prove $\exists$ a continuous function g s.t. $|g(x)-f(x)|<\epsilon$ for all x in a subset E of (a,b) and $\mu[(a,b)-E]<\delta$

Question

I am having trouble starting with ths problem. Let f be a Lebesgue-integrable function over a bounded interval (a,b). Prove that for any $\epsilon >0$, $\delta >0$, there exists a continuous function g in $[a,b]$ such that $|g(x)-f(x)|<\epsilon$ for all x in a subset E of (a,b) and $\mu[(a,b)-E]<\delta$

It seems to me that this should show that all integrable functions can be somehow approximated by almost everywhere continuous functions. i.e all integrable functions are continuous a.e.

I am struggeling to start the proof - Im glad for any sort of help!

Thanks

Answer 1

The answer depends on what you know about integrable functions. For example you can first prove that continuous functions are dense in $L_1[a,b]$. Then if $||f-g||_1 < \varepsilon$ then there difference can't be too big on a big subset. Of course, this result also follows from Lusin's theorem that is a very standard yet more subtle result, see here