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Find a sequence of integrable functions $f_n:[0,1]\to\mathbb{R}$ such that $f_n\to0$ almost everywhere in $[0,1]$ and $\lim_{n\to\infty}\int^1_0 f_n(x)dx=0$ but there exists a set $E\subset [0,1]$ such that $\int_E f_n(x)dx=1$ for all $n\in\mathbb{N}$

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Hint. Consider

$$f_n(x) = 2n \mathbf{1}_{(0,\frac{1}{2n}]}(x) - 2n \mathbf{1}_{[1-\frac{1}{2n}, 1)}(x) $$

together with $E = [0, \frac{1}{2}]$.

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