Terrence Tao's measure theory text is one of the best undergraduate math textbooks I've ever read. It teaches you not just about the subject, but about how to approach it the way a mathematician does.
Suggested edit:
"... the way a world-class mathematician does."
(because it may have a lot to do with why his book is so much better than other texts on this topic like Rudin, Royden, and Cohn.)