# Random measure

In probability theory, a random measure is a measure-valued random element.[1][2] Let X be a complete separable metric space and ${\displaystyle {\mathfrak {B}}(X)}$ the σ-algebra of its Borel sets. A Borel measure μ on X is boundedly finite if μ(A) < ∞ for every bounded Borel set A. Let ${\displaystyle M_{X}}$ be the space of all boundedly finite measures on ${\displaystyle {\mathfrak {B}}(X)}$. Let (Ω, ℱ, P) be a probability space, then a random measure maps from this probability space to the measurable space (${\displaystyle M_{X}}$, ${\displaystyle {\mathfrak {B}}(M_{X})}$).[3]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.

## Random counting measure

A random measure of the form:

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

where ${\displaystyle \delta }$ is the Dirac measure, and ${\displaystyle X_{n}}$ 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 ${\displaystyle X_{n}}$. 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 the measurable space (${\displaystyle N_{X}}$, ${\displaystyle {\mathfrak {B}}(N_{X})}$) a measurable space. Here ${\displaystyle N_{X}}$ is the space of all boundedly finite integer-valued measures ${\displaystyle N\in M_{X}}$ (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.[4]