Let $(\mathbb{R},B,\mu)$ be measure space where $B$ is Borel sigma algebra, and $\mu$ is probability measure.
Let $g :(\mathbb{R},B,\mu) \to (\mathbb{R},B)$ be nonnegative bounded measurable function such that $g$ is almost everywhere ($\mu$) continuous.
Is there sequence of continous function $g_n$ such that $g_n(x) \to g(x)$ at continuity of $g$?
My try: 1. there are sequence $g_n$ of simple-function converges to $g$ a.e $\mu$.
Since Borel measurable set is approximated by finite union of open intervals, simple function has form $\sum \chi _I$ where $i$ is open interval.
If simple function $g_n$ is discontinous, then I transform graph of $g_n$ so that $g_n$ is continuous.(BY linear function)
But I think $3$ is impossible.
Could you help me?