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 infinity [math]\displaystyle{ \infty }[/math] if the subset is infinite.[1] The counting measure can be defined on any measurable space (that is, any set [math]\displaystyle{ X }[/math] along with a sigma-algebra) but is mostly used on countable sets.[1]
In formal notation, we can turn any set [math]\displaystyle{ X }[/math] into a measurable space by taking the power set of [math]\displaystyle{ X }[/math] as the sigma-algebra [math]\displaystyle{ \Sigma; }[/math] that is, all subsets of [math]\displaystyle{ X }[/math] are measurable sets. Then the counting measure [math]\displaystyle{ \mu }[/math] on this measurable space [math]\displaystyle{ (X,\Sigma) }[/math] is the positive measure [math]\displaystyle{ \Sigma \to [0,+\infty] }[/math] defined by [math]\displaystyle{ \mu(A) = \begin{cases} \vert A \vert & \text{if } A \text{ is finite}\\ +\infty & \text{if } A \text{ is infinite} \end{cases} }[/math] for all [math]\displaystyle{ A\in\Sigma, }[/math] where [math]\displaystyle{ \vert A\vert }[/math] denotes the cardinality of the set [math]\displaystyle{ A. }[/math][2]
The counting measure on [math]\displaystyle{ (X,\Sigma) }[/math] is σ-finite if and only if the space [math]\displaystyle{ X }[/math] is countable.[3]
Integration on [math]\displaystyle{ \mathbb{N} }[/math] with counting measure
Take the measure space [math]\displaystyle{ (\mathbb{N}, 2^\mathbb{N}, \mu) }[/math], where [math]\displaystyle{ 2^\mathbb{N} }[/math] is the set of all subsets of the naturals and [math]\displaystyle{ \mu }[/math] the counting measure. Take any measurable [math]\displaystyle{ f : \mathbb{N} \to [0,\infty] }[/math]. As it is defined on [math]\displaystyle{ \mathbb{N} }[/math], [math]\displaystyle{ f }[/math] can be represented pointwise as [math]\displaystyle{ f(x) = \sum_{n=1}^\infty f(n) 1_{\{n\}}(x) = \lim_{M \to \infty} \underbrace{ \ \sum_{n=1}^M f(n) 1_{\{n\}}(x) \ }_{ \phi_M (x) } = \lim_{M \to \infty} \phi_M (x) }[/math]
Each [math]\displaystyle{ \phi_M }[/math] is measurable. Moreover [math]\displaystyle{ \phi_{M+1}(x) = \phi_M (x) + f(M+1) \cdot 1_{ \{M+1\} }(x) \geq \phi_M (x) }[/math]. Still further, as each [math]\displaystyle{ \phi_M }[/math] is a simple function [math]\displaystyle{ \int_\mathbb{N} \phi_M d\mu = \int_\mathbb{N} \left( \sum_{n=1}^M f(n) 1_{\{n\}} (x) \right) d\mu = \sum_{n=1}^M f(n) \mu (\{n\}) = \sum_{n=1}^M f(n) \cdot 1 = \sum_{n=1}^M f(n) }[/math]Hence by the monotone convergence theorem [math]\displaystyle{ \int_\mathbb{N} f d\mu = \lim_{M \to \infty} \int_\mathbb{N} \phi_M d\mu = \lim_{M \to \infty} \sum_{n=1}^M f(n) = \sum_{n=1}^\infty f(n) }[/math]
Discussion
The counting measure is a special case of a more general construction. With the notation as above, any function [math]\displaystyle{ f : X \to [0, \infty) }[/math] defines a measure [math]\displaystyle{ \mu }[/math] on [math]\displaystyle{ (X, \Sigma) }[/math] via [math]\displaystyle{ \mu(A):=\sum_{a \in A} f(a)\quad \text{ for all } A \subseteq X, }[/math] where the possibly uncountable sum of real numbers is defined to be the supremum of the sums over all finite subsets, that is, [math]\displaystyle{ \sum_{y\,\in\,Y\!\ \subseteq\,\mathbb R} y\ :=\ \sup_{F \subseteq Y,\, |F| \lt \infty} \left\{ \sum_{y \in F} y \right\}. }[/math] Taking [math]\displaystyle{ f(x) = 1 }[/math] for all [math]\displaystyle{ x \in X }[/math] gives the counting measure.
See also
- Pip (counting) – Easily countable items
- Set function – Function from sets to numbers
References
- ↑ 1.0 1.1 Counting Measure at PlanetMath.org.
- ↑ 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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