Suppose we are given a measure space $(X,\mathfrak{M},\mu)$, but we know nothing more than this information. (Assume that $\mu$ is a positive, extended real-valued function.) Is there any "nice" way to tell whether or not there exists a topological space $Y$ and a measure $\nu$ on the Borel sets $B(Y)$ of $Y$ such that there is a bimeasurable bijection between $X$ and $Y$ i.e. a measure space isomorphism of $(X,\mathfrak{M},\mu)$ with $(Y,B(Y),\nu)$?
Either this question is ridiculous as asked, or there is probably some kind of set-theoretic business connected to it...sorry if it's the former!