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Counting measure - Wikipedia

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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 {\displaystyle \infty } if the subset is infinite.[1]

The counting measure can be defined on any measurable space (that is, any set {\displaystyle X} along with a sigma-algebra) but is mostly used on countable sets.[1]

In formal notation, we can turn any set {\displaystyle X} into a measurable space by taking the power set of {\displaystyle X} as the sigma-algebra {\displaystyle \Sigma ;} that is, all subsets of {\displaystyle X} are measurable sets. Then the counting measure {\displaystyle \mu } on this measurable space {\displaystyle (X,\Sigma )} is the positive measure {\displaystyle \Sigma \to [0,+\infty ]} 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}}} for all {\displaystyle A\in \Sigma ,} where {\displaystyle \vert A\vert } denotes the cardinality of the set {\displaystyle A.}[2]

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

Integration on the set of natural numbers with counting measure

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

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

The counting measure is a special case of a more general construction. With the notation as above, any function {\displaystyle f:X\to [0,\infty )} defines a measure {\displaystyle \mu } on {\displaystyle (X,\Sigma )} via {\displaystyle \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, {\displaystyle \sum _{y\,\in \,Y\!\ \subseteq \,\mathbb {R} }y\ :=\ \sup _{F\subseteq Y,\,|F|<\infty }\left\{\sum _{y\in F}y\right\}.} Taking {\displaystyle f(x)=1} for all {\displaystyle x\in X} gives the counting measure.

  1. ^ a b 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.