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Intensity (measure theory)

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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

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

if the limit exists and is independent of for all .

Example

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

so

Therefore the Lebesgue measure has intensity one.

Properties

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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.

References

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  • Kallenberg, Olav (2017). Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling. Vol. 77. Switzerland: Springer. p. 173. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.