Intensity measure


In probability theory, an intensity measure is a measure that is derived from a random measure. The intensity measure is a non-random measure and is defined as the expectation value of the random measure of a set, hence it corresponds to the average volume the random measure assigns to a set. The intensity measure contains important information about the properties of the random measure. A Poisson point process, interpreted as a random measure, is for example uniquely determined by its intensity measure.

Definition

Let be a random measure on the measurable space and denote the expected value of a random element with.
The intensity measure
of is defined as
for all.
Note the difference in notation between the expectation value of a random element, denoted by and the intensity measure of the random measure, denoted by.

Properties

The intensity measure is always s-finite and satisfies
for every positive measurable function on.