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

${\displaystyle {\overline {\mu }}:=\lim _{|t|\to \infty }{\frac {\mu ((-s,t-s])}{t}}}$

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

Example

Look at the Lebesgue measure ${\displaystyle \lambda }$. Then for a fixed ${\displaystyle s}$, it is

${\displaystyle \lambda ((-s,t-s])=(t-s)-(-s)=t,}$

so

${\displaystyle {\overline {\lambda }}:=\lim _{|t|\to \infty }{\frac {\lambda ((-s,t-s])}{t}}=\lim _{|t|\to \infty }{\frac {t}{t}}=1.}$

Therefore the Lebesgue measure has intensity one.

Properties

The set of all measures ${\displaystyle M}$ for which the intensity is well defined is a measurable subset of the set of all measures on ${\displaystyle \mathbb {R} }$. The mapping

${\displaystyle I\colon M\to \mathbb {R} }$

defined by

${\displaystyle I(\mu )={\overline {\mu }}}$

is measurable.

References

• Kallenberg, Olav (2017). Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling. Vol. 77. Switzerland: Springer. p. 173. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.