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Let a sequence of Borel probability measures $\mu_n$ converges to $\mu$ in the weak*-topology. Are there any related results on the evolution of the support of $\mu_n$?

For example, in which case, the following holds: for all $\epsilon>0$, there is $N\geq 0$ such that for all $n\geq N$, the support of $\mu_n$ is contained in the $\epsilon$-neighborhood of the support of $\mu$?

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    $\begingroup$ No, consider $(1-\varepsilon)\delta_{x_1} + \varepsilon \delta_{x_2}$ for two distinct points $x_1, x_2$. $\endgroup$
    – Nate River
    Commented Jan 15 at 23:02
  • $\begingroup$ You might be looking for the Portmanteau Theorem: en.wikipedia.org/wiki/Convergence_of_measures $\endgroup$
    – Augusto Santos
    Commented Jan 15 at 23:04
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    $\begingroup$ What is true, though, is that for every $\varepsilon, \delta > 0$, there exists $N \geq 0$ and such that for all $n \geq N$, $\mu_n (E_\varepsilon) > 1 - \delta$, where $E_\varepsilon$ is the $\varepsilon$-neighbourhood of the support of $\mu$. $\endgroup$
    – Nate River
    Commented Jan 15 at 23:05
  • $\begingroup$ The fundamental problem is that if you take an infinitesimal amount of the mass of a measure $\mu$ and spread it somewhere else, it changes the measure only infinitesimally with respect to the weak-* topology, but it can make its support tremendously larger. $\endgroup$
    – Nate Eldredge
    Commented Jan 16 at 0:12

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There's a "semicontinuity" property: the support of $\mu$ is not larger than the "limit" of the supports of the $\mu_n$. Specifically, letting $S(\mu_n), S(\mu)$ denote the supports, we have $$S(\mu) \subseteq \bigcap_{N=1}^\infty \overline{\bigcup_{n=N}^\infty S(\mu_n)}.$$ This follows directly from the portmanteau theorem.

It's not really possible to have any result in the other direction. For instance, let $\mu_n$ be a normal distribution on $\mathbb{R}$ with mean 0 and variance $1/n$. Then the support of each $\mu_n$ is all of $\mathbb{R}$, but they converge weak-* to a point mass at 0.

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