Intensity (measure theory) explained

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

\mu

be a measure on the real numbers. Then the intensity

\overline\mu

\mu

is defined as

\overline\mu:=\lim_|t|

	\mu((-s,t-s])
	t

if the limit exists and is independent of

for all

s\in\R

Example

. Then for a fixed

, it is

λ((-s,t-s])=(t-s)-(-s)=t,

\overlineλ:=\lim_|t|

	λ((-s,t-s])
	t

=\lim_|t|

	t
	t

=1.

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

I\colonM\toR

defined by

I(\mu)=\overline\mu

is measurable.

References

Book: Kallenberg . Olav . Olav Kallenberg . 2017 . Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling . 77 . Switzerland . Springer . 10.1007/978-3-319-41598-7. 173. 978-3-319-41596-3.