# 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 if the subset is infinite.[1]

The counting measure can be defined on any measurable space (i.e. 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 ${\displaystyle \Sigma }$, i.e. all subsets of ${\displaystyle X}$ are measurable. Then the counting measure ${\displaystyle \mu }$ on this measurable space ${\displaystyle (X,\Sigma )}$ is the positive measure ${\displaystyle \Sigma \rightarrow [0,+\infty ]}$ 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]