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 [math]\displaystyle{ \mu }[/math] be a measure on the real numbers. Then the intensity [math]\displaystyle{ \overline \mu }[/math] of [math]\displaystyle{ \mu }[/math] is defined as
- [math]\displaystyle{ \overline \mu:= \lim_{|t| \to \infty} \frac{\mu((-s,t-s])}{t} }[/math]
if the limit exists and is independent of [math]\displaystyle{ s }[/math] for all [math]\displaystyle{ s \in \R }[/math].
Example
Look at the Lebesgue measure [math]\displaystyle{ \lambda }[/math]. Then for a fixed [math]\displaystyle{ s }[/math], it is
- [math]\displaystyle{ \lambda((-s,t-s])=(t-s)-(-s)=t, }[/math]
so
- [math]\displaystyle{ \overline \lambda:= \lim_{|t| \to \infty} \frac{\lambda((-s,t-s])}{t}= \lim_{|t| \to \infty} \frac t t =1. }[/math]
Therefore the Lebesgue measure has intensity one.
Properties
The set of all measures [math]\displaystyle{ M }[/math] for which the intensity is well defined is a measurable subset of the set of all measures on [math]\displaystyle{ \R }[/math]. The mapping
- [math]\displaystyle{ I \colon M \to \mathbb R }[/math]
defined by
- [math]\displaystyle{ I(\mu) = \overline \mu }[/math]
is measurable.
References
- Kallenberg, Olav (2017). Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling. 77. Switzerland: Springer. p. 173. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.
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