Integral of Increasing Function Composed with Measurable Function

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Theorem

Let $\struct {X, \Sigma, \mu}$ 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 $\map \phi 0 = 0$.


Then:

$\ds \int \phi \circ f \rd \mu = \int_0^\infty \map {\phi'} t \map F t \rd t$

where:

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


Corollary

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


Then:

$\ds \norm f_p^p = \int_0^\infty p t^{p - 1} \map F t \rd t$

where $F$ is the survival function of $\size f$.


Proof


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Sources

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