# 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 ${\displaystyle \Sigma }$ of measurable subsets to consist of all subsets of ${\displaystyle X}$. Then the counting measure ${\displaystyle \mu }$ on this measurable space ${\displaystyle (X,\Sigma )}$ is the positive measure ${\displaystyle \Sigma \rightarrow [0,+\infty ]}$ defined by

${\displaystyle \mu (A)={\begin{cases}\vert A\vert &{\text{if }}A{\text{ is finite}}\\+\infty &{\text{if }}A{\text{ is infinite}}\end{cases}}}$

for all ${\displaystyle A\in \Sigma }$, where ${\displaystyle \vert A\vert }$ denotes the cardinality of the set ${\displaystyle A}$.[2]

The counting measure on ${\displaystyle (X,\Sigma )}$ is σ-finite if and only if the space ${\displaystyle X}$ is countable.[3]

## Discussion

The counting measure is a special case of a more general construct. With the notation as above, any function ${\displaystyle f\colon X\to [0,\infty )}$ defines a measure ${\displaystyle \mu }$ on ${\displaystyle (X,\Sigma )}$ via

${\displaystyle \mu (A):=\sum _{a\in A}f(a)\,\forall A\subseteq X,}$

where the possibly uncountable sum of real numbers is defined to be the sup of the sums over all finite subsets, i.e.,

${\displaystyle \sum _{y\in Y\subseteq \mathbb {R} }y:=\sup _{F\subseteq Y,|F|<\infty }\left\{\sum _{y\in F}y\right\}.}$

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

## Notes

1. ^ a b Counting Measure at PlanetMath.org.
2. ^ Schilling (2005), p.27
3. ^ 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.