I am trying to show that if $u : \mathbb{R} \to \mathbb{R}$ is a bounded, measurable, periodic function with period $1$ and $g$ is Lebesgue integrable over $\mathbb{R}$ then $$ \lim_{n \to \infty} \int_{\mathbb{R}} g(x) u(nx) \, dx = \left(\int_{0}^{1} u(x) \, dx \right) \left( \int_{\mathbb{R}} g(x) \, dx \right). $$ My Attempt: It is enough to prove the limit for $g$ being a nonnegative integrable function. I thought it might also suffice to prove it for simple and hence for characteristic functions but then there is an issue of changing the order of double limits (I am not sure if we can switch the order of limits in that case). I tried to use Fubini's Theorem but was stuck on how to introduce an integral inside the integral above on the left hand-side (a standard trick is to write $u(nx)$ as something like $\int_{0}^{u(nx)} dt$).
Any hint towards the solution will be immensely appreciated. Thanks in advance.
