Let $X$ and $Y$ be topological spaces. By a simple function $\phi: X\to Y$ we mean a finite range Borel measurable function.


Q. Is the point-wise limit of a sequence of simple functions a Borel measurable function?



Is the point-wise limit of simple functions a measurable function?
The answer to your question is Yes provided the topology of $Y$ is such that for each non-empty open set $O\subset Y$ there is a strictly increasing sequence $(O\_k)$ of open sets:
$$
\overline O\_k\subset O\_{k+1}\subset O,\quad k=1,2,\ldots,
$$
with $\bigcup\_kO\_k=O$. For suppose $(f\_n)$ is a sequence of Borel functions from $X$ to $Y$ with pointwise limit $f$. With $O$ and the $O\_k$ as above,
$$
f^{-1}(O)=\bigcup\_{k=1}^\infty\bigcup\_{n=1}^\infty\bigcap\_{m=n}^\infty f\_m^{-1}(O\_k).
$$
This shows that $f^{-1}(O)$ is a measurable subset of $X$. Because the Borel $\sigma$-field on $Y$ is generated by the open sets, it follows that $f$ is Borel measurable.


In particular, the condition above is true if $Y$ is metrizable.


