Counting Measure
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Counting Measure

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 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 ?-finite if and only if the space ${\displaystyle X}$ is countable.[3]

## Discussion

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

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

where the possibly uncountable sum of real numbers is defined to be the supremum 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 gives the counting measure.

## References

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