Inprobability theory, arandom measure is ameasure-valuedrandom element.^[1][2] Random measures are for example used in the theory ofrandom processes, where they form many importantpoint processes such asPoisson point processes andCox processes.

Random measures can be defined astransition kernels or asrandom elements. Both definitions are equivalent. For the definitions, let${\displaystyle E}$ be aseparablecomplete metric space and let${\displaystyle \mathcal{E}}$ be itsBorel${\displaystyle \sigma }$-algebra. (The most common example of a separable complete metric space is${\displaystyle {\mathbb{R}}^n}$.)

A random measure${\displaystyle \zeta }$ is a (a.s.)locally finitetransition kernel from an abstractprobability space${\displaystyle (\mathrm{\Omega },\mathcal{A},P)}$ to${\displaystyle (E,\mathcal{E})}$.^[3]

Being a transition kernel means that

• For any fixed${\displaystyle B\in \mathcal{E}}$, the mapping

${\displaystyle \omega \mapsto \zeta (\omega ,B)}$

ismeasurable from${\displaystyle (\mathrm{\Omega },\mathcal{A})}$ to${\displaystyle (\mathbb{R},\mathcal{B}(\mathbb{R}))}$

• For every fixed${\displaystyle \omega \in \mathrm{\Omega }}$, the mapping

${\displaystyle B\mapsto \zeta (\omega ,B)(B\in \mathcal{E})}$

is ameasure on${\displaystyle (E,\mathcal{E})}$

Being locally finite means that the measures

${\displaystyle B\mapsto \zeta (\omega ,B)}$

satisfy for all bounded measurable sets${\displaystyle \tilde{B}\in \mathcal{E}}$and for all${\displaystyle \omega \in \mathrm{\Omega }}$ except some${\displaystyle P}$-null set

In the context ofstochastic processes there is the related concept of astochastic kernel, probability kernel, Markov kernel.

Define

${\displaystyle \tilde{\mathcal{M}}:=\{\mu \mid \mu \text{ is measure on }(E,\mathcal{E})\}}$

and the subset of locally finite measures by

For all bounded measurable${\displaystyle \tilde{B}}$, define the mappings

${\displaystyle I_{\tilde{B}}:\mu \mapsto \mu (\tilde{B})}$

from${\displaystyle \tilde{\mathcal{M}}}$ to${\displaystyle \mathbb{R}}$. Let${\displaystyle \tilde{\mathbb{M}}}$ be the${\displaystyle \sigma }$-algebra induced by the mappings${\displaystyle I_{\tilde{B}}}$ on${\displaystyle \tilde{\mathcal{M}}}$ and${\displaystyle \mathbb{M}}$ the${\displaystyle \sigma }$-algebra induced by the mappings${\displaystyle I_{\tilde{B}}}$ on${\displaystyle \mathcal{M}}$. Note that${\displaystyle \tilde{\mathbb{M}}{|}_{\mathcal{M}}=\mathbb{M}}$.

A random measure is a random element from${\displaystyle (\mathrm{\Omega },\mathcal{A},P)}$ to${\displaystyle (\tilde{\mathcal{M}},\tilde{\mathbb{M}})}$ that almost surely takes values in${\displaystyle (\mathcal{M},\mathbb{M})}$^[3][4][5]

For a random measure${\displaystyle \zeta }$, the measure${\displaystyle \mathrm{E}\zeta }$ satisfying

${\displaystyle \mathrm{E}\left[\int f(x)\zeta (\mathrm{d}x)\right]=\int f(x)\mathrm{E}\zeta (\mathrm{d}x)}$

for every positive measurable function${\displaystyle f}$ is called the intensity measure of${\displaystyle \zeta }$. The intensity measure exists for every random measure and is as-finite measure.

For a random measure${\displaystyle \zeta }$, the measure${\displaystyle \nu }$ satisfying

${\displaystyle \int f(x)\zeta (\mathrm{d}x)=0\text{ a.s. }\text{ iff }\int f(x)\nu (\mathrm{d}x)=0}$

for all positive measurable functions is called thesupporting measure of${\displaystyle \zeta }$. The supporting measure exists for all random measures and can be chosen to be finite.

For a random measure${\displaystyle \zeta }$, theLaplace transform is defined as

${\displaystyle {\mathcal{L}}_{\zeta }(f)=\mathrm{E}\left[\exp \left(-\int f(x)\zeta (\mathrm{d}x)\right)\right]}$

for every positive measurable function${\displaystyle f}$.

For a random measure${\displaystyle \zeta }$, the integrals

${\displaystyle \int f(x)\zeta (\mathrm{d}x)}$

and${\displaystyle \zeta (A):=\int {\mathbf{1}}_A(x)\zeta (\mathrm{d}x)}$

for positive${\displaystyle \mathcal{E}}$-measurable${\displaystyle f}$ are measurable, so they arerandom variables.

The distribution of a random measure is uniquely determined by the distributions of

${\displaystyle \int f(x)\zeta (\mathrm{d}x)}$

for all continuous functions with compact support${\displaystyle f}$ on${\displaystyle E}$. For a fixedsemiring${\displaystyle \mathcal{I}\subset \mathcal{E}}$ that generates${\displaystyle \mathcal{E}}$ in the sense that${\displaystyle \sigma (\mathcal{I})=\mathcal{E}}$, the distribution of a random measure is also uniquely determined by the integral over all positivesimple${\displaystyle \mathcal{I}}$-measurable functions${\displaystyle f}$.^[6]

A measure generally might be decomposed as:

${\displaystyle \mu ={\mu }_d+{\mu }_a={\mu }_d+\sum _{n=1}^N{\kappa }_n{\delta }_{X_n},}$

Here${\displaystyle {\mu }_d}$ is a diffuse measure without atoms, while${\displaystyle {\mu }_a}$ is a purely atomic measure.

A random measure of the form:

${\displaystyle \mu =\sum _{n=1}^N{\delta }_{X_n},}$

where${\displaystyle \delta }$ is theDirac measure and${\displaystyle X_n}$ are random variables, is called apoint process^[1][2] orrandom counting measure. This random measure describes the set ofN particles, whose locations are given by the (generally vector valued) random variables${\displaystyle X_n}$. The diffuse component${\displaystyle {\mu }_d}$ is null for a counting measure.

In the formal notation of above a random counting measure is a map from a probability space to themeasurable space(${\displaystyle N_X}$, ${\displaystyle \mathfrak{B}(N_X)}$). Here${\displaystyle N_X}$ is the space of all boundedly finite integer-valued measures${\displaystyle N\in M_X}$ (calledcounting measures).

The definitions of expectation measure, Laplace functional, moment measures and stationarity for random measures follow those ofpoint processes. Random measures are useful in the description and analysis ofMonte Carlo methods, such asMonte Carlo numerical quadrature andparticle filters.^[7]

• Poisson random measure
• Vector measure
• Ensemble

• Measures (measure theory)
• Stochastic processes

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• Short description matches Wikidata