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 $\mu$ be a measure on the real numbers. Then the intensity $\overline \mu$ of $\mu$ is defined as

:$\overline \mu:= \lim_ \frac$

if the limit exists and is independent of $s$ for all $s \in \R$.

Example

Look at the Lebesgue measure $\lambda$. Then for a fixed $s$, it is
:$\lambda((-s,t-s])=(t-s)-(-s)=t,$

so
:$\overline \lambda:= \lim_ \frac= \lim_ \frac t t =1.$

Therefore the Lebesgue measure has intensity one.

Properties

The set of all measures $M$ for which the intensity is well defined is a measurable subset of the set of all measures on $\R$. The mapping
:$I \colon M \to \mathbb R$
defined by

:$I(\mu) = \overline \mu$

is Measurable function, measurable.

References

*{{cite book , last1=Kallenberg , first1=Olav , author-link1=Olav Kallenberg , year=2017 , title=Random Measures, Theory and Applications, series=Probability Theory and Stochastic Modelling , volume=77 , location= Switzerland , publisher=Springer , doi= 10.1007/978-3-319-41598-7, page=173, isbn=978-3-319-41596-3

Measure theory