Suppose $\mu_n\implies\mu$, i.e. $\mu_n$ converges weakly to $\mu$ where $\mu_n$, $\mu$ are probability measures on some metric space $(X,d)$. Given a Borel set $B$, define $\mu^B$ to be the conditional probability given $B$, i.e. $\mu^B(A)=\mu(A\cap B)/\mu(B)$, and similarly for $\mu_n^B$.

Under what conditions (e.g. on $X$, $\mu$, and/or $B$) does it follow that $\mu_n^B\implies\mu^B$?