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 $\bar{\mu }$ of $\mu$ is defined as

$\bar{\mu }:=\underset{|t|\to \mathrm{\infty }}{\lim }\frac{\mu ((-s,t-s])}{t}$

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

$\bar{\lambda }:=\underset{|t|\to \mathrm{\infty }}{\lim }\frac{\lambda ((-s,t-s])}{t}=\underset{|t|\to \mathrm{\infty }}{\lim }\frac{t}{t}=1.$

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

$I:M\to ℝ$

defined by

$I(\mu )=\bar{\mu }$

is measurable.

References

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