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 ${\displaystyle \operatorname {E} \zeta }$ .

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]