A: Banach-Tarski Banach-Tarski.

I did my undergraduate project on the question of finitely-additive, isometry-invariant measures that extend the Lebesgue Measure and which are defined on all possible bounded subsets of R^n.

And yes, that's relevant. There is (perhaps surprisingly) such a measure on R,and there is such a measure on R^2, but the Banach-Tarski "paradox" shows that there is no such measure on R^3 (and therefore trivially, R^n for n>3).

For contrast, we'd usually like to have a measure that's countably additive, but there is a simple construction (albeit using the axiom of choice) that shows such an object is impossible.

This has, by the way, been submitted and discussed several times in the past. In case you're interested in reading other descriptions, and previous discussions, you can find some of them here:

http://www.hnsearch.com/search#request/all&q=banach+tars...