Measure Densities

Definition

Suppose $(X, \mathcal{F} )$ is a measurable space. A measure $\mu : \mathcal{F} \to \Rbar$ is said to have a density with respect to a measure $\nu : \mathcal{F} \to \Rbar$ if there exists a measurable function $f: X \to \R _+$

\[ \mu (A) = \int _A f d\nu \quad \text{for all } A \in \mathcal{F} \]

In this case $f$ is called a density of $\mu $ with respect to $\nu $.

Examples

Probability on finite sets. Suppose $P$ is a probability measure for a finite set $\Omega $. Define $p: \Omega \to [0,1]$ by

\[ p(\omega ) = P(\set{\omega }) \quad \text{for all } \omega \in \Omega \]

Then $p$ is a probability distribution. Moreover, $p$ is a density for $P$ with respect to the counting measure $\#: \powerset{\Omega } \to \R $. Witness, for every $A \subset \Omega $,

\[ \int_{A} p d\# = \sum_{a \in A} p(\omega ) \]

We recognize the right hand side as $P(A)$ by using the additivity of $P$.