Let $\{f_n\}_{n=1}^{\infty}$ be a sequence of real valued measurable functions on $[0,1]$. Show that there is a sequence of positive real numbers $\{a_n\}_{n=1}^{\infty}$ such that $a_nf_n \rightarrow 0$ a.e. on $[0,1]$.
Here is my idea. Let $E=[0,1]$. Let $F_n \subseteq [0,1]$ and closed such that $m(E\setminus F_n)<\epsilon$. Also, let $g_n$ be a continuous function on $F_n$, and thus bounded since $F$ is closed. Each $g_n$ is bounded by a corresponding $M_n$.
My goal was to use Lusin's theorem, but I'm starting to think that my idea is a dead end.