# Math Help - Lebesgue Integral

1. ## Lebesgue Integral

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

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

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

$\int_F |f_n|^p dm <\epsilon^p$, for all $n\in \mathbb{N}$
Note that it suffices to prove the bound for $g$, since $\int |f_n|^p dm\leq \int |g|^p dm$.

I guess $X$ is supposed to be $\sigma$-finite: there is an increasing sequence $(E_k)_{k\geq 0}$ of measurable subsets with $\mu(E_k)<\infty$ and $X=\bigcup_k E_k$. Then you should justify that $\int_{E_k} |g|^p dm\to \int |g|^p dm$ and use this to conclude.