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 $\infty$ if the subset is infinite.

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.

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 $A.$

The counting measure on $(X,\Sigma )$ is σ-finite if and only if the space $X$ is countable.

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

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

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.

See also

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