Frechet's Theorem states that every measurable function $f$ on $\mathbb{R}$ is the limit of a sequence of continuous functions converging almost everywhere.
Frechet's Theorem is then used to prove Lusin's Theorem.
Is it possible to "use" Lusin's Theorem to derive Frechet's Theorem as a special case?
What I tried is use Lusin's Theorem to get a continuous function $f_k$ and a closed set $F_k\subseteq\mathbb{R}$ such that $f_k|_{F_k}=f$ and $m(\mathbb{R}\setminus F_k)<\frac{1}{k}$. Then let $F=\bigcup_{k=1}^\infty F_k$. This looks promising as $m(\mathbb{R}\setminus F)=0$.
However I can't show that on $F$, $|f_k-f|\to 0$ as $k\to\infty$.
Thanks for any help.