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 dx)\right]= \int f(x) \operatorname E\zeta(dx)

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

See also

References

  1. ^ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 528. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
  2. ^ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 526. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
  3. ^ Kallenberg, Olav (2017). Random Measures, Theory and Applications. Vol. 77. Probability Theory and Stochastic Modelling. Switzerland: Springer. p. 53. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.