# 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.

## Notes

- ↑
^{1.0}^{1.1}Template:PlanetMath - ↑ Schilling (2005), p.27
- ↑ Hansen (2009) p.47

## References

- Schilling, René L. (2005)."Measures, Integral and Martingales". Cambridge University Press.
- Hansen, Ernst (2009)."Measure theory, Fourth Edition". Department of Mathematical Science, University of Copenhagen.