Seeking a layman's guide to Measure Theory

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