Lebesgue integral of $f^{p}$

Question

$f : \mathbb{R}^d\rightarrow [0,\infty], p\geq1$ show that:

$\int_{\mathbb{R}^d}f^p = p \int_0^\infty y^{p-1}m(\{x \in \mathbb{R}^d: f(x)>y\})\,dy$.

Any hints? Thanks in advance!

Answer 1

You can use this (general) identity for positive functions :

$\int_X f(x) d\mu = \int_0^{\infty} \mu( x\in X, f(x) > t) dt$