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Let $X$ be a compact metric space and $\{\mu_n\}_{n \in \mathbb{N}}$ a sequence of regular borel probability measures in $X$ weakly converging to some $\mu$. such that for each $n$ the support of $\mu_n$ lies inside the support of $\mu_{n+1}$.

What can we say about the support of $\mu$? Can we say that $\bigcup\limits_{n \in \mathbb{N}}supp \;\mu_n \subseteq supp \; \mu$?

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No. Let $X = [0,1]$ and let $\mu_n$ be absolutely continuous to Lebesgue measure with density $f_n(x) = (n+1) x^n$. Then every $\mu_n$ has $[0,1]$ as its support, but they converge weakly to a point mass $\mu = \delta_1$ whose support is $\{1\}$.

You may also like to think about the example on $\mathbb{R}$ where $\mu_n \sim N(0, 1/n)$ are normal distributions with decreasing variance. If you want to work on a compact space, you can project them to the circle or something like that.

I think the most you can say is that $\operatorname{supp} \mu \subseteq \overline{\bigcup_n \operatorname{supp} \mu_n}$. This should be easy to verify using the portmanteau theorem.

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