Intensity measure

Let ${\displaystyle \zeta }$ be a random measure on the measurable space ${\displaystyle (S,{\mathcal {S}})}$ 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 ${\displaystyle \operatorname {E} \zeta }$.

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]