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Let $f_n\in L^1_{loc}(\mathbb{R})$ be a sequence of a locally integrable functions such that for all $a<b$ $$\int_a^b|f_n(x)|dx\to 0,$$ when $n\to\infty$. We know that for each interval $[a,b]$ there exists a subsequence $(f_{p_n})$ which converges pointwise a.e. on $[a,b]$ to $0$. But can we construct a subsequence which converges a.e. on $\mathbb{R}$. My problem is that the subsequence $(f_{p_n})$ depends on the interval $[a,b]$. I don't even know if it is possible to do this or not.

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Use Cantor's diagonalization argument.

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