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.