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 ∞ if the subset is infinite.^{[1]}

The counting measure can be defined on any measurable space (i.e. 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 , i.e. all subsets of are measurable. 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 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, i.e.,

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

## References

- ^
^{a}^{b}Counting Measure at PlanetMath. **^**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.