TITLE: $L^p$-norm of a non-negative measurable function
QUESTION [5 upvotes]: Can I ask a homework question here?
Let $f$ be measurable and non-negative in $\mathbb R^d.$ 
Using Fubini's theorem, show that for $1 \leq p \lt \infty,$
$$\lVert f\rVert^p_p = \int^{\infty}_{0}pt^{p-1}\lambda(\{x:f(x)\gt t\}) \ dt.$$

REPLY [3 votes]: $$f(x)^p=\int_0^{f(x)}pt^{p-1}\,\mathrm dt=\int_0^{+\infty}pt^{p-1}\,\mathbf 1_{f(x)\gt t}\,\mathrm dt$$