Let $\mu\_n, \mu$ be a sequence of probability measures on a Polish space $S$ and $\mu\_n', \mu'$ be some kind of extension of $\mu\_n, \mu$ on $\bar{S}$ such that all the boundary points of $S$ gets a zero measure. Now if 


$$\int fd\mu\_n' \to \int fd\mu'$$ for every $f \in C(\bar{S})$


then I think this claims that 


for any bounded $g \in C(S)$ 


$$\int gd\mu\_n \to \int gd\mu$$


is true. I think to claim that we need continuous extension of $g$ over $\bar{S}$. Am I correct ? 



Problem on convergence in space of probability measures
It seems that in the paper, $S$ is a Polish space, which is homeomorphic to a dense subset of a compact metric space denoted by $\overline S$. Without loss of generality, we shall work with this subspace instead of the original $S$.


By portmanteau theorem, it suffices to show the wanted convergence when $g$ is bounded and uniformly continuous on $S$. Such a function can be extended to a continuous (and bounded) function on $\overline S$, denoted $\overline g$. Then we deduce convergence 
$$\lim\_{n\to \infty } \int\_{\overline S} \overline g\mathrm d\mu\_n=\int\_{\overline S} \overline g\mathrm d\mu,$$
and since for each $n$, $\mu\_n(\overline S\setminus S)=\mu(\overline S\setminus S)=0$ (by definition of measures on $\overline S$), we get the wanted convergence.


