Topological property of the space of probability measures

Question

Suppose that $\mathbb{P}$ is the metric space of Borel probability measures on the interval $[0,1]$ equipped with the topology of $w^*$ convergence.

Consider also $\mathbb{P}_{ac}, \mathbb{P}_{s}$ the subsets of absolutely continuous and singular measures respectively (with respect to the Lebesgue measure). I was wandering how different are these subsets from the topological point of view. That is, does there exist a homeomorphism $\theta : \mathbb{P} \to \mathbb{P}$ such that $\theta(\mathbb{P}_{ac}) = \mathbb{P}_s $ and $\theta(\mathbb{P}_{s}) = \mathbb{P}_{ac}$ ?

Of course this is impossible if $\mathbb{P}_{ac}, \mathbb{P}_s$ are not homeomorphic but I do not know how to exclude this possibility either.

Answer 1

I'm not sure whether or not these subsets are homeomorphic, but there can not be such a map $\theta$ because $P_s$ is a $G_{\delta}$ set. Compare Theorem 1.2 here.

If we had a $\theta$ as desired, then $P_{ac}$ would be a $G_{\delta}$ set also, but this impossible because then $\emptyset =P_{ac}\cap P_s$ would have to be dense by Baire's theorem.