This is the book I used to learn measure theory. There is now a second edition featuring a slightly more friendly format, and a whole new chapter on probability.
I don't remember any lacking proofs with one exception - the approximation-by-simple-functions proposition. I guess this is because Cohn somewhat delays the introduction of the general definition of measurability. Rudin gives a concise (and unusually clear) proof in Real and Complex Analysis, Thm 1.17.
This is also where I learned measure theory from (2nd edition). I found this text great for both learning and as a reference. I haven't used any other measure theory textbooks, but I didn't feel the need to with this book handy. There is a nice chapter on probability theory, and a proof of the Banach-Tarski paradox in the appendix.