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