Random measure

A random measure is the measure and probability theory, a random variable whose values ​​are measures. Random geometric structures, such as those studied in stochastic geometry can be modeled by random measures. Thus, a point process, such as a general Poisson process are regarded as random counting measure, which assigns the random number of points contained in it a lot. In the non- parametric statistics appear random measures on as an empirical measure.

Definition

Let the -dimensional Euclidean space with the Borel σ - algebra and the set of all locally finite dimensions ( Borel measures) on. Next denote the smallest σ - algebra such that all the pictures, with a limited Borel set is measurable. A random measure on is then a random variable on a probability space with values ​​in the measuring room.

So a random measure assigns each random result to a measure on that takes on bounded measurable amounts of finite values ​​. For any Borel set

A non-negative random variable, called the random measure of the amount.

Identifies the expected value of, then by mapping

A measure is to be given the intensity measure called. If again locally finite, is called integrable.

Example

A random arrangement of points in the plane or in space can be modeled as a random measure: Are the positions of points, interpreted as - valued random variable, then by

A random measure defined on. Herein, the Diracmaß in place. For a Borel set then the (random) number of points that lie in the set.