The following is a standard result of real-valued measurable functions:
Theorem: let $\{f_n\}$ be a sequence of measurable functions $(X,\Sigma_X)\to(\mathbb{R},\mathcal{B}_{\mathbb{R}})$, where $\mathcal{B}_{\mathbb{R}}$ is the Borel $\sigma$-algebra generated by the Euclidean topology on $\mathbb{R}$. Then $$f_n\to f \text{ (pointwise)} \implies f \text{ is measurable}.$$
More generally, given a measurable function $(X,\Sigma_X)\to(Y,\Sigma_Y)$, if the codomain measure space is Borel we can already talk (because there is a topology on $Y$) about limits of sequences i.e. questions such as $$\text{Does $f_n(x)$ have any limits as $n\to\infty$?}$$ make sense. If the topology in question is Hausdorff, the limits -if they exist- must be unique, and we can talk about the function $$f:X\to Y:x\mapsto \lim_{n\to \infty}f_n(x).$$
With all that in mind, I'm wondering if the following, more general result holds:
Theorem (?): let $\{f_n\}$ be a sequence of measurable functions $(X,\Sigma_X)\to(Y,\mathcal{B}_{Y})$, where $\mathcal{B}_Y$ is the Borel $\sigma$-algebra generated by some Hausdorff topology $\tau$ on $Y$. Then $$f_n\to f \text{ (pointwise)} \implies f \text{ is measurable}.$$
I know two proofs of the theorem involving real-valued functions: one which uses $\liminf f$ and $\limsup f$, and another which works with $\pi$-systems and the fact that $f$ is measurable if and only if $$\{f\le c\} := \{r\in\mathbb{R} : f(r) \le c\}\in\Sigma_X \ \text{for any } c\in\mathbb{R}.$$ Neither proof seems generalizable.
In this post the OP asks whether the above result is true for first countable Hausdorff spaces, and it seems they recieve no solid answer on the matter.
I'm asking whether it holds for any Hausdorff space.
