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 ?