Definition:Counting Measure

Definition
The counting measure is a mapping taking a set to its cardinality, that is, the number of elements it has.

Formal Definition
Let $\left({X, \Sigma}\right)$ be a measurable space.

The counting measure (on $X$), denoted $\left\vert{\cdot}\right\vert$, is the measure defined by:


 * $\left\vert{\cdot}\right\vert: \Sigma \to \overline{\R}, \ \left\vert{E}\right\vert := \begin{cases}\#\left({E}\right) & \text{if $E$ is finite} \\ +\infty & \text{if $E$ is infinite}\end{cases}$

where $\overline{\R}$ denotes the extended real numbers, and $\#$ denotes cardinality.

That $\left\vert{\cdot}\right\vert$ is actually a measure is shown on Counting Measure is Measure.

Also defined as
The phrase counting measure on $X$ is sometimes taken to imply that $\Sigma = \mathcal P \left({X}\right)$, the power set of $X$.

Also see

 * Natural Numbers
 * Cardinality