Let $f$ be a non-negative measurable function defined on a measurable set $E \subset \mathbb{R}$. Prove that for any $\delta > 0$ there exists a closed, measurable set $F$ such that $m(E/F) < \delta$ and the restriction of $f$ on $F$ is continuous.
I get what it's asking but I don't know at all how I would go about solving this.
I tried thinking of strange examples but I can't even find $F$ in special cases. For example, the rational indicator function on $[0,1]$. What closed $F$ is there? The obvious thing is to take out the rationals, but $[0,1] \setminus \mathbb{Q}$ is not closed.