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 .

The intensity measure
: \operatorname E \zeta \colon \mathcal A \to ,\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 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 $(S, \mathcal A)$.

References

{{Measure theory

Measures (measure theory)
Probability theory