Let $\epsilon>0$ be given. Let $Y$ be a compact, Hausdorff space and let $U\subseteq Y$ be an open subset. Assume that $(\mu_n)_{n\in\mathbb{N}}$ is a sequence of regular Borel probability measures on $Y$. I want to show the existence of a continuous function $f$, supported on $U$, with $0\leq f\leq 1$ and s.t. $\mu_n(\{x\in U: f(x)\neq 1\})<\epsilon$ for all $n\in \mathbb{N}$.

What have I tried:

Clearly, we can not take just the characteristic function on $U$, since this function will not be continuous on $Y$.

By regularity of the measures, we can choose for every $n$, a compact subset $K_n\subseteq U$ s.t. the measure $\mu_n(K_n)>\mu(U)-\epsilon$. Now, define $K:=\bigcup_{n\in\mathbb{N}}K_n$. $K\subseteq U$ and it is not clear anymore that $K$ is closed! But if it was closed, by Urysohn's lemma there exists the function that takes the value $1$ on $K$ and $0$ outside $U$.

I don't know how to fix this idea.

Thanks for any help