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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 $\zeta$ 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 $A\in {\mathcal {A}}$ .^[2] ^[3]

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 $\operatorname {E} \zeta$ .