Intensity measure

Last updated December 15, 2024

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 $\operatorname {E} \zeta$ .

Properties

The intensity measure $\operatorname {E} \zeta$ is always s-finite and satisfies

\operatorname {E} \left[\int f(x)\;\zeta (\mathrm {d} x)\right]=\int f(x)\operatorname {E} \zeta (dx)

for every positive measurable function $f$ on $(S,{\mathcal {A}})$ .^[3]

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References

↑ Klenke, Achim (2008). Probability Theory . Berlin: Springer. p. 528. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
↑ Klenke, Achim (2008). Probability Theory . Berlin: Springer. p. 526. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
1 2 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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[Klenke528-1] Klenke, Achim (2008). Probability Theory . Berlin: Springer. p. 528. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.

[Klenke526-2] Klenke, Achim (2008). Probability Theory . Berlin: Springer. p. 526. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.

[Kallenberg53-3] 1 2 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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Intensity measure

Contents

Definition

Properties

Related Research Articles

References