Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that

(i)	$f : X \to [0, \infty]$ is an D5610: Unsigned basic Borel function on $M$

Let $p \in (0, \infty)$ be a D993: Real number.

Then \begin{equation} \int_X f^p \, d \mu = \int^{\infty}_0 \mu (\{ x \in X : f(x) > t \}) \, p t^{p - 1} d t \end{equation}

Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that

(i)	$f : X \to [0, \infty]$ is an D5610: Unsigned basic Borel function on $M$

Let $p \in (0, \infty)$ be a D993: Real number.

Then \begin{equation} \Vert f \Vert_{L^p}^p = \int^{\infty}_0 \mu (f > t) \, d t^p \end{equation}