Mathematical concept
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
if the subset is infinite.[1]
The counting measure can be defined on any measurable space (that is, any set
along with a sigma-algebra) but is mostly used on countable sets.[1]
In formal notation, we can turn any set
into a measurable space by taking the power set of
as the sigma-algebra
that is, all subsets of
are measurable sets.
Then the counting measure
on this measurable space
is the positive measure
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8681d8e9a092f05faaaf47db0f8d4cbea447752)
for all
![{\displaystyle A\in \Sigma ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1aae6b26b1fb89e81482b908ca3f164985e58ea)
where
![{\displaystyle \vert A\vert }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e452c00372a18f4de3b3abf77377bfa176d628d)
denotes the
cardinality of the set
[2]
The counting measure on
is σ-finite if and only if the space
is countable.[3]
Integration on
with counting measure[edit]
Take the measure space
, where
is the set of all subsets of the naturals and
the counting measure. Take any measurable
. As it is defined on
,
can be represented pointwise as
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08b0b080156ac4f07eb949b7997f2c517dcbc7ba)
Each
is measurable. Moreover
. Still further, as each
is a simple function
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f0cce8bbdcadc8264d1276efd5d3efa23cc4ba4)
Hence by the monotone convergence theorem
![{\displaystyle \int _{\mathbb {N} }fd\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82f47ed2f3dfc36144214b1ad9c55b53d6212996)
Discussion[edit]
The counting measure is a special case of a more general construction. With the notation as above, any function
defines a measure
on
via
![{\displaystyle \mu (A):=\sum _{a\in A}f(a)\quad {\text{ for all }}A\subseteq X,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c39e84949b904cd9dcb9a5c6a147347019fbaa9a)
where the possibly uncountable sum of real numbers is defined to be the
supremum of the sums over all finite subsets, that is,
![{\displaystyle \sum _{y\,\in \,Y\!\ \subseteq \,\mathbb {R} }y\ :=\ \sup _{F\subseteq Y,\,|F|<\infty }\left\{\sum _{y\in F}y\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05d444378123c1d9aa248c37aab38eea1711f20e)
Taking
![{\displaystyle f(x)=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea78f54e69b72f398cf6077e61c50a05b532d4c0)
for all
![{\displaystyle x\in X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d)
gives the counting measure.
See also[edit]
References[edit]