If $F$ is the distribution function of an $L^p$ function, then $\lambda^p F(\lambda)\to0$ as $\lambda\to0$

Question

Let $1\leq p<\infty$, $f\in L^p (\mathbb{R}^n)$. Let $F(\lambda)=m\{|f(x)|>\lambda\}$, show that:

$$\lim_{\lambda\to 0} \lambda^{p}F(\lambda)=0$$

What I only know about distribution function is:

$$\int_{0}^{\infty} p\lambda^{p-1}F(\lambda)=\int |f|^p dx$$

Answer 1

This is Lebesgue dominated convergence theorem for the functions $f_\lambda:x\mapsto\lambda^p\,\mathbf 1_{|f(x)|\gt\lambda}$ such that $f_\lambda\to0$ pointwisely when $\lambda\to0$ and the domination $|f_\lambda|\leqslant g$ with $g=|f|^{1/p}$.

Answer 2

Take a simple function $g$ such that $\lVert f-g\rVert_p\lt\varepsilon$. Then $$\lambda^pF(\lambda)\leqslant \lambda^pm\{|f-g|>\lambda/2\}+\lambda^pm\{|g|>\lambda/2\}\leqslant 2^p\varepsilon+2^p\chi_{(0,\lVert g\rVert_\infty]}(\lambda).$$