In probability theory, a random measure is a measure-valued random element.
Random measures can be defined as transition kernels or as random elements.
be a separable complete metric space and let
(The most common example of a separable complete metric space is
is a (a.s.) locally finite transition kernel from an abstract probability space
[3] Being a transition kernel means that Being locally finite means that the measures satisfy
for all bounded measurable sets
-null set In the context of stochastic processes there is the related concept of a stochastic kernel, probability kernel, Markov kernel.
Define and the subset of locally finite measures by For all bounded measurable
that almost surely takes values in
satisfying for every positive measurable function
is called the intensity measure of
satisfying for all positive measurable functions is called the supporting measure of
The supporting measure exists for all random measures and can be chosen to be finite.
, the Laplace transform is defined as for every positive measurable function
are measurable, so they are random variables.
The distribution of a random measure is uniquely determined by the distributions of for all continuous functions with compact support
, the distribution of a random measure is also uniquely determined by the integral over all positive simple
[6] A measure generally might be decomposed as: Here
is a diffuse measure without atoms, while
is a purely atomic measure.
A random measure of the form: where
are random variables, is called a point process[1][2] or random counting measure.
This random measure describes the set of N particles, whose locations are given by the (generally vector valued) random variables
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 (
is the space of all boundedly finite integer-valued measures
(called counting measures).
The definitions of expectation measure, Laplace functional, moment measures and stationarity for random measures follow those of point processes.
Random measures are useful in the description and analysis of Monte Carlo methods, such as Monte Carlo numerical quadrature and particle filters.