[throwaway81523]

The Banach-Tarski theorem motivated the idea of amenable groups in topological group theory. Understanding exactly what that means is on my todo list, but I think the basic idea is that a given space can have additive measures invariant under some transformation groups but not others. Particularly, the Banach-Tarski paradox shows that regular old 3-dimensional Euclidean space doesn't have an additive measure invariant under rotation and translation. On the other hand, 2-dimensional Euclidean space does have it.