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 space (i.e. any set $X$ along with a sigma-algebra) but is mostly used on countable sets.[1]

In formal notation, we can turn any set $X$ into a measurable space by taking the power set of $X$ as the sigma-algebra $\Sigma$ , i.e. all subsets of $X$ are measurable. 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 construction. 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)\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\

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

References

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

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[pm-1] Counting Measure at PlanetMath.

[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.