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. [1]
Definition
Let [math]\displaystyle{ \zeta }[/math] be a random measure on the measurable space [math]\displaystyle{ (S, \mathcal A) }[/math] and denote the expected value of a random element [math]\displaystyle{ Y }[/math] with [math]\displaystyle{ \operatorname E [Y] }[/math].
The intensity measure
- [math]\displaystyle{ \operatorname E \zeta \colon \mathcal A \to [0,\infty] }[/math]
of [math]\displaystyle{ \zeta }[/math] is defined as
- [math]\displaystyle{ \operatorname E \zeta(A)= \operatorname E[\zeta(A)] }[/math]
for all [math]\displaystyle{ A \in \mathcal A }[/math].[2] [3]
Note the difference in notation between the expectation value of a random element [math]\displaystyle{ Y }[/math], denoted by [math]\displaystyle{ \operatorname E [Y] }[/math] and the intensity measure of the random measure [math]\displaystyle{ \zeta }[/math], denoted by [math]\displaystyle{ \operatorname E\zeta }[/math].
Properties
The intensity measure [math]\displaystyle{ \operatorname E\zeta }[/math] is always s-finite and satisfies
- [math]\displaystyle{ \operatorname E \left[ \int f(x) \; \zeta(\mathrm dx)\right]= \int f(x) \operatorname E\zeta(dx) }[/math]
for every positive measurable function [math]\displaystyle{ f }[/math] on [math]\displaystyle{ (S, \mathcal A) }[/math].[3]
References
- ↑ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 528. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6. https://archive.org/details/probabilitytheor00klen_646.
- ↑ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 526. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6. https://archive.org/details/probabilitytheor00klen_646.
- ↑ 3.0 3.1 Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. p. 53. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.
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