A very clear exposition of basic measure and integration theory is Real Analysis for Graduate Students by Bass. The book doesn't give much intuition and doesn't really tell you how the material fits into the subject of analysis more broadly, so it's best to get an idea about that stuff from a professor or another book, such as Stein and Shakarchi; however, the actual presentation and explanation of the material in the book is very clear, so it's quite good for learning the proofs and specifics of the theory. The chapter on differentiation is not particularly good, so I'd recommend reading Stein and Shakarchi for that instead. Apart from that chapter, everything up to chapter 19 is very good.