let$\{f_n\}$ be a sequence of Lebesgue measurable functions on $[0,\infty)$ suth that $\vert {f_n (x)}\vert \le e^{-x}$

Question

let$\{f_n\}$ be a sequence of Lebesgue measurable functions on $[0,\infty)$ suth that $\vert {f_n (x)}\vert \le e^{-x}$ for all $x \in [0,\infty)$. if $f_n \rightarrow 0 [a.e]$, then $f_n \rightarrow 0 [a.u] $

Answer 1

Hint: Let $\epsilon>0.$ Choose $M>0$ such that $e^{-x}<\epsilon$ on $[M,\infty).$ Use Egorov on $[0,M].$