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Let $\langle f_n\rangle$ be a sequence of nonnegative measurable functions on $(-\infty,\infty)$ such thta $f_n\rightarrow f$ a.e., and suppose that $\int f_n\rightarrow\int f<\infty$. Then for each measurable set $E$ we have $\int_E f_n \rightarrow \int_E f$.

I know that by Fatou Lemma we have $\int_E f\leq\liminf_{n\rightarrow \infty} \int_E f_n$, but I have no idea about how to use the finiteness of $\int f$ to conclude the problem.

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You have $|f - f_n| \le |f| + |f_n| = f + f_n$ so that $f + f_n - |f - f_n| \ge 0$. Apply Fatou to this sequence. Since $f + f_n - |f - f_n| \to 2f$ you have $$\int 2f \le \liminf \int (f + f_n - |f - f_n|).$$

You have to be a bit careful with the $\liminf$ because it is generally not additive, but by hypothesis you have $\int f_n \to \int f$ so that $$\liminf \int (f + f_n - |f - f_n|) = 2 \int f - \limsup \int |f - f_n|.$$ Thus $$\limsup \int |f - f_n| \le 0.$$

Consequently for any set $E$ you get $$\left| \int_E f_n - \int_E f \right| \le \int_E |f - f_n| \le \int |f - f_n| \to 0.$$

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