Radon–Nikodym Derivative

Let $\mu$ and $\nu$ be $\sigma$ -finite positive measures on the same measurable space. If $\nu\ll\mu$ , the Radon–Nikodym derivative $d\nu/d\mu$ is the $\mu$ -almost-everywhere unique measurable function $X\geq 0$ such that

\nu(S)=\int_S X\,d\mu

for every measurable set $S$ .

Remarks

Existence and almost-everywhere uniqueness follow from the Radon–Nikodym theorem.

In Shale's notation for a Gaussian pushforward, $X(T)=dn(T)/dn$ plays the role of a measure-theoretic Jacobian.

Examples

For $\nu=f\mu$ with $f\ge0$ , one has $d\nu/d\mu=f$ .