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 **R**^{n} (or **C**^{n}), 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* = (*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 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...
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is finite. This space is often written as . |