Counting Measure

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

• See: Sigma-Algebra, Countable, Mathematics, Measure Theory, Measure (Mathematics), Set (Mathematics), Subset, Infinity Symbol, Infinite Set, Measurable Space, Power Set, Cardinality.

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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.

    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.

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