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Let $f_n$ be a sequence of nonnegative functions defined on $\mathbb{R}^N$ such that

$f_n \rightarrow f $ almost everywhere on $\mathbb{R}^N$ and such that $$\int_{\mathbb{R}^N} f_n \rightarrow \int_{\mathbb{R}^N} f $$ If $f$ belongs to $L^1(\mathbb{R}^N)$ show that for every Borel set $B$ $$\int_B f_n \rightarrow \int_B f $$

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up vote 1 down vote accepted

Use Fatou's Lemma on $f + f_n - |f - f_n|$ to show that actually $f_n \rightarrow f$ in $L^1$. This easily implies your claim.

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Note that $|f_n\cdot\chi_B|\leq f_n$ and $|f\cdot\chi_B|\leq f$. So by the Lebesgue Dominated Convergence Theorem, we have that $$\lim\limits_{n\to\infty}\int_Bf_n=\lim\limits_{n\to\infty}\int_{\mathbb{R}^N}f_n\cdot \chi_B=\int_{\mathbb{R}^N}f\cdot\chi_B=\int_Bf.$$

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