On page 7 of Villani's Optimal Transport: Old and New (page 19 in this preprint), he states that any two polish atomless probability spaces $(\mathcal{X},\mu)$ and $(\mathcal{Y},\nu)$ are measure-theoretically isomorphic and that, moreover, the isomorphism can be made from all of $\mathcal{X}$ to all of $\mathcal{Y}$ (not just almost all of each). Where can I find a proof of this fact?
In particular, I'm interested in the descriptive-set-theoretic complexity of such a map in a certain case.
Less importantly, he says
Experience shows that it is quite easy to fall into logical traps when working with the measurable isomorphism, and my advice is to never use it.
What logical traps is he talking about?
This is due to Halmos and von Neumann, see this page for references. Not sure they deal with the "descriptive-set-theoretic complexity" of such an isomorphism, though.
One could also check the references mentioned on the obvious page.