Suggestions for a good Measure Theory book 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.
 A: Have a look at " An Introduction to Measure and Integration" by Inder K. Rana, Graduate Series in Mathematics 45, American Mathematical Society,2002
A: 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.
A: 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'.
A: 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.
A: Check M.M Rao's "Measure and Integration Theory", it is very good.
A: 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)
A: 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.
A: "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.
A: 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!  
A: *

*Principles of Real Analysis by Charlambos Aliprantis and Owen Burkinshaw

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

*Rudin, Real and Complex Analysis.

*Royden, Real Analysis.

*Halmos, Measure Theory.
A: Bartle, The Elements of Integration and Lebesgue Measure
A: 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.
A: D. Cohn, Measure Theory, Birhkäuser
A: 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.


*

*It is extremely well-written.

*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.

*It includes a lot of classical material that many books ignore.

*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.
A: Real Analysis:
Measure Theory, Integration, and Hilbert Spaces by Elias Stein.  Lots of problems.
A: measure theory should be learned first from Saks 
A: Why don't you look at this newly published book:
https://www.crcpress.com/Measure-and-Integration-A-First-Course/Nair/p/book/9780367348397
