# 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.

## References

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