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Encyclopedia > Counting measure

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 Lp spaces into more familiar settings. If Ω = {1,...,n} and S is the measure space with the counting measure on Ω, then Lp(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,...,xn).

Similarly, if Ω is taken to be the natural numbers and S is the measure space with the counting measure on Ω, then Lp(S) consists of those sequences x = (xn) 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 . Results from FactBites:

 Measure (mathematics) - Wikipedia, the free encyclopedia (1077 words) A measure can be extended to a complete one by considering the σ-algebra of subsets Y which differ by a negligible set from a measurable set X, that is, such that the symmetric difference of X and Y is contained in a null set. The Haar measure for a locally compact topological group is a generalization of the Lebesgue measure and has a similar uniqueness property. Measures that take values in Banach spaces have been studied extensively.
 Measure (2343 words) Measure informs analysis and is one of the key building blocks of the modern theory of analysis and probability. Measure Property 2: The value of μ under any finite or countably infinite disjoint union of subsets of X that are also elements of M is equal to the finite or countably infinite sum respectively of the value of μ under each of the subsets. Measure allows the expansion of the definition of the integral to functions whose domain is any arbitrary set with a corresponding σ-algebra and measure.
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