Is the pointwise limit of a uniformly bounded net of increasing positive continuous functions on a compact Hausdorff space Borel measurable?

Question

Let $X$ be a compact and Hausdorff space. Let $\{f_i\}$ be a uniformly bounded net of increasing positive continuous functions on $X$. Let $f$ be the pointwise limit of the net $\{f_i\}$. Is $f$ a Borel measurable function?

Answer 1

Yes, in fact, $f$ is lower semicontinuous (i.e., for all $r\in\mathbb{R}$, $f^{-1}((r,\infty))$ is open; since these generate the Borel sets on $\mathbb{R}$, this implies $f$ is Borel). For if $f(x)>r$, there is an $f_i$ such that $f_i(x)>r$, and then $f_i(y)>r$ for all $y$ in some neighborhood of $x$ by continuity, so then $f(x)>r$ for all $y$ in that neighborhood since the $f_i$ are increasing.

Conversely, if $f:X\to\mathbb{R}$ is bounded and lower semicontinuous, it is easy to show using Urysohn's lemma that $f$ is the limit of the net consisting of all continuous functions that are $\leq f$.