$1 \leq p < \infty, \mu$ is a Lebesgue measure on $\mathbb R^n.$ $f: \mathbb R^n \rightarrow \mathbb R$ is Lebesuge measurable, and $\int_{\mathbb R^n} |f(x)|^p d\mu < \infty.$
Define $\mu_f$ by
$$\mu_f (\lambda) := \mu(\{x \in \mathbb R^n \mid |f(x)|> \lambda \}) \quad (\lambda>0).$$
I want to prove
$$\int_{\mathbb R^n} |f(x)|^p d \mu = p \int_0^\infty \lambda^{p-1} \mu_f(\lambda) d\lambda.$$
Somehow, to infer from $p\lambda^{p-1} $, I think that something is differentiated. However, the functions that appear are not always differentiable. It's been a few hours, but it doesn't work. Would you please help me?
