Martin Väth's Integration Theory: A Second Course does an admirable job at living up to its title. The basic approach is to start from a very abstract notion of integral and then slowly add more and more conditions so that in the end one gets the Lebesgue integral. So in the first part of the book we start with abstract measure spaces and measurable functions. Then we add a topological structure and learn about Radon measures. Then we add a group structure and discuss the Haar measure. Apply that to the real numbers and we've got Lebesgue measure.
The second half of the book feels more like a topics course. The chapters discuss the Lp spaces, convolutions, connections with logic and set theory, special properties of Lebesgue measure, and other related ideas. All in all, it would make for a good "second course" in integration theory.
Except for a few minor (but irritating) copyediting problems, the book is nicely produced. It's not clear to me that the cover has anything to do with the subject matter, but it looks pretty nice. The writing is not too friendly, but it's friendly enough that it keeps me reading.
I like the author's general approach. Not having used the book with students, I don't know how well it would work in class. In any case, it's worth a look; I think most mathematicians will find it interesting.