Of course it involves sets, but the kind of set theory you need for that is rather limited, because measure theory deals with very special cases, compared to which set theory proper looks pathological. So, it's a prerequisite to know about countable and uncountable, union and intersection, open and closed and compact, but not the contents of a course in set theory, which gets quite a bit more complicated. I know very little set theory and never studied logic.

Edit: nhamann is right, you need a first course in analysis before you'll understand measure theory. I used Rudin, but I'm not partisan in favor of it. But you've got to learn analysis first. Or you will get confused. Do not pass Go, do not collect $200.

The thing about foundations is that they're mysterious, even at the research level, but when you're working on classical special cases it doesn't matter. In the same way that you can do arithmetic even though you can't prove the axioms of arithmetic are consistent. Measure spaces are very much nicer than sets. And often it's safe to think of the reals as a guiding example. In set theory, thinking of familiar sets as guiding examples is dangerous, which is why I think it's a harder subject, but that may be my idiosyncrasy. The point is, measure theory and set theory are independent from the point of view of the learner -- learn whichever you like first, but one isn't a prerequisite for the other.