let$\{f_n\}$ be a sequence of Lebesgue measurable functions on $[0,\infty)$ suth that $\vert {f_n}\vert \le e^{-x}$ for all $x \in [0,\infty)$. if $f_n \rightarrow 0 [a.e]$, then $f_n \rightarrow 0 [a.u] $
i don't have any idea about problem, please help me. it is very valuable for me that be solved it.
Fix $0<\varepsilon<1$. Then $e^{-x}\le \varepsilon$ for all $x\ge |\log\varepsilon|$ and thus $|f_n(x)|\le \varepsilon$ for all $x\ge |\log\varepsilon|$. On the other hand, since the interval $[0,|\log\varepsilon|]$ has finite measure and $f_n$ converges almost everywhere, it follows by Egoroff's theorem (which says that if a set has finite measure then pointwise convergence implies almost uniform convergence), you have almost uniform convergence in $[0,|\log\varepsilon|]$.