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Radon-Nikodym derivative
Formulation 0
Let $M = (X, \mathcal{F})$ be a D1108: Measurable space such that
(i) $\mu, \nu : \mathcal{F} \to [0, \infty]$ are each an D85: Unsigned basic measure on $M$
(ii) $f : X \to [0, \infty]$ is an D5610: Unsigned basic Borel function on $M$
Then $f$ is a Radon-Nikodym derivative of $\mu$ with respect to $\nu$ if and only if \begin{equation} \forall \, E \in \mathcal{F} : \mu(E) = \int_E f \, d \nu \end{equation}
Also known as
Density function
Conventions
Convention 0 (Notation for a Radon-Nikodym derivative) : If $\mu, \nu : M \to [0, \infty]$ are each an D85: Unsigned basic measure and $f : M \to [0, \infty]$ is a D2888: Radon-Nikodym derivative of $\mu$ with respect to $\nu$, then we may denote $f$ by $\frac{d \mu}{d \nu}$.
Child definitions
» D209: Probability density function
Results
» R4714: Sufficient condition for the existence of density function for a random euclidean real number