Definition:Radon-Nikodym Derivative

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

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


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

Let $\map {\mathcal M} X$ be the space of $\Sigma$-measurable functions $f : X \to \overline \R$.

Define the equivalence $\sim$ by:


 * $f \sim g$ if $f = g$ $\mu$-almost everywhere.

From the Radon-Nikodym Theorem, there exists a $\Sigma$-measurable function $g : X \to \hointr 0 \infty$ such that:


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

for each $A \in \Sigma$.

We define the Radon-Nikodym derivative $\dfrac {\d \nu} {\d \mu}$ by:


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