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
if the subset is infinite.
[1] The counting measure can be defined on any measurable space (that is, any set
along with a sigma-algebra) but is mostly used on countable sets.
[1] In formal notation, we can turn any set
into a measurable space by taking the power set of
are measurable sets.
Then the counting measure
on this measurable space
is the positive measure
{\displaystyle \mu (A)={\begin{cases}\vert A\vert &{\text{if }}A{\text{ is finite}}\\+\infty &{\text{if }}A{\text{ is infinite}}\end{cases}}}
denotes the cardinality of the set
[2] The counting measure on
is σ-finite if and only if the space
[3] Take the measure space
, μ )
is the set of all subsets of the naturals and
μ
the counting measure.
Take any measurable
can be represented pointwise as
ϕ
ϕ
is a simple function
Hence by the monotone convergence theorem
The counting measure is a special case of a more general construction.
With the notation as above, any function
defines a measure
where the possibly uncountable sum of real numbers is defined to be the supremum of the sums over all finite subsets, that is,
gives the counting measure.