Intensity (measure theory)

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

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

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

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

:$\backslash lambda((-s,t-s])=(t-s)-(-s)=t,$

so

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

Therefore the Lebesgue measure has intensity one.

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

:$I\; \backslash colon\; M\; \backslash to\; \backslash mathbb\; R$

defined by

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

is measurable.

• {{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}}

Category:Measure theory