Counting measure

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 infinity $\infty$ if the subset is infinite.

The counting measure can be defined on any measurable space (that is, 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 $\Sigma;$ that is, all subsets of $X$ are measurable sets.
Then the counting measure $\mu$ on this measurable space $(X,\Sigma)$ is the positive measure \Sigma \to ,+\infty/math> defined by
$\mu(A) = \begin \vert A \vert & \text A \text\\ +\infty & \text A \text \end$
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.

Discussion

The counting measure is a special case of a more general construction. With the notation as above, any function $f : X \to [0, \infty)$ defines a measure $\mu$ on $(X, \Sigma)$ via
$\mu(A):=\sum_ f(a)\quad \text A \subseteq X,$
where the possibly uncountable sum of real numbers is defined to be the supremum of the sums over all finite subsets, that is,
$\sum_ y\ :=\ \sup_ \left\.$
Taking $f(x) = 1$ for all $x \in X$ gives the counting measure.

See also

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References

Measures (measure theory)

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