Given a Polish space $X$, I note $C\_b(X)$ the set of the continuous bounded functions with the norm of the uniform convergence, and $(C\_b(X))^\star$ its topological dual with the $\*-$weak convergence $\sigma((C\_b(X))^\star, C\_b(X))$. 


To a Borel probability $P$ on $X$ we can associate a $l(P) \in (C\_b(X))^\star$ by $$E\_P[\phi]=\langle l(P) , \phi \rangle$$for every $\phi \in C\_b(X)$, where $\langle,\rangle$ denotes the duality bracket. 


I further consider a sequence $(P\_n)\_{n\in \mathbb{N}}$ of Borel probabilities on $X$ which is such that $(l(P\_n))$ converges in the $\*-$weak topology of $(C\_b(X))^\star$ to a $m \in (C\_b(X))^\star$. Obviously $$\langle m,G\_1 \rangle=1$$ (where $G\_1$ is the function $G\_1(x)=1$ for any $x\in X$), and for any positive $\phi$ $$\langle m,\phi \rangle\geq 0$$ (the positive cone is closed). 


However, is it true that one can find a unique probability $P^\star$ on $X$ such that $$m=l(P^\star) \, ?$$ The problem could come from the sigma additivity.



Convergence of probability measure and the *-weak convergence ?
I believe that the answer is affirmative. Weak$^{\ast}$ topology on probability measures can be metrized (e.g. via [Lévy–Prokhorov metric](http://en.wikipedia.org/wiki/L%C3%A9vy%E2%80%93Prokhorov_metric)). When we deal with a Polish space $X$, then this metric is complete. Using this, I would like to conclude that the set of probability measures is **sequentially closed**; indeed, weak$^{\ast}$ convergent sequence of probability measures satisfies Cauchy condition, so it has to converge to some probability measure. Nevertheless, this space is definitely not weak$^{\ast}$ closed, as the example of $(\delta\_{n}) \subset \ell\_{\infty}^{\ast}$ shows.


I hope that it is correct.


EDIT The above argument is completely wrong: of course completely metrizable subset isn't necessarily closed, e.g. $(0,1) \subset \mathbb{R}$. Nevertheless, I found a paper by Dimitris Gatzouras, *On weak convergence of probability measures in metric spaces*, in which he claims that the set of separably supported Borel probability measures is sequentially closed. I haven't read it, so I can't tell if his argument is correct.


