A professor posed me a problem a few days ago, and I have not been able to find an answer to it.
Let $(X_1,\Sigma_1,\mu_1)$ and $(X_2,\Sigma_2,\mu_2)$ be probability spaces. Suppose the following conditions hold.
$\mu_1$ is atomless, i.e., $\not\exists A\in\Sigma_1$ such that $\mu(A)>0$ and for any measurable $B\subset A$, $\mu(B)=0$.
$X_2$ is a complete, separable and metrizable topological space. (I think these spaces are called Polish spaces, although I might be mistaken.)
$\Sigma_2$ is a Borel $\sigma$-algebra on $X_2$.
Does there exist a measure-preserving map $f:(X_1,\Sigma_1,\mu_1)\rightarrow(X_2,\Sigma_2,\mu_2)$?
I really appreciate it if someone is able to provide a rough proof sketch (with hints) and provide a reference for this result. Thank you!