Editing Counting measure (section)

{{Short description|Mathematical concept}}
In [[mathematics]], specifically [[measure theory]], the '''counting measure''' is an intuitive way to put a [[Measure (mathematics)|measure]] on any [[Set (mathematics)|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 symbol|infinity <math>\infty</math>]] if the subset is [[Infinite set|infinite]].<ref name="pm">{{PlanetMath|urlname=CountingMeasure|title=Counting Measure}}</ref>

The counting measure can be defined on any [[measurable space]] (that is, any set <math>X</math> along with a sigma-algebra)  but is mostly used on [[countable]] sets.<ref name="pm" />

In formal notation, we can turn any set <math>X</math> into a measurable space by taking the [[power set]] of <math>X</math> as the [[sigma-algebra]] <math>\Sigma;</math> that is, all subsets of <math>X</math> are measurable sets.  
Then the counting measure <math>\mu</math> on this measurable space <math>(X,\Sigma)</math> is the positive measure <math>\Sigma \to [0,+\infty]</math> defined by
<math display=block>
\mu(A) = \begin{cases}
\vert A \vert & \text{if } A \text{ is finite}\\
+\infty & \text{if } A \text{ is infinite}
\end{cases}
</math>
for all <math>A\in\Sigma,</math> where <math>\vert A\vert</math> denotes the [[cardinality]] of the set <math>A.</math><ref>{{cite book |first=René L. |last=Schilling |year=2005 |title=Measures, Integral and Martingales |publisher=Cambridge University Press |isbn=0-521-61525-9 |page=27}}</ref>

The counting measure on <math>(X,\Sigma)</math> is [[σ-finite]] if and only if the space <math>X</math> is [[countable]].<ref>{{cite book |first=Ernst |last=Hansen |year=2009 |title=Measure Theory |edition=Fourth |publisher=Department of Mathematical Science, University of Copenhagen |isbn=978-87-91927-44-7 |page=47}}</ref>