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Let $\{f_n\}$ be a sequence of $L^1(\mathbb R)$ functions converging a.e. to zero. Does $$ \lim_{n\to \infty} \int_{\mathbb R} \sin(f_n(x)) dx = 0? $$

I think the answer is no, but I can't find a counterexample.

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You are correct. For example, consider $f_n=\chi_{[n, n+1]}$.

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Try $f_n:=\frac{\pi}2\chi_{(n,n+1)}$.

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