which is why I recommended instead the even better book by Taylor below. Stien and Shakarchi's Lectures are all excellent,but they're quite a bit more challenging. –  Mathemagician1234 Oct 11 '11 at 4:33

This book, Problems in Mathematical Analysis III, has plenty of exercises (with solutions!) on The Lebesgue Integration.

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I found Robert Bartle's book: Elements of Integration and Lebesgue Measure to be very helpful.

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I really like Foundations of Modern Analysis by Avner Friedman. Excellent text on the essentials plus it is a "worker's book on analysis" in the sense that it shows you how many of the tools you learn in a measure theory course are actually used to tackle problems in PDE, functional analysis, etc. Plus it is cheap ($12).

My other recommendation is a second nod to Lebesgue Integration on Euclidean Spaces by Frank Jones. Very accessible but astronomically expensive. Perhaps you can get a copy from the library (or interlibrary loan).

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Thinking back very far, to when I was a student learning measure theory, I really liked "Introduction to measure and probability" by Kingman and Taylor. The measure theory part was also published as a separate book, "Introduction to measure and integration" by (only) Taylor.

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