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 $(S,{𝒜})$ and denote the expected value of a random element $Y$ with $\mathrm{E}[Y]$.

The intensity measure

$\mathrm{E}\zeta :{𝒜}\to [0,\mathrm{\infty }]$

of $\zeta$ is defined as

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

for all $A\in {𝒜}$.

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

Properties

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

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

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