Suppose that a sequence $(\mu_{k})_{k=1}^{\infty}$ converges weakly to some measure $\mu$, where $(X,d)$ is a polish space and the measures are Borel probability measures.
If for some Borel set $A$ we have $\mu_{k}(A)=c$ (where c is a constant) for all $k$, do we know anything about $\mu(A)$? I know $\mu_{k}(A)\to \mu(A)$ if $\mu(\partial A)=0$, but I would be interested to know if there's anything we can do in this case.
Thanks in advance.