In probability theory, an intensity measure is a measure that is derived from a random measure. The intensity measure is a non-random measure and is defined as the expectation value of the random measure of a set, hence it corresponds to the average volume the random measure assigns to a set. The intensity measure contains important information about the properties of the random measure. A Poisson point process, interpreted as a random measure, is for example uniquely determined by its intensity measure. [1]


Let be a random measure on the measurable space and denote the expected value of a random element with .

The intensity measure

of is defined as

for all .[2] [3]

Note the difference in notation between the expectation value of a random element , denoted by and the intensity measure of the random measure , denoted by .


The intensity measure is always s-finite and satisfies

for every positive measurable function on .[3]


  1. ^ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 528. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
  2. ^ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 526. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
  3. ^ a b Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. p. 53. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.