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.

DefinitionEdit

Let   be a measure on the real number. Then the intensity   of   is defined as

 

if the limit exists and is independent of   for all  

ExampleEdit

Look at the Lebesgue measure  . Then for a fixed  , it is

 

so

 

Therefore the Lebesgue measure has intensity one.

PropertiesEdit

The set of all measures   for which the intensity is well defined is a measurable subset of the set of all measures on  . The mapping

 

defined by

 

is measurable.

ReferencesEdit

  • Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. p. 173. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.