Counting measure

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In mathematics, 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 subset is finite, and ∞ if the subset is infinite.

Formally, start with a set Ω and consider the sigma algebra Σ on Ω consisting of all subsets of Ω. Define a measure μ on this sigma algebra by setting μ(A) = |A| if A is a finite subset of Ω and μ(A) = ∞ if A is an infinite subset of Ω, where |A| denotes the cardinality of set A. Then (Ω, Σ, μ) is a measure space.

The counting measure allows one to translate many statements about L^p spaces, such as the Cauchy–Schwarz inequality, Hölder's inequality or the Minkowski inequality, to more familiar settings. If Ω = {1,...,n} and S = (Ω, Σ, μ) is the measure space with the counting measure μ on Ω, then L^p(S) is the same as Rⁿ (or Cⁿ), with norm defined by

for x = (x₁,...,x_n). Dividing the counting measure μ by the number n of elements in Ω gives the discrete uniform distribution.

Similarly, if Ω is taken to be the set of natural numbers and S is the measure space with the counting measure on Ω, then L^p(S) consists of those sequences x = (x_n) for which

is finite. This space is often written as .

The counting measure on countable sets is also helpful to apply theorems from Lebesgue integration theory (like monotone convergence theorem, Fatou's lemma, dominated convergence theorem, Fubini's theorem, etc.) to series.