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.
and denote the expected value of a random element
The intensity measure of
[2] [3] Note the difference in notation between the expectation value of a random element
and the intensity measure of the random measure
The intensity measure
is always s-finite and satisfies for every positive measurable function