Let $(X,\Sigma)$ be a measurable space, where $X$ is an infinite set, and denote by $(X^2,\Sigma^2)$ its product space.
Under which conditions it is true that there exists a measurable bijection $f:X \to X^2$?
[I know this a general question and it may not admit an easy answer if it is not "always"; in particular, I am only interested in the case where $X$ is a locally convex topological vector space, and $\Sigma$ stands for its Borel $\sigma$-field]