Question about functions in $L^2$ space

Question

Let $f,g \in L^2 (\mathbb{R})$. For $n \in \mathbb{N}$, define $f_n \in L^2(\mathbb{R})$ as $f_n(x) = f(x-n)$ for all $x \in \mathbb{R}$.

Prove that $\lim_{n \rightarrow \infty} \int_{\mathbb{R}} f_n g dm = 0$. ($m$ is a Lebesgue measure.)

I completely lose my way to solve this problem.

I thought that since we don't what $lim f_n(x)$ is, we need to bound the $\int_\mathbb{R} f_n g dm$ to solve this problem.

But, how?

Answer 1

Try this way:

• $g \in C^{\infty}_c(\mathbb{R})$, then you should be able to compute this limit
• approximate $g \in L^2$ with $g_n \in C^{\infty}_c$ by density