[adroniser]

Essentially the difficulty arises from attempting to assign a measure (area) to every single subset of the sphere, where you say that rotations need to preserve this measure. The paradox can be viewed as a proof that you cannot assign a measure to every subset of the sphere in a consistent way.

The way measure theory resolves this is by showing that if you restrict to appropriate subsets, called measurable subsets, you can get all the nice properties you would expect.

It turns out that basically everything is measurable. In fact the existence of a non-measurable set is independent of ZF. This means that you need the axiom of choice, which was used here in the Banach-Tarski paradox, in order to construct a non-measurable set. So measure theory doesn't really lose a great deal by restricting in this way, which is why it gives such a great theory of integration.