Integral of Increasing Function Composed with Measurable Function/Corollary

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Corollary to Integral of Increasing Function Composed with Measurable Function

Let $\left({X, \Sigma, \mu}\right)$ be a measure space, and let $p \in \R$, $p \ge 1$.

Let $f: X \to \R$ be a $p$-integrable function.


Then:

$\displaystyle \left\Vert{f}\right\Vert_p^p = \int_0^\infty p t^{p-1} F \left({t}\right) \, \mathrm d t$

where $F$ is the survival function of $\left\vert{f}\right\vert$.


Proof