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General Poisson point process

In measure theory, the Poisson point process can be further generalized to what is sometimes known as the general Poisson point process or general Poisson process by using a Radon measure Λ {\displaystyle \textstyle \Lambda }, which is a locally finite measure. In general, this Radon measure Λ {\displaystyle \textstyle \Lambda } can be atomic, which means multiple points of the Poisson point process can exist in the same location of the underlying space. In this situation, the number of points at x {\displaystyle \textstyle x} is a Poisson random variable with mean Λ (x) {\displaystyle \textstyle \Lambda ({x})}. But sometimes the converse is assumed, so the Radon measure Λ {\displaystyle \textstyle \Lambda } is diffuse or non-atomic.

A point process N {\displaystyle \textstyle {N}} is a general Poisson point process with intensity Λ {\displaystyle \textstyle \Lambda } if it has the two following properties:

The Radon measure Λ {\displaystyle \textstyle \Lambda } maintains its previous interpretation of being the expected number of points of N {\displaystyle \textstyle {N}} located in the bounded region B {\displaystyle \textstyle B}, namely

Furthermore, if Λ {\displaystyle \textstyle \Lambda } is absolutely continuous such that it has a density (which is the Radon–Nikodym density or derivative) with respect to the Lebesgue measure, then for all Borel sets B {\displaystyle \textstyle B} it can be written as:

where the density λ (x) {\displaystyle \textstyle \lambda (x)} is known, among other terms, as the intensity function.