Oddly enough, an important part of recent set theory is two-person zero-sum infinite games. The question of solvability of games defined on certain types of sets turn out to depend on delicate set-theoretical questions (most of which turn out to be independent of ZFC). Solvability of particular types of games on a set can be thought of a refinement of measurability for that set.

Jindrich Zapletal has some lecture notes on the subject.