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 number. Then the intensity of is defined as
if the limit exists and is independent of for all

Example

Look at the Lebesgue measure. Then for a fixed, it is
so
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 mapping
defined by
is measurable.