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.

Integration on the set of natural numbers with counting measure

Take the measure space $(\mathbb, 2^\mathbb, \mu)$, where $2^\mathbb$ is the set of all subsets of the naturals and $\mu$ the counting measure. Take any measurable f : \mathbb \to ,\infty/math>. As it is defined on $\mathbb$, $f$ can be represented pointwise as $f(x) = \sum_^\infty f(n) 1_(x) = \lim_ \underbrace_ = \lim_ \phi_M (x)$

Each $\phi_M$ is measurable. Moreover $\phi_(x) = \phi_M (x) + f(M+1) \cdot 1_(x) \geq \phi_M (x)$. Still further, as each $\phi_M$ is a simple function $\int_\mathbb \phi_M d\mu = \int_\mathbb \left( \sum_^M f(n) 1_ (x) \right) d\mu = \sum_^M f(n) \mu (\) = \sum_^M f(n) \cdot 1 = \sum_^M f(n)$Hence by the monotone convergence theorem
$\int_\mathbb f d\mu = \lim_ \int_\mathbb \phi_M d\mu = \lim_ \sum_^M f(n) = \sum_^\infty f(n)$

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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