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.
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.
I would recommend "Lebesgue Integration on Euclidean space" by Frank Jones. The analysis texts by Stein and Shakarchi are also very accessible.
My favourite book on measure theory is Cohn's. It has a manageable size and yet, it covers all the basics.
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.
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.
I very much enjoyed these lecture notes.
It's written in a style that is suitable for self-study.
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).
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.
This book, Problems in Mathematical Analysis III, has plenty of exercises (with solutions!) on The Lebesgue Integration.