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I have taken analysis and have looked at different measures, but I am currently looking at realizing a certain problem in a different light and feel that I need a better background in various measures that have been used / discovered / et cetera in order to really move my (very basic) research forward. So, I am curious if anyone can suggest a good book on Measure Theory that has theory and perhaps a NUMBER of examples and uses of various measures.

Thanks for any help; contact privately if you feel that you need more info so as to recommend better; I will explain what I am thinking about for my research -- I'm a n00b to research in math so it's probably not that interesting ;-) but who knows.

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Community wiki, please!(edit and check the box). –  Anweshi Jan 12 '10 at 21:59
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You might try Folland's "Real Analysis: Modern Techniques and Their Applications". –  Ben Linowitz Jan 13 '10 at 13:04
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I am surprised that this got so many answers without anyone asking (publicly) the questioner what s/he was looking for in a measure theory text. Without that information, the question becomes "Please list some measure theory books that some people have liked", which is pretty close to just "Please list some measure theory books". Even a community wiki question should have more of a focus than this, IMO. –  Pete L. Clark Feb 1 '10 at 8:59
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17 Answers

  1. Rudin, Real and Complex Analysis.

  2. Royden, Real Analysis.

  3. Halmos, Measure Theory.

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Bartle, The Elements of Integration and Lebesgue Measure

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Really nice book-my old analysis mentor Gerald Itzkowitz learned the subject from the first edition of that book and the lecture notes of Eberlien at the University of Rochester. –  Andrew L Apr 7 '10 at 21:49
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Folland's Real Analysis is nice and has some pretty good exercises which often elucidate important examples. Also, it contains some applications to other fields.

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I "learned" measure theory and analysis from this book and I have to say it is horrible. The level of sophistication is way to high. It's nice if you've seen the material once before and as a second pass it probably wouldn't be so bad but for a beginner it's plain horrible. –  davidk01 Feb 1 '10 at 8:13
    
I agree with you in that this is not a begginer's book, but I don't think this justifies saying the book is horrible. I mentioned it because Andrew asked for a reference with examples, which can be found, if not in the text, in the exercises. This is probably not the best book to start learning measure theory (more basic references were already cited before) but it is certainly a valuable reference, IMHO. –  Rodrigo Barbosa Feb 25 '10 at 13:44
    
It's actually a very good book,Folland-the second edition is really the best.The bibligraphical notes really help one assimilate the enormous amount of material by providing all important motivation and context. "adult Rudin" has SOME of this,but it's so concise and packed,the ideas have no room to breathe. It's almost like Rudin was delibrately trying to see how much he could pack into a one year course. –  Andrew L Apr 8 '10 at 3:34
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Jürgen Elstrodt - Maß- und Integrationstheorie (only in German)

Fremlin - Measure Theory (freely available in the web space, contains pretty much every significant aspect of measure theory in appropriate depth)

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The second one is great. Not able to look into the first one. –  Anweshi Jan 14 '10 at 0:13
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+1. Elstrodt is one of the best math books ever written. –  Johannes Hahn Mar 5 '10 at 13:59
    
There is a typo: It should be Maß- und Integrationstheorie (without the extra "g") - the link to google books: books.google.de/… Unfortunately I don't have the rights to include this into the above article, perhaps somebody could do it... Thanks –  vonjd Mar 5 '10 at 19:05
    
You are right, there was a typo, I corrected it. Thanks. –  efq Mar 6 '10 at 15:04
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I am a huge fan of Frank Jones's book "Lebesgue Integration on Euclidean Space". It's not as well known as most of the other books mentioned, but I like it for the following reasons.

  1. It is extremely well-written.
  2. On the one hand, it works in $\mathbb{R}^n$ from the offset rather than starting with $\mathbb{R}$ (though really this is no more difficult, and it is good to train students not to be scared of higher dimensions; also, it makes it a bit easier to draw pictures). However, it is still quite a bit more gentle than most other books, and its perspective is extremely concrete.
  3. It includes a lot of classical material that many books ignore.
  4. Its exercises are fantastic.

I learned the subject from this book back when I was a 2nd year undergraduate (back in 1999!). However, though I now own many other books it is still the one I go back to when I want to remind myself about the basic facts of life about integration theory or measure theory or Fourier analysis.

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GREAT BOOK FOR UNDERGRADUATES,ANDY. If you can afford it and you're learning it on your own,no better choice. –  Andrew L Apr 7 '10 at 21:37
    
Agreed. One of the things I really like about Jones' book is that many exercises are given right after the definitions and the theorems (rather than at the end of the chapter), which allows you to grasp the concepts taught just after they've been presented. I really don't understand why many books in mathematics prefer the "long list of exercises at the end of the chapter" approach. All in all, it's a great introduction to measure theory. For the more advanced stuff (generalities on Radon measures, $L^p$ spaces, etc.), I recommend Folland's book, as was mentioned here already. –  Mark Aug 11 '10 at 11:10
    
This is an outstanding book. I learned a tremendous amount from it as an undergrad. That said, it doesn't say much about measures other the one referenced in the title (presumably to keep the prerequisites minimal). –  Frank Thorne Jul 29 '11 at 21:27
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Real Analysis: Measure Theory, Integration, and Hilbert Spaces by Elias Stein. Lots of problems.

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Another VERY good book.It's kind of expensive for what you're getting,though,sadly.See if you can borrow a copy. –  Andrew L Apr 7 '10 at 21:37
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D. Cohn, Measure Theory, Birhkäuser

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"A Modern Theory of Integration" by Robert G. Bartle is an excellent introduction to the theory of gauge integrals which subsumes and generalizes the usual measure theory of Lebesgue.

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A GREAT book,but it's really kind of a seperate subject,David. A subject well worth learning and this is the book to do it with.But I don't think this is exactly what's being asked for. –  Andrew L Apr 7 '10 at 21:36
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Have a look at " An Introduction to Measure and Integration" by Inder K. Rana, Graduate Series in Mathematics 45, American Mathematical Society,2002

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measure theory should be learned first from Saks

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A very good book is "Measure and Integration Theory" from Heinz Bauer, especially if you are planning to study probability theory. One of its strengths is that the theory is first developed without using topology and then applied to topological spaces. In my opinion this leads to a better understanding of Radon measures for example. Its style is also very concise and precise.

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Well, I personally HATE Halmos' Measure Theory, even though an entire generation grew up on it. My favorite book on measure and integration is available in Dover paperback and is one of my all time favorite analysis texts: Angus Taylor's General Theory Of Functions And Integration. Lots of wonderful examples and GREAT exercises along with discussions of point set topology, measure theory both on $\mathbb{R}$ and in abstract spaces and the Daneill approach. And all written by a master analyst with lots of references for further reading. It's one of my all time favorites and I heartily recommend it. Folland's Real Analysis is a fine book, but it's much harder and it's really more of a general first year graduate analysis course. On the plus side, it does have many applications, including probability and harmonic analysis. It's definitely worth having, but it's going to take a lot more effort then Taylor. The IDEAL thing to do would be to work through both books simultaneously for a fantastic course in first year graduate analysis. And please don't torture yourself with Rudin's Real And Complex Analysis. It's sole purpose seems to be to see how much analysis can be crammed incomprehensibly into a single text. Folland is the same level and is much more accessible. That should get you started. Good luck!

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Rudin's book is a classic; deservedly so IMO. The proofs emphasize the important ideas, so it is particularly good for someone to go back to after becoming a professional. –  Bill Johnson Apr 8 '10 at 1:40
    
Maybe so,Bill-but it's ridiculously concise and abstract and I just don't think it ends up teaching you a lot when you're a beginner unless you have a really good teacher who helps you with The Big Picture. –  Andrew L Apr 8 '10 at 3:28
    
@Andrew L: I don't think Rudin (resp. Halmos) wrote Real and Complex (resp. Measure Theory) as bedtime reading for beginners, but during its four decades in print it has proven to be an important resource for mature students with a serious interest in analysis and (as Bill mentions) as a reference for analysts. I don't think it's productive to dismiss a standard (and valuable) text, even though it almost certainly isn't the ideal one for this stage of the OP's analysis education. –  Zach N Jan 25 '11 at 4:12
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  • Principles of Real Analysis by Charlambos Aliprantis and Owen Burkinshaw

  • Introduction to Measure theory by the great Terence Tao :) which is available online here

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also introduction to radical theory of lebesgue integration. D.M.Bressoud published by the MAA. –  Chandrasekhar Jul 16 '10 at 11:04
    
The online version of Tao's book is merely a first draft,Chandrasekhar. The AMS will publish a finished version early next year. From the draft,though,it looks like a terrific addition to the textbook literature-the influence of Elias Stein on Tao's book is very evident. –  Andrew L Jul 29 '11 at 22:33
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If the focus is on measures on $\mathbb{R}^n$, Measure theory and fine properties of functions by Lawrence C. Evans and Ronald F. Gariepy. Otherwise Linear Operators. Part I: General Theory by Nelson Dunford and Jacob T. Schwartz, chapter 'Integration and Set Functions'.

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I am late to the party, but let me say that while Evans & Gariepy is a great book, it is much more about fine properties of functions than it is about measure theory, so it hardly fits the bill for this question. On the other hand, it does provide a concise introduction to Hausdorff measures, which is very useful if that is what you need. –  Harald Hanche-Olsen Jun 13 '11 at 7:49
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If you want to go deeper into probability theory: I would also recommend Heinz Bauer's "Measure and integration theory". But I found it a bit dry (in German). Also Kai Lai Chung's "A Course in Probability Theory" is excellent.

If you're more interested in (functional) analysis and want just a short intro to measue theory: As an alternative to Rudin's "Real and Complex Analysis" I warmly recommend the recent book by Jürgen Jost "Postmodern Analysis", which includes an intro to PDE. I wish I had that when I was young... Right next to these in my library is Segal & Kunze "Integrals and Operators" and Robert Geroch's "Mathematical Physics" (no physics inside).

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I would surprisingly point to Shiryaev's "Probability", which doesn't go into details but definitely motivates the introduction of the notion of measure and explains of what use are the different properties ; the other suggestions given in the other answers will then plug the holes, if need be.

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Personal favorites, in suggested reading order:

  • Bartle, The Elements of Integration. Exercises are all doable and at about the same level. The best first-time text.

  • Royden. I loved this book: the exercises vary from easy to quite tricky (when you need to sleep on it!or take a looong shower) and working as many exercises as possible, especially the hard ones, is a great way to really understand the real numbers.

  • Rudin. The proofs are at times way too slick, formulas popping out with no motivation, but that should be no real (or complex?) problem after Royden, and it gives a beautiful overview of the essentials; the topology part is great too. The problems are great and often quite challenging.

  • Oxtoby, Measure and Category. This is just a fantastic little book. After you have studied the others, you can read through this like a novel and everything will start to fit together much more. Pure inspiration.

  • Dunford and Schwarz. Some encounter with this is necessary, especially after you've also been through Rudin's Functional Analysis.

  • Lamperti's "Probability". This could be called "Probability for Analysts" and it is a beautiful little book.

  • Billingsley's Ergodic Theory and Information. Now you're ready to see what some of that abstract stuff is good for, and this beautiful text is an excellent choice.

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