We are forced to relax one of the four named properties that we desire for the measure function. (And, well, Banach-Tarski rules out one of those options.) So we relax the requirement that mu be defined on all inputs.
But I wonder, isn't there an implicit 5th property that could be relaxed? That's the property that the codomain of mu is the reals. Is it viable to use, say, the hyperreals instead, or some other exotic extension that would allow us to name the (nonzero and nonreal!) number mu(V) such that a countable sum of mu(V) comes out to 1?
But I wonder, isn't there an implicit 5th property that could be relaxed? That's the property that the codomain of mu is the reals. Is it viable to use, say, the hyperreals instead, or some other exotic extension that would allow us to name the (nonzero and nonreal!) number mu(V) such that a countable sum of mu(V) comes out to 1?