# Lower semicontinuous non-negative function on a locally compact Hausdroff space with a countable base

An extended real number is an element of $\mathbb R \cup \{-\infty, +\infty\}$. Let $X$ be a locally compact Hausdorff space with a countable base. An extended real valued function $f$ on $X$ is called lower semicontinuous if $\{x\in X | f(x) \gt a\}$ is open for every $a\in \mathbb R$. We denote by $\mathcal L$ the set of complex valued continuous functions of compact support on $X$. We denote $\{f \in \mathcal L | f \ge 0\}$ by $\mathcal L_+$. Let $f \ge 0$ be a lower semicontinous function on $X$. Is the following assertion true?

There exists a non-decreasing sequence $(f_n)$, $f_n\in \mathcal L_+$ such that $f(x)$ = lim $f_n(x)$ for every $x\in X$.

The motivation is as follows. Let $\mu: \mathcal L \rightarrow \mathbb C$ be a $\mathbb C$-linear map such that $\mu(f) \ge 0$ for every $f\in \mathcal L_+$. Let $f\ge 0$ be a lower semicontinuous function on $X$. Bourbaki defined the integral $\int f d\mu$ as sup $\mu(g)$ where the sup runs through $g \in \mathcal L_+$ such that $g\le f$. If the assertion is true, the definition becomes much more down-to-earth.

The assertion is true. I have the following idea of the proof. Let $F$ be a monotonic homeomorphism between the extended real line and the segment $[-1,1]$ such that $F(0)=0$ (for instance, we can put $F^{-1}(x)|_{(-1,1)}=\tan\frac{\pi x}2$). An extended real valued function $f$ on $X$ is lower semicontinuous iff the composition $F\circ f$ is lower semicontinuous.
Let $f\ge 0$ be a lower semicontinous function on $X$. Then $F\circ f\ge 0$ is a lower semicontinous function on $X$ too. Since $X$ is a locally compact Hausdorff space, by [Eng, Theorem 3.3.1], $X$ is a regular (even a completely regular) space. Since the space $X$ has a countable base, it is metrizable (by [Eng, 4.4.7] (Nagata-Smirnov metrization theorem) or by [Eng, 4.4.8] (Bing metrization theorem) :-) ) and (or by [Eng, Lemma 4.4.5]) hence, by [Eng, Corollary 4.1.3] perfectly normal. By [Eng, Example 1.7.15.c], there exists a non-decreasing sequence $\{f_n’’\}$ of continuous real valued functions on the space $X$ which pointwise converges to the function $F\circ f$. Since the space $X$ is a locally compact space with a countable base, it is a union $X=\bigcup K_n$ of a non-decreasing sequence $\{K_n\}$ of its compact subsets. Now for each point $x\in X$ and each number $n$ put $f’_n(x)=\max\{0, f_n’’(x)\}$ if $x\in K_n$ and $f_n’=0$ if $x\in X\setminus K_n$. Since the function $F\circ f$ is non-negative, a sequence $\{f_n’\}$ of non-negative continuous real valued functions with compact support on the space $X$ pointwise converges to the function $F\circ f$. For each $n$ put $f_n=F^{-1}\circ f_n’$. Then $\{f_n\}$ is a non-decreasing sequence of non-negative continuous real valued functions with compact support on the space $X$, which pointwise converges to the function $f$.