# Does $f_n \to 0$, a.e., implies $\int_{\mathbb R} \sin(f_n(x)) dx \to 0$, when each $f_n \in L^1$

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.

You are correct. For example, consider $f_n=\chi_{[n, n+1]}$.
Try $f_n:=\frac{\pi}2\chi_{(n,n+1)}$.