Proposition 4.22.3. Let $X$ be a LCH space and $f: X \to [0, \infty]$ be lower semicontinuous, then
\[f = \sup_{\substack{\phi \in C_c(X) \\ 0 \le \phi \le f}}\phi\]
Proof. Let $x \in X$ such that $f(x) > 0$ and $a \in (0, f(x))$, then $\bracs{f > a}$ is open. By Urysohn’s lemma, there exists $\phi \in C_{c}(X; [0, a])$ such that $\phi(x) = a$ and $\supp{\phi}\subset \bracs{f > a}$. As this holds for all $a \in (0, f(x))$,
\[f(x) = \sup_{\substack{\phi \in C_c(X) \\ 0 \le \phi \le f}}\phi(x)\]
$\square$