Definition:Counting Measure

Definition

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.

Also defined as

The phrase counting measure on $X$ is sometimes taken to imply that $\Sigma = \powerset X$, the power set of $X$.

Also see

• Definition:Natural Numbers
• Definition:Cardinality

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $4.7 \ \text{(iii)}$