Intensity measure

In probability theory, an intensity measure is a measure that is derived from a random measure. The intensity measure is a non-random measure and is defined as the expectation value of the random measure of a set, hence it corresponds to the average volume the random measure assigns to a set. The intensity measure contains important information about the properties of the random measure. A Poisson point process, interpreted as a random measure, is for example uniquely determined by its intensity measure.

Definition

Let $\zeta$ be a random measure on the measurable space $(S,{\mathcal {A}})$ and denote the expected value of a random element $Y$ with $\operatorname {E} [Y]$.

The intensity measure

$\operatorname {E} \zeta \colon {\mathcal {A}}\to [0,\infty ]$

of $\zeta$ is defined as

$\operatorname {E} \zeta (A)=\operatorname {E} [\zeta (A)]$

for all $A\in {\mathcal {A}}$.

Note the difference in notation between the expectation value of a random element $Y$, denoted by $\operatorname {E} [Y]$ and the intensity measure of the random measure $\zeta$, denoted by $\operatorname {E} \zeta$.

Properties

The intensity measure $\operatorname {E} \zeta$ is always s-finite and satisfies

$\operatorname {E} \left[\int f(x)\;\zeta (\mathrm {d} x)\right]=\int f(x)\operatorname {E} \zeta (dx)$

for every positive measurable function $f$ on $(S,{\mathcal {A}})$.