integral of $f^p$ on a $\sigma$-finite measure

Question

I have a σ-finite measure on $(X,\mathbb{F},\mu)$ and a non negative function $f$ that is measurable on X. Further there exists a $p>0$ such that $$ \int_X f^p d\mu\:\text{ exists.} $$

It is required to show that $$ \int_X f^p d\mu=p\int_{[0,\infty]}t^{p-1}\mu\big(\{x\in X: f(x)\geq t\}\big)dt $$

I would be grateful for some hints, because I am not really coming a step forward.

Answer 1

Hint

$$\mu\{x\mid f(x)>t\}=\int_X\boldsymbol 1_{\{x\mid f(x)>t\}}\mathrm d\mu(x),$$ and use Fubini.