The relationship is that measure theory provides the theoretical framework for making probability theory rigorous.

The only formal prerequisite for learning measure theory is that you should know series and sequences. For a reference, I'm not so sure, maybe Halmos's book. The important parts are probably:

- Monotone convergence theorem

- Dominated convergence theorem

- The construction of the Lebesgue integral

- Fubini's theorem and Tonelli's theorem

I would probably try not to get bogged down in details of construction of measures (unless you like that) and take the Lebesgue measure (essentially length) as given. Also check out the Radon-Nikodym theorem which states that we can always (ish) work with density functions.

The typical prerequisite for measure theory is a two-semester real analysis course, a la Rudin or any of its alternatives (I particularly like Pugh's book). A solid topological background is also a good idea, although you can probably get away with whatever you learned in real analysis. Two standard measure theory texts are Folland's Real Analysis and the first half of Rudin's Real and Complex Analysis.

Probability theory is the study of distributions of constant measure in measure theoretic terms. There are some good resources that mtzet mentions, but I just wanted to note that a lot of the integration terminology which you take for granted reading about probability theory is formally defined in measure theory. It's also very nice for making signal processing math more formal.

I'm taking a measure theory course right now, and we primarily use some set theory and some topology of R^n.