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

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

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

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

Discussion

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

\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.,

\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 2 Counting Measure at PlanetMath.org.
↑ 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.

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