Counting Measure

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



References

2020

  • (Wikipedia, 2020) ⇒ https://en.wikipedia.org/wiki/Counting_measure 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] X [/math] along with a sigma-algebra) but is mostly used on countable sets.

      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] , i.e. all subsets of [math] X [/math] are measurable. Then the counting measure [math] \mu [/math] on this measurable space [math] (X,\Sigma) [/math] is the positive measure [math] \Sigma\rightarrow[0,+\infty] [/math] defined by : [math] \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] . The counting measure on [math] (X,\Sigma) [/math] is σ-finite if and only if the space [math] X [/math] is countable.

  1. Counting Measure at PlanetMath.org.