We denote by $\mathcal{M}(X)$ the set of all Borel probability measure of $X$. We know that this set is a compact metrizable convex space and its topology is weak $^\*$ topology defined by the convergence $\mu\_n\to \mu$ if and only if $\int \varphi d \mu\_n\to \int \varphi d\mu$ for every continuous map $\varphi:X\to \mathbb{R}$.

Let $\mu\_m\to \mu$ and $\mu(A)>0$ for some a non-empty subset $A\subseteq X$. 
Can we say that there is $N\in\mathbb{N}$ such that $\mu\_m(A)>0$ for $m>N$?