Reference book on measure theory I post this question with some personal specifications. I hope it does not overlap with old posted questions.
Recently I strongly feel that I have to review the knowledge of measure theory for the sake of starting my thesis.
I am not totally new with measure theory, since I have taken and past one course at the graduate level. Unfortunately, because the lecturer was not so good at teaching, I followed the course by self-study. Now I feel that all the knowledge has gone after the exam and still don’t have a clear overview on the structure of measure theory.
And here come my specified requirements for a reference book.


*

*I wish the book elaborates the proofs, since I will read it on my own again, sadly. And this is the most important criterion for the book.

*I wish the book covers most of the topics in measure theory. Although the topic of my thesis is on stochastic integration, I do want to review measure theory at a more general level, which means it could emphasize on both aspects of analysis and probability. If such a condition cannot be achieved, I'd like to more focus on probability.

*I wish the book could deal with convergences and uniform integrability carefully, as Chung’s probability book. 
My expectation is after thorough reading, I could have strong background to start a thesis on stochastic integration at an analytic level.
Sorry for such a tedious question.
P.S: the textbook I used is Schilling’s book: measures, integrals and martingales. It is a pretty good textbook, but misprints really ruin the fun of reading.

It seems unnecessary to add to this long list of great books, but
Real Analysis and Probability by R.M. Dudley is wonderful. His book fits your need to emphasize on both aspects of analysis and probability.  
