PlanetMath: Radon-Nikodym theorem

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Radon-Nikodym theorem

(Theorem)

Let $\mu$ and $\nu$ be two $\sigma$ -finite measures on the same measurable space $(\Omega, \mathscr{S})$ , such that $\nu\ll \mu$ (i.e. $\nu$ is absolutely continuous with respect to $\mu$ .) Then there exists a measurable function , which is nonnegative and finite, such that for each $A\in \mathscr{S}$ ,

$\displaystyle \nu(A)=\int_A fd\mu.$

This function is unique (any other function satisfying these conditions is equal to

$\mu$ -almost everywhere,) and it is called the Radon-Nikodym derivative of $\nu$ with respect to $\mu$ , denoted by $f = \frac{d\nu}{d\mu}$ .

Remark. The theorem also holds if $\nu$ is a signed measure. Even if $\nu$ is not $\sigma$ -finite the theorem holds, with the exception that is not necessarely finite.

Some properties of the Radon-Nikodym derivative

Let $\nu$ , $\mu$ , and $\lambda$ be $\sigma$ -finite measures in $(\Omega,\mathscr{S})$ .

If $\nu \ll \lambda$ and $\mu\ll\lambda$ , then
$\displaystyle \frac{d(\nu+\mu)}{d\lambda} = \frac{d\nu}{d\lambda}+\frac{d\mu}{d\lambda}\;\; \mu$ -almost everywhere $\displaystyle ;$
If $\nu\ll\mu\ll\lambda$ , then
$\displaystyle \frac{d\nu}{d\lambda}=\frac{d\nu}{d\mu}\frac{d\mu}{d\lambda} \;\; \mu$ -almost everywhere $\displaystyle ;$
If $\mu\ll\lambda$ and is a $\mu$ -integrable function, then
$\displaystyle \int_\Omega gd\mu = \int_\Omega g\frac{d\mu}{d\lambda}d\lambda;$
If $\mu\ll\nu$ and $\nu \ll\mu$ , then
$\displaystyle \frac{d\mu}{d\nu}=\left(\frac{d\nu}{d\mu}\right)^{-1}.$

"Radon-Nikodym theorem" is owned by Koro.

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