In the mathematical discipline of

measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...

, 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

\mu

is defined as :

\overline \mu:= \lim_ \frac

if the limit exists and is independent of

s

for all

s \in \R

Example

Look at the

Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...

\lambda

. Then for a fixed

s

, it is :

\lambda((-s,t-s])=(t-s)-(-s)=t,

so :

\overline \lambda:= \lim_ \frac=  \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 \colon M \to \mathbb R

defined by :

I(\mu) = \overline \mu

is Measurable function, measurable.

References

*{{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 Measure theory