# Counting measure

In mathematics, 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 ∞ if the subset is infinite.[1]

The counting measure can be defined on any measurable set but is mostly used on countable sets.[1]

In formal notation, we can make any set *X* into a measurable space by taking the sigma-algebra
of measurable subsets to consist of all subsets of
. 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]

## Discussion

The counting measure is a special case of a more general construct. 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 sup of the sums over all finite subsets, i.e.,

Taking *f(x)=1* for all *x* in *X* produces the counting measure.

## References

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