Let $(\Omega,\mathscr{F})$ be a measurable space, $X$ a topological space, $\mathscr{B}$ the borel sets of $X$, $f:\Omega\longrightarrow X$ a $\mathscr{F}/\mathscr{B}$-measurable function. As per usual, by simple function I understand a $\mathscr{F}/\mathscr{B}$-measurable function whose range is finite.

Question: what are necessary and sufficient conditions for the existence of a sequence $(f_n)_{n\in\mathbb{N}}$ of simple functions such that $f_n\rightarrow f$ pointwise?

I already know that it is sufficient for $X$ to be metrizable and separable, and I suspect it is also sufficient if $X$ is separable, first-countable and regular...

Many thanks!



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