Intensity (measure theory)
In the mathematical discipline of measure theory, the intensity of a measure is the average value the measure assigns to an interval of length one.
Definition
Let be a measure on the real numbers. Then the intensity of is defined asif the limit exists and is independent of for all.
Example
Look at the Lebesgue measure. Then for a fixed, it isso
Therefore the Lebesgue measure has intensity one.
Properties
The set of all measures for which the intensity is well defined is a measurable subset of the set of all measures on. The mappingdefined by
is measurable.