Lusin's Theorem: Suppose $f$ is measurable and finite valued on $E$ with $E$ of finite measure. Then for every $\epsilon >0$ there exists a set $F_{\epsilon}$,with $$F_{\epsilon} \subset E, \hspace{2pt} and \hspace{5pt} m(E-F_{\epsilon}) < \epsilon$$ and such that $f|_{F_{\epsilon}}$ is continuous.(Reference:Real Analysis by EM Stein)
My question is if we assume $f$(defined on $E$) to be a characteristic function on a measurable set $F \subset E$ then for what subsets $F$ will there exist a set $G \subset E$ such that $m(E-G)=0$ and $f|_{G}$ is continuous. For example if $E=[0,1]$ and $F=[0,1] \cap \mathbb{Q}^c$ then $f|_F$ is continuous and $m(E-F)=0$.
But I was unable to do the same if $F$ is a cantor like set with positive measure. I have a feeling that it is not possible in this case but I am unable to prove it. Thanks.