Inmathematics, specificallymeasure theory, thecounting measure is an intuitive way to put ameasure on anyset – the "size" of asubset is taken to be the number of elements in the subset if the subset has finitely many elements, andinfinity${\displaystyle \mathrm{\infty }}$ if the subset isinfinite.^[1]

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

In formal notation, we can turn any set${\displaystyle X}$ into a measurable space by taking thepower set of${\displaystyle X}$ as thesigma-algebra${\displaystyle \mathrm{\Sigma };}$ that is, all subsets of${\displaystyle X}$ are measurable sets. Then the counting measure${\displaystyle \mu }$ on this measurable space${\displaystyle (X,\mathrm{\Sigma })}$ is the positive measure${\displaystyle \mathrm{\Sigma }\to [0,+\mathrm{\infty }]}$ defined byfor all${\displaystyle A\in \mathrm{\Sigma },}$ where${\displaystyle |A|}$ denotes thecardinality of the set${\displaystyle A.}$^[2]

The counting measure on${\displaystyle (X,\mathrm{\Sigma })}$ isσ-finite if and only if the space${\displaystyle X}$ iscountable.^[3]

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${\displaystyle f:\mathbb{N}\to [0,\mathrm{\infty }]}$. As it is defined on${\displaystyle \mathbb{N}}$,${\displaystyle f}$ can be represented pointwise as$${\displaystyle f(x)=\sum _{n=1}^{\mathrm{\infty }}f(n)1_{\{n\}}(x)=\underset{M\to \mathrm{\infty }}{lim}\underset{{\varphi }_M(x)}{\underbrace{\sum _{n=1}^{M}f(n)1_{\{n\}}(x)}}=\underset{M\to \mathrm{\infty }}{lim}{\varphi }_M(x)}$$

Each${\displaystyle {\varphi }_M}$ is measurable. Moreover${\displaystyle {\varphi }_{M+1}(x)={\varphi }_M(x)+f(M+1)\cdot 1_{\{M+1\}}(x)\ge {\varphi }_M(x)}$. Still further, as each${\displaystyle {\varphi }_M}$ is a simple function$${\displaystyle {\int }_{\mathbb{N}}{\varphi }_{M}d\mu ={\int }_{\mathbb{N}}\left(\sum _{n=1}^{M}f(n)1_{\{n\}}(x)\right)d\mu =\sum _{n=1}^{M}f(n)\mu (\{n\})=\sum _{n=1}^{M}f(n)\cdot 1=\sum _{n=1}^{M}f(n)}$$Hence by the monotone convergence theorem$${\displaystyle {\int }_{\mathbb{N}}fd\mu =\underset{M\to \mathrm{\infty }}{lim}{\int }_{\mathbb{N}}{\varphi }_{M}d\mu =\underset{M\to \mathrm{\infty }}{lim}\sum _{n=1}^{M}f(n)=\sum _{n=1}^{\mathrm{\infty }}f(n)}$$

The counting measure is a special case of a more general construction. With the notation as above, any function${\displaystyle f:X\to [0,\mathrm{\infty })}$ defines a measure${\displaystyle \mu }$ on${\displaystyle (X,\mathrm{\Sigma })}$ via$${\displaystyle \mu (A):=\sum _{a\in A}f(a)\text{ for all }A\subseteq X,}$$where the possibly uncountable sum of real numbers is defined to be thesupremum of the sums over all finite subsets, that is,Taking${\displaystyle f(x)=1}$ for all${\displaystyle x\in X}$ gives the counting measure.

• Pip (counting) – Easily countable items
• Random counting measure
• Set function – Function from sets to numbers

• Measures (measure theory)

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