Counting_measure Knowpia

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 $\infty$ if the subset is infinite.^[1]

The counting measure can be defined on any measurable space (that is, 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 $\Sigma ;$ that is, all subsets of $X$ are measurable sets. Then the counting measure $\mu$ on this measurable space $(X,\Sigma )$ is the positive measure $\Sigma \to [0,+\infty ]$ 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]

Integration on $\mathbb {N}$ with counting measure edit

Take the measure space $(\mathbb {N} ,2^{\mathbb {N} },\mu )$ , where $2^{\mathbb {N} }$ is the set of all subsets of the naturals and $\mu$ the counting measure. Take any measurable $f:\mathbb {N} \to [0,\infty ]$ . As it is defined on $\mathbb {N}$ , $f$ can be represented pointwise as

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)

Each $\phi _{M}$ is measurable. Moreover $\phi _{M+1}(x)=\phi _{M}(x)+f(M+1)\cdot 1_{\{M+1\}}(x)\geq \phi _{M}(x)$ . Still further, as each $\phi _{M}$ is a simple function

\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)

Hence by the monotone convergence theorem

\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)

Discussion edit

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

\mu (A):=\sum _{a\in A}f(a)\quad {\text{ for all }}A\subseteq X,

where the possibly uncountable sum of real numbers is defined to be the supremum of the sums over all finite subsets, that is,

\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 edit

^ ^a ^b 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.