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 $$ (S, \mathcal A) $$ 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 $$ A \in \mathcal A$$.

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 $$ \operatorname E\zeta $$.

Properties
The intensity measure $$ \operatorname E\zeta $$ is always s-finite and satisfies
 * $$\operatorname E \left[ \int f(x) \; \zeta(\mathrm dx)\right]= \int f(x) \operatorname E\zeta(dx)$$

for every positive measurable function $$ f $$ on $$ (S, \mathcal A) $$.