# weak-converging sequence of measures with ascending supports

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 $$\bigcup\limits_{n \in \mathbb{N}}supp \;\mu_n \subseteq supp \; \mu$$?

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.