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**problem** Let $\displaystyle f \in L_p(X)$,1 $\displaystyle \le p < \infty $ and let $\displaystyle \epsilon > 0$. Show that there exists 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 $\displaystyle \int_F |f|^p dm < \epsilon^p$

I let $\displaystyle E_\epsilon = \{ x \in X : |f(x)| \ge \delta_\epsilon \}$,where $\displaystyle \delta_\epsilon > 0$ and $\displaystyle F = \{ x \in X : |f(x)| < \delta_\epsilon \}$.

$\displaystyle \int_F |f|^P dm < \int_F (\delta_\epsilon)^p dm = (\delta_\epsilon)^p m(F)$

I am not sure whether my construction of the sets are correct because I can not make any conclusion regarding $\displaystyle m(F)$.

Can anyone comment on this?