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