Let $X$ be a separable completely metrizable space, let $\mathscr{B}(X)$ denote the Borel $\sigma$-algebra on $X$, and let $\mathscr{P}(X)$ denote the space of all probability measures on $(X, \mathscr{B}(X))$.

Let $\tau$ denote the topology of setwise convergence on $\mathscr{P}(X)$, i.e. the smallest topology on $\mathscr{P}(X)$ such that for every $B \in \mathscr{B}(X)$ the mapping $\nu \mapsto \nu(B)$ on $\mathscr{P}(X)$ is continuous.

**Question:** Is $\tau$ metrizable?

Measure Theory. $\endgroup$