Prove that for almost every $x\in$ $\mathbb{R}$ , $\lim_{n\to\infty}n^{-p}f(nx)=0$ .

Question

Let $f\in$$L^1(\mathbb{R})$ and let $p>0$ . Prove that for almost every $x\in$$\mathbb{R}$ , $\lim_{n\to\infty}n^{-p}f(nx)=0$.

Answer 1

Hint: If $f_n$ is a sequence of measurable functions on $(X,\mu)$ and

$$ \sum_{n=1}^{\infty}\int_X |f_n|\,d\mu < \infty,$$

then $\lim_{n\to\infty} f_n(x)=0$ for $\mu$-a.e. $x\in X.$ Why?