Editing Counting measure

{{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>

==Integration on the set of natural numbers with counting measure==

Take the measure space <math>(\mathbb{N}, 2^\mathbb{N}, \mu)</math>, where <math>2^\mathbb{N}</math> is the set of all subsets of the naturals and <math>\mu</math> the counting measure. Take any measurable <math>f : \mathbb{N} \to [0,\infty]</math>. As it is defined on <math>\mathbb{N}</math>, <math>f</math> can be represented pointwise as <math display=block> f(x) = \sum_{n=1}^\infty f(n) 1_{\{n\}}(x) = \lim_{M \to \infty} \underbrace{ \ \sum_{n=1}^M f(n) 1_{\{n\}}(x) \ }_{  \phi_M (x) } = \lim_{M \to \infty} \phi_M (x)  </math> 

Each <math>\phi_M</math> is measurable. Moreover <math>\phi_{M+1}(x) = \phi_M (x) + f(M+1) \cdot 1_{ \{M+1\} }(x) \geq \phi_M (x) </math>. Still further, as each <math>\phi_M</math> is a simple function <math display="block"> \int_\mathbb{N} \phi_M d\mu  =
\int_\mathbb{N} \left( \sum_{n=1}^M f(n) 1_{\{n\}} (x) \right) d\mu
= \sum_{n=1}^M f(n) \mu (\{n\}) 
= \sum_{n=1}^M f(n) \cdot 1 = \sum_{n=1}^M f(n) </math>Hence by the monotone convergence theorem
<math display=block>
\int_\mathbb{N} f d\mu = \lim_{M \to \infty} \int_\mathbb{N} \phi_M d\mu = \lim_{M \to \infty} \sum_{n=1}^M f(n) = \sum_{n=1}^\infty f(n)
</math>

==Discussion==

The counting measure is a special case of a more general construction.  With the notation as above, any function <math>f : X \to [0, \infty)</math> defines a measure <math>\mu</math> on <math>(X, \Sigma)</math> via
<math display=block>\mu(A):=\sum_{a \in A} f(a)\quad \text{ for all } A \subseteq X,</math>
where the possibly uncountable sum of real numbers is defined to be the [[supremum]] of the sums over all finite subsets, that is, 
<math display=block>\sum_{y\,\in\,Y\!\ \subseteq\,\mathbb R} y\ :=\ \sup_{F \subseteq Y,\, |F| < \infty} \left\{ \sum_{y \in F} y  \right\}.</math>
Taking <math>f(x) = 1</math> for all <math>x \in X</math> gives the counting measure. 

==See also==

* {{annotated link|Pip (counting)}}
* {{annotated link|Random counting measure}}
* {{annotated link|Set function}}

==References==

{{reflist}}

{{Measure theory}}

[[Category:Measures (measure theory)]]

{{DEFAULTSORT:Counting Measure}}