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


 * $$ \overline \mu:= \lim_{|t| \to \infty} \frac{\mu((-s,t-s])}{t} $$

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

Example
Look at the Lebesgue measure $$ \lambda $$. Then for a fixed $$ s $$, it is
 * $$ \lambda((-s,t-s])=(t-s)-(-s)=t, $$

so
 * $$ \overline \lambda:= \lim_{|t| \to \infty} \frac{\lambda((-s,t-s])}{t}= \lim_{|t| \to \infty} \frac t t =1. $$

Therefore the Lebesgue measure has intensity one.

Properties
The set of all measures $$ M $$ for which the intensity is well defined is a measurable subset of the set of all measures on $$ \R $$. The mapping
 * $$ I \colon M \to \mathbb R $$

defined by


 * $$ I(\mu) = \overline \mu $$

is measurable.