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. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a measure is a function that assigns a number, e. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ...

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. In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S that is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ... In mathematics, a measure is a function that assigns a number, e. ...

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 In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ...

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 Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... This is a page about mathematics. ...

is finite. This space is often written as .