Suppose $\displaystyle g\in L_p(X)$ and $\displaystyle |f_n| \le g$,show that for each $\displaystyle \epsilon >0$, there is a set $\displaystyle E_\epsilon \subseteq X$ with $\displaystyle m(E_\epsilon) < \infty$ such that if $\displaystyle F \subseteq X$ and $\displaystyle F\cap E_\epsilon = \phi$ ,then

$\displaystyle \int_F |f_n|^p dm <\epsilon^p$, for all $\displaystyle n\in \mathbb{N}$