# Weak$^*$ convergence of measures vs. convergence of supports

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?

• I am really interested in this question in the context of shift invariant measures, therefore I have added dynamical'' tags to it. – Dominik Kwietniak Jan 24 '17 at 11:30
• There are no assumption on the underlying space that are going to help you. Let $x$ be a point in your space and $\delta_x$ be the corresponding Dirac measure concentrated on $x$. Let $\nu$ be an atomless probability measure, such a probability measure exists on the Borel $\sigma$-algebra of every Polish space and its support cannot be a single point. Then the sequence of probability measures $\langle (n-1)/(n) \delta_x+1/n\nu\rangle$ weak*-converges to $\delta_x$, but the support does not converge to $\{x\}$. – Michael Greinecker Jan 24 '17 at 12:21
• Sure, I am aware of such counterexamples, but I hope that there is a general condition on the sequence of measures which may have a nice formulation when phrased in the setting on shift invariant measures on the product space of two-point set. – Dominik Kwietniak Jan 24 '17 at 12:25
• I don't understand downvote somebody gave. It is well-known that the support do not have to converge, but on some instances they do. For example, in Bonatti, Christian, Lorenzo J. Díaz, and Anton Gorodetski. "Non-hyperbolic ergodic measures with large support." Nonlinearity 23, no. 3 (2010): 687 there is a criterion implying that some measures supported on periodic points converge weak$^*$ and their supports converge as well. So the question is: can you characterize when it happens? – Dominik Kwietniak Jan 24 '17 at 14:34
• Consider a converging sequence of shift-invariant $\mu_n$ on $X$. Each of their supports $Y_n$ will be a shift-invariant closed subset of $X$. The limit of the sequence $(Y_n)_n$ is going to be a shift-invariant closed subset $Y$, satisfying the simple property that for every $k$ the projections of the $Y_n$'s on $\{0,1\}^{[-k,k]}$ eventually stabilize to equal the projection of $Y$. So in this case your condition will be satisfied iff every cylinder set that has measure 0 under $\mu$ also has measure 0 eventually under $\mu_n$. – Vladimir Apr 26 '17 at 7:36