# 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.

## 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.