Random measure

 ( Measures ) 

Being a transition kernel means that

• For any fixed $B\; \backslash in\; \backslash mathcal\; \backslash mathcal\; E$, the mapping

$\backslash omega\; \backslash mapsto\; \backslash zeta(\backslash omega,B)$

is measurable from $(\backslash Omega,\; \backslash mathcal\; A)$ to $(\backslash R,\; \backslash mathcal\; B(\backslash R))$

• For every fixed $\backslash omega\; \backslash in\; \backslash Omega$, the mapping

$B\; \backslash mapsto\; \backslash zeta(\backslash omega,\; B)\; \backslash quad\; (B\; \backslash in\; \backslash mathcal\; E)$

is a measure on $(E,\; \backslash mathcal\; E)$

Being locally finite means that the measures

$B\; \backslash mapsto\; \backslash zeta(\backslash omega,\; B)$

In the context of stochastic processes there is the related concept of a Markov kernel .

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

and the subset of locally finite measures by

For all bounded measurable $\backslash tilde\; B$, define the mappings

$I\_\{\backslash tilde\; B\; \}\; \backslash colon\; \backslash mu\; \backslash mapsto\; \backslash mu(\backslash tilde\; B)$

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

A random measure is a random element from $(\backslash Omega,\; \backslash mathcal\; A,\; P)$ to $(\backslash tilde\; \backslash mathcal\; M,\; \backslash tilde\; \backslash mathbb\; M)$ that almost surely takes values in $(\backslash mathcal\; M,\; \backslash mathbb\; M)$

$\backslash operatorname\; E\; \backslash left\; =\; \backslash int\; f(x)\; \backslash ;\; \backslash operatorname\; E\; \backslash zeta\; (\backslash mathrm\; dx)$

for every positive measurable function $f$ is called the intensity measure of $\backslash zeta$. The intensity measure exists for every random measure and is a s-finite measure.

$\backslash int\; f(x)\; \backslash ;\; \backslash zeta(\backslash mathrm\; dx\; )=0\; \backslash text\{\; a.s.\; \}\; \backslash text\{\; iff\; \}\; \backslash int\; f(x)\; \backslash ;\; \backslash nu\; (\backslash mathrm\; dx)=0$

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

$\backslash mathcal\; L\_\backslash zeta(f)=\; \backslash operatorname\; E\; \backslash left$

for every positive measurable function $f$.

$\backslash int\; f(x)\; \backslash zeta(\backslash mathrm\; dx)$

for positive $\backslash mathcal\; E$-measurable $f$ are measurable, so they are .

$\backslash int\; f(x)\; \backslash zeta(\backslash mathrm\; dx)$

for all continuous functions with compact support $f$ on $E$. For a fixed semiring $\backslash mathcal\; I\; \backslash subset\; \backslash mathcal\; E$ that generates $\backslash mathcal\; E$ in the sense that $\backslash sigma(\backslash mathcal\; I)=\backslash mathcal\; E$, the distribution of a random measure is also uniquely determined by the integral over all positive simple function $\backslash mathcal\; I$-measurable functions $f$.

$\backslash mu=\backslash mu\_d\; +\; \backslash mu\_a\; =\; \backslash mu\_d\; +\; \backslash sum\_\{n=1\}^N\; \backslash kappa\_n\; \backslash delta\_\{X\_n\},$

$\backslash mu=\backslash sum\_\{n=1\}^N\; \backslash delta\_\{X\_n\},$

where $\backslash delta$ is the Dirac measure and $X\_n$ are random variables, is called a point process or random counting measure. This random measure describes the set of N particles, whose locations are given by the (generally vector valued) random variables $X\_n$. The diffuse component $\backslash mu\_d$ is null for a counting measure.

In the formal notation of above a random counting measure is a map from a probability space to the measurable space . Here $N\_X$ is the space of all boundedly finite integer-valued measures $N\; \backslash in\; M\_X$ (called ).

The definitions of expectation measure, Laplace functional, moment measures and stationarity for random measures follow those of . Random measures are useful in the description and analysis of Monte Carlo methods, such as Monte Carlo numerical quadrature and .

• Poisson random measure
• Vector measure
• Ensemble