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 defined by for all A\in\Sigma, where 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 \mathbb{N} with counting measure

Take the measure space, where is the set of all subsets of the naturals and \mu the counting measure. Take any measurable. As it is defined on \mathbb{N}, f can be represented pointwise as Each \phi_M is measurable. Moreover. Still further, as each \phi_M is a simple function Hence by the monotone convergence theorem

Discussion

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