Definition:Radon-Nikodym Derivative

Definition
Let $\struct {X, \Sigma}$ be a measurable space.

Let $\map {\mathcal M^+} {X, \Sigma, \R}$ be the space of positive real-valued measurable functions.

Let $\mu$ and $\nu$ be $\sigma$-finite measures on $\struct {X, \Sigma}$ such that:


 * $\nu$ is absolutely continuous with respect to $\mu$.

Let $\sim$ be the $\mu$-almost everywhere equality relation on $\map {\mathcal M} {X, \Sigma, \R}$ restricted to the space of positive real-valued measurable functions $\map {\mathcal M^+} {X, \Sigma, \R}$.

We say that a $\Sigma$-measurable function $g : X \to \hointr 0 \infty$ is a Radon-Nikodym derivative of $\nu$ with respect to $\mu$, :


 * $\ds \map \nu A = \int_A g \rd \mu$

for each $A \in \Sigma$.

We also define an element of $\map {\mathcal M^+} {X, \Sigma, \R}/\sim$ by:


 * $\ds \frac {\d \nu} {\d \mu} = \eqclass g \sim$

where $\eqclass g \sim$ is the equivalence class of $g$ under $\sim$, which may be called the Radon-Nikodym derivative of $\nu$ with respect to $\mu$.

Also see

 * The Radon-Nikodym Theorem establishes the existence of a Radon-Nikodym derivative, and shows that any two Radon-Nikodym derivatives coincide almost everywhere, so the object $\dfrac {\d \nu} {d \mu}$ is also well-defined.