Integral of Increasing Function Composed with Measurable Function

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Theorem

Let $\left({X, \Sigma, \mu}\right)$ be a $\sigma$-finite measure space.

Let $f: X \to \R_{\ge 0}$ be a positive measurable function.

Let $\phi: \R_{\ge 0} \to \R_{\ge 0}$ be a continuously differentiable, increasing function such that $\phi \left({0}\right) = 0$.


Then:

$\displaystyle \int \phi \circ f \rd \mu = \int_0^\infty \phi' \left({t}\right) F \left({t}\right) \rd t$

where:

$F$ is the survival function of $f$
$\displaystyle \int_0^\infty$ denotes an improper integral.


Corollary

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


Sources