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Suppose $f_n \to f$ almost everywhere on $X$. Let $\epsilon > 0$ and choose $\delta > 0$ such that for all measurable sets $E\subseteq X$ such that $ \mu(E)< \delta $, we have $\int_E |f_n| < \epsilon$ for every $n$. Using Fatou's Lemma, how can prove that $f$ is integrable on any measurable set $E\subseteq X$ such that $\mu(E) < \delta$ and $\int_E |f| < \epsilon$.

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