It is known from Putnam 1967 problem, integration we have \begin{align*} \lim \limits_{n \rightarrow \infty} \int_0^1 f(x)g(nx)dx=\int_0^1g(x)dx\int_0^1f(x)dx \end{align*} for $g$ is a bounded, continuous function with period $1$ in $\mathbb{R}$ and $f$ is continuous in $[0,1]$. Can we generalise this as follows?
Let $g : \mathbb{R} \rightarrow \mathbb{C}$ be a bounded, measurable, periodic function with period $1$ and $f \in L^1(\mathbb{R})$. Then the similar result is true, that is \begin{align*} \lim \limits_{n \rightarrow \infty} \int_{\mathbb{R}} f(x)g(nx) dx=\left( \int_0^1g(x)dx \right) \left( \int_{\mathbb{R}} f(x)dx \right) \end{align*}