Radon–Nikodym theorem

The Radon-Nikodym theorem is a result in functional analysis that states that if a measure $Q$ is absolutely continuous with respect to a positive measure $P$ then there is a function, often writen ${\frac {\delta Q}{\delta P}}$, on the underlying space such that

$Q(A)=\int _{A}{\frac {\delta Q}{\delta P}}dP$

The function ${\frac {\delta Q}{\delta P}}$ is called the Rado-Nikodym derivative. The choice of notation and the name of the function reflects the fact that the function is analogous to a derivative in calculus in the sense that it describes the rate of change of probability density of one measure with respect to another. It follows trivially from the definition of the derivative that

${\mathcal {(}}E)_{P}(A)={\mathcal {(}}E)_{Q}({\frac {\delta Q}{\delta P}}A)$

where ${\mathcal {(}}E)$ is the expectation operator.