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 R^{n} (or C^{n}), with norm defined by
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
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