Template:Short description In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any 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 <math>\infty</math> if the subset is infinite.<ref name="pm">Template:PlanetMath</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>Template:Cite book</ref>
The counting measure on <math>(X,\Sigma)</math> is σ-finite if and only if the space <math>X</math> is countable.<ref>Template:Cite book</ref>
Integration on the set of natural numbers with counting measureEdit
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>
DiscussionEdit
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 alsoEdit
ReferencesEdit