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 ${\displaystyle \infty }$ if the subset is infinite.^[1]

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