Let $X$ be a compact metric space and let $\mathcal M(X)$ denote the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\text{supp} \mu$ for the *support of $\mu$*. It is easy to see that if a sequence of measures $(\mu_n)_{n=1}^\infty$ weak$^*$ converges to some $\mu\in\mathcal M(X)$, then
$$
\text{supp}\, \mu\subset \liminf_{n\to\infty}(\text{supp}\,\mu_n).
$$
(By $\liminf$ above I mean the Kuratowski limit inferior, for a definition see wikipedia entry.)
My question is: are there any useful criteria guaranteeing that
$$
\text{supp}\, \mu = \lim_{n\to\infty}(\text{supp}\,\mu_n).
$$
(That is, I am looking for conditions which imply that the convergence of supports in the Hausodrff metric on the set of nonempty closed subsets of $X$ follows from the weak$^*$ convergence of measures.)

What if $X=\{0,1\}^\mathbb{Z}$, where $\{0,1\}$ is discrete and $\{0,1\}^\mathbb{Z}$ is given the product topology?