Counting Measure

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 $X$ along with a sigma-algebra) but is mostly used on countable sets.

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

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

1. Counting Measure at PlanetMath.org.