This is the statement of Lusin's Theorem (taken from Royden):
Let $f$ be a real-valued measurable function on $E$. Then for each $\epsilon>0$, there is a continuous function $g$ on $\mathbb{R}$ and a closed set $F\subseteq E$ for which $f=g$ on $F$ and $m(E\setminus F)<\epsilon$.
Q1) Taking $\epsilon=\frac 1k$, where $k\in\mathbb{N}$, we get a continuous function $g_k$ and a closed set $F_k\subseteq E$, for which $f=g_k$ on $F_k$ and $m(E\setminus F_k)<\frac 1k$.
What I am curious is, can we further assume that the $F_k$ are nested, i.e. $$F_1\subseteq F_2\subseteq F_3\subseteq\dots$$? This may be useful for some proofs. Intuitively, since $F_k$ are getting bigger, it seems possible but I am not sure how to prove it.
Update: I thought of replacing $G_k=\bigcup_{i=1}^k F_i$, but it doesn't work.
Thanks!