3
$\begingroup$

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?

$\endgroup$
2
$\begingroup$

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 $\leq f$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.