Consider $f:\mathbb{R}\to \mathbb{C}$ a bounded and $1$-periodic function, and $g \in L^1(R)$ then
$\lim_{n\to \infty} \int _{R}g(x)f(nx)dx=\int_0^1f(s)ds\int_R g(t)dt.$
I think the fact that $f$ is complex valued as well $g$ is not a big deal but I couldn't solve it in any case.