Is the set of probability measures with strong convergence compact and/or metrizable?

Question

Given a compact metrizable space $X$ endowed with its Borel sigma algebra, call $P$ the set of probability measures on the resulting measurable space. Endow $P$ with the topology of strong convergence i.e. a net $(m_{\alpha})$ converges to a point $m$ iff $m_{\alpha}(B)$ converges to $m(B)$ for any measurable set B. I would like to know whether P is compact and/or Polish. Of course we know that this is the case with weak convergence.

Answer 1

No. Let $X=[0,1]$, $\mu_n=\delta_{1/n}$. Then $(\mu_n)$ has no strongly convergent subsequence.

(If $\mu_{n_j}\to\mu$ strongly then $\mu((0,1])=1$ but $\mu((1/n,1])=0$ for all $n$.)