Counting measure - Definition and Overview

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 X 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. Then (Ω, X, μ) is a measure space.

The counting measure allows to translate many statements about L^p spaces into more familiar settings. If Ω = {1,...,n} and S is the measure space with the counting measure on Ω, then L^p(S) is the same as Rⁿ (or Cⁿ), with norm defined by

<math>\|x\|_p = \left ( \sum_{i=1}^n |x_i|^p \right )^{1/p}<math>

for x = (x₁,...,x_n).

Similarly, if Ω is taken to be the 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

<math>\|x\|_p = \left ( \sum_{i=1}^\infty |x_i|^p \right)^{1/p}<math>

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