Counting measure

In mathematics, the counting measure is an intuitive way to put a measure (mathematics) 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 Lp space 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 R^''n'' (or C^''n''), with normed vector space defined by :

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

for ''x'' = (''x''₁,...,''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 :

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

is finite. This space is often written as

\ell^p

. Measure theory

Counting measure

I do not see the connection between the 2nd and the 3rd paragraph. Apparently, the |.|-notation is used in two different meanings. Thanks in advance for any improved explanation of how Lp spaces relate to the counting measure. - User:Weialawaga 20:47, 29 Mar 2004 (UTC)

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