Misplaced Pages

This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages) This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (June 2018) (Learn how and when to remove this message) This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2018) (Learn how and when to remove this message) (Learn how and when to remove this message)

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.

• Measure theory