1. ## Lebesgue Integral

Let $(f_n) \in L^p(X)$ for all $n \in \mathbb{N}$ and $1\le p <\infty$.Suppose there exist a function $g \in L^p(X)$ such that $|f_n| \le g$for all $n \in \mathbb{N}$.

Prove that for each $\epsilon>0$, there exist 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
Let $(f_n) \in L^p(X)$ for all $n \in \mathbb{N}$ and $1\le p <\infty$.Suppose there exist a function $g \in L^p(X)$ such that $|f_n| \le g$for all $n \in \mathbb{N}$.

Prove that for each $\epsilon>0$, there exist 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 $n \in \mathbb{N}.$
Since $|f_n|\leqslant g$ for all n, it suffices to find a set $E_\varepsilon \subseteq X$ with $m(E_\varepsilon)< \infty$ such that if $F\subseteq X$ and $F \cap E_\epsilon =\emptyset$, then $\int_F |g|^p dm < \varepsilon^p$ for all $n \in \mathbb{N}.$ That is a question that you have raised in this forum previously, and you'll find the answer here.