Intensity measure explained

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, l A) and denote the expected value of a random element Y with \operatorname E [Y] .

The intensity measure

\operatorname E \zeta \colon l A \to [ 0 , inf ty] of \zeta is defined as\operatorname E \zeta(A) = \operatorname E[\zeta(A)] for all A \in l 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( dx)\right]= \int f(x) \operatorname E\zeta(dx) f on (S, l A) .

References

^{[1] [2] [3]}

Notes and References

1. Book: Kallenberg . Olav . Olav Kallenberg . 2017 . Random Measures, Theory and Applications. Switzerland . Springer . 53. 10.1007/978-3-319-41598-7. 978-3-319-41596-3.
2. Book: Klenke . Achim . 2008 . Probability Theory . limited . Berlin . Springer . 10.1007/978-1-84800-048-3 . 978-1-84800-047-6 . 526.
3. Book: Klenke . Achim . 2008 . Probability Theory . limited . Berlin . Springer . 10.1007/978-1-84800-048-3 . 978-1-84800-047-6 . 528.