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 $\mu$.

  1. Since Borel measurable set is approximated by finite union of open intervals, simple function has form $\sum \chi _I$ where $i$ is open interval.

  2. 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?

  • 1
    $\begingroup$ What's your definition of continuous a.e.? $\endgroup$ – Jacky Chong Oct 3 '16 at 0:45
  • $\begingroup$ $\mu$ is probability measure. g is continuous a.e $\mu$ means that there is borel set $B$ g is continuous at all points on$B$ and $\mu(B)$ $\endgroup$ – Planche Oct 3 '16 at 1:04
  • $\begingroup$ So is 1 on rational and 0 on irrational continuous? $\endgroup$ – Jacky Chong Oct 3 '16 at 1:52
  • $\begingroup$ What's your mean? $\endgroup$ – Planche Oct 3 '16 at 15:32
  • $\begingroup$ The function you give is not continuous. But Is it related to my question? why? $\endgroup$ – Planche Oct 3 '16 at 15:34

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