Counting measure

In mathematics, 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 if the subset is infinite.

The counting measure can be defined on any measurable set, but is mostly used on countable sets.

In formal notation, we can make any set X into a measurable space by taking the sigma-algebra $\Sigma$ of measurable subsets to consist of all subsets of $X$ . Then the counting measure $\mu$ on this measurable space $(X,\Sigma )$ is the positive measure $\Sigma \rightarrow [0,+\infty ]$ 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 construct. With the notation as above, any function $f\colon X\to [0,\infty )$ defines a measure $\mu$ on $(X,\Sigma )$ via

$\mu (A):=\sum _{a\in A}f(a)\,\forall A\subseteq X,$ where the possibly uncountable sum of real numbers is defined to be the sup of the sums over all finite subsets, i.e.,

$\sum _{y\in Y\subseteq \mathbb {R} }y:=\sup _{F\subseteq Y,|F|<\infty }\left\{\sum _{y\in F}y\right\}.$ Taking f(x)=1 for all x in X produces the counting measure.