for every positive measurable function is called the intensity measure of . The intensity measure exists for every random measure and is a
s-finite measure.
Supporting measure
For a random measure , the measure 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.
for positive -measurable are measurable, so they are
random variables.
Uniqueness
The distribution of a random measure is uniquely determined by the distributions of
for all continuous functions with compact support on . For a fixed
semiring that generates in the sense that , the distribution of a random measure is also uniquely determined by the integral over all positive
simple-measurable functions .[6]
Decomposition
A measure generally might be decomposed as:
Here is a diffuse measure without atoms, while is a purely atomic measure.
Random counting measure
A random measure of the form:
where is the
Dirac measure, and 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 . The diffuse component 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 (, ) a
measurable space. Here is the space of all boundedly finite integer-valued measures (called counting measures).
^
abKallenberg, O., Random Measures, 4th edition. Academic Press, New York, London; Akademie-Verlag, Berlin (1986).
ISBN0-12-394960-2MR854102. An authoritative but rather difficult reference.
^
abJan Grandell, Point processes and random measures, Advances in Applied Probability 9 (1977) 502-526.
MR0478331JSTOR A nice and clear introduction.
^Daley, D. J.; Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes. Probability and its Applications.
doi:
10.1007/b97277.
ISBN0-387-95541-0.
^"Crisan, D., Particle Filters: A Theoretical Perspective, in Sequential Monte Carlo in Practice, Doucet, A., de Freitas, N. and Gordon, N. (Eds), Springer, 2001,
ISBN0-387-95146-6