\begin{definition}[Definition:Counting Measure]
Let $\struct {X, \Sigma}$ be a measurable space.
The '''counting measure (on $X$)''', denoted $\size {\, \cdot \,}$, is the measure defined by:
:$\size {\, \cdot \,}: \Sigma \to \overline \R, \ \size E := \begin {cases} \map \# E & : \text {$E$ is finite} \\ +\infty & : \text {$E$ is infinite} \end{cases}$
where $\overline \R$ denotes the extended real numbers, and $\#$ denotes cardinality.
That $\size {\, \cdot \,}$ is actually a measure is shown on Counting Measure is Measure.
\end{definition}