Counting Measure

(Redirected from counting measure)
Jump to navigation Jump to search

A Counting Measure is a Mathematical Operation that can be defined as Counting Function.



  • (Wikipedia, 2020) ⇒ Retrieved:2020-10-11.
    • 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 if the subset is infinite.[1]

      The counting measure can be defined on any measurable space (i.e. any set [math]\displaystyle{ X }[/math] along with a sigma-algebra) but is mostly used on countable sets.

      In formal notation, we can turn any set [math]\displaystyle{ X }[/math] into a measurable space by taking the power set of [math]\displaystyle{ X }[/math] as the

       sigma-algebra [math]\displaystyle{ \Sigma }[/math] , i.e. all subsets of [math]\displaystyle{ X }[/math] are measurable. Then the counting measure [math]\displaystyle{ \mu }[/math] on this measurable space [math]\displaystyle{ (X,\Sigma) }[/math] is the positive measure [math]\displaystyle{ \Sigma\rightarrow[0,+\infty] }[/math] defined by : [math]\displaystyle{ \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]\displaystyle{ A\in\Sigma }[/math] , where [math]\displaystyle{ \vert A\vert }[/math] denotes the cardinality of the set [math]\displaystyle{ A }[/math] . The counting measure on [math]\displaystyle{ (X,\Sigma) }[/math] is σ-finite if and only if the space [math]\displaystyle{ X }[/math] is countable.

  1. Counting Measure at