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 μ {\displaystyle \mu } be a measure on the real numbers. Then the intensity μ ¯ {\displaystyle {\overline {\mu }}} of μ {\displaystyle \mu } is defined as
μ ¯ := lim | t | → ∞ μ ( ( − s , t − s ] ) t {\displaystyle {\overline {\mu }}:=\lim _{|t|\to \infty }{\frac {\mu ((-s,t-s])}{t}}}
if the limit exists and is independent of s {\displaystyle s} for all s ∈ R {\displaystyle s\in \mathbb {R} } .
Example Look at the Lebesgue measure λ {\displaystyle \lambda } . Then for a fixed s {\displaystyle s} , it is
λ ( ( − s , t − s ] ) = ( t − s ) − ( − s ) = t , {\displaystyle \lambda ((-s,t-s])=(t-s)-(-s)=t,}
so
λ ¯ := lim | t | → ∞ λ ( ( − s , t − s ] ) t = lim | t | → ∞ t t = 1. {\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 M {\displaystyle M} for which the intensity is well defined is a measurable subset of the set of all measures on R {\displaystyle \mathbb {R} } . The mapping
I : M → R {\displaystyle I\colon M\to \mathbb {R} }
defined by
I ( μ ) = μ ¯ {\displaystyle I(\mu )={\overline {\mu }}}
is measurable.
See also Intensity measure
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.
