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.^[1]

Definition

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

The intensity measure

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

of ${\displaystyle \zeta }$ is defined as

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

for all ${\displaystyle A\in {\mathcal {A}}}$.^[2][3]

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

Properties

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

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

for every positive measurable function ${\displaystyle f}$ on ${\displaystyle (S,{\mathcal {A}})}$.^[3]