Definition:Counting Measure
From ProofWiki
in the process of being refactored, or:
has been identified as a candidate for refactoring. In particular:
page needs to be smoothened
Until this has been finished, please leave {{refactor}} in the code.
Contents |
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
Sources
- René L. Schilling: Measures, Integrals and Martingales (2005)... (previous)... (next) $4.7 \ \text{(iii)}$
