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 the subset's elements if this 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 Lp 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 Lp(S) is the same as Rn (or Cn), with norm defined by

\|x\|_p = \biggl ( \sum_{i=1}^n |x_i|^p \biggr )^{1/p}

for x = (x1,...,xn). 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 Lp(S) consists of those sequences x = (xn) for which

\|x\|_p = \biggl ( \sum_{i=1}^\infty |x_i|^p \biggr)^{1/p}

is finite. This space is often written as \ell^p.

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.