In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinity if the subset is infinite.[1]

The counting measure can be defined on any measurable space (that is, any set along with a sigma-algebra) but is mostly used on countable sets.[1]

In formal notation, we can turn any set into a measurable space by taking the power set of as the sigma-algebra that is, all subsets of are measurable sets. Then the counting measure on this measurable space is the positive measure defined by

for all where denotes the cardinality of the set [2]

The counting measure on is σ-finite if and only if the space is countable.[3]

Integration on with counting measure

Take the measure space , where is the set of all subsets of the naturals and the counting measure. Take any measurable . As it is defined on , can be represented pointwise as

Each is measurable. Moreover . Still further, as each is a simple function

Hence by the monotone convergence theorem


The counting measure is a special case of a more general construction. With the notation as above, any function defines a measure on via

where the possibly uncountable sum of real numbers is defined to be the supremum of the sums over all finite subsets, that is,
Taking for all gives the counting measure.

See also


  1. ^ a b Counting Measure at PlanetMath.
  2. ^ Schilling, René L. (2005). Measures, Integral and Martingales. Cambridge University Press. p. 27. ISBN 0-521-61525-9.
  3. ^ Hansen, Ernst (2009). Measure Theory (Fourth ed.). Department of Mathematical Science, University of Copenhagen. p. 47. ISBN 978-87-91927-44-7.