# 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 $\displaystyle{ X }$ along with a sigma-algebra) but is mostly used on countable sets.

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

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

1. Counting Measure at PlanetMath.org.