Locally compact metric space, Urysohn, approximation

Question

Let $E$ be a locally compact separable metric space, $\mathcal{B}(E)$ be the $\sigma$-algebra of $E$ and $m$ be a $\sigma$-finite borel measure on $(E,\mathcal{B}(E))$.

Assumtion

There exists a dense subset $S$ of $L^{2}(E;m)$ and $S \cap C_{0}(E)$ is dense in $C_{0}(E)$ (:= all continuous functions on $E$ with compact support) w.r.t uniform norm.

My Question

For every relatively compact open subset $U$ of $E$, does there exists a $f\in S$ such that $f\geq1\,m$-a.e. on $U$ ?

My idea:

By Urysohn's lemma, there exists a $g\in C_{0}(E)$ such that $g|_{\bar{U}}=1$. Since $S\cap C_{0}(E)$ is dence in $C_{0}(E)$, there exists a $(f_{n})\subset S \cap C_{0}(E) \subset S $ s.t. $f_{n}\to g$ in $C_{0}(E)$. But I can't find a $f$ as in my question.

Do you know a nice example? Please let me know.

Answer 1

Use Urysohn to construct $g$ with $g \equiv 2$ on $U$ (multiply your $g$ by $2$).

Then take a sequence $(f_n)$ with$f_n \to g$ uniformly. For $n$ large, you will then have $f_n \geq 1$ on $U$.