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- Measurable_function abstract "In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration. Specifically, a function between measurable spaces is said to be measurable if the preimage of each measurable set is measurable, analogous to the situation of continuous functions between topological spaces.This definition can be deceptively simple, however, as special care must be taken regarding the σ-algebras involved. In particular, when a function f: R → R is said to be Lebesgue measurable what is actually meant is that is a measurable function—that is, the domain and range represent different σ-algebras on the same underlying set (here is the sigma algebra of Lebesgue measurable sets, and is the Borel algebra on R). As a result, the composition of Lebesgue-measurable functions need not be Lebesgue-measurable.By convention a topological space is assumed to be equipped with the Borel algebra generated by its open subsets unless otherwise specified. Most commonly this space will be the real or complex numbers. For instance, a real-valued measurable function is a function for which the preimage of each Borel set is measurable. A complex-valued measurable function is defined analogously. In practice, some authors use measurable functions to refer only to real-valued measurable functions with respect to the Borel algebra. If the values of the function lie in an infinite-dimensional vector space instead of R or C, usually other definitions of measurability are used, such as weak measurability and Bochner measurability.In probability theory, the sigma algebra often represents the set of available information, and a function (in this context a random variable) is measurable if and only if it represents an outcome that is knowable based on the available information. In contrast, functions that are not Lebesgue measurable are generally considered pathological, at least in the field of analysis.".
- Measurable_function wikiPageExternalLink www.encyclopediaofmath.org.
- Measurable_function wikiPageExternalLink Borel_function.
- Measurable_function wikiPageExternalLink Measurable_function.
- Measurable_function wikiPageID "44055".
- Measurable_function wikiPageRevisionID "605938615".
- Measurable_function hasPhotoCollection Measurable_function.
- Measurable_function subject Category:Measure_theory.
- Measurable_function subject Category:Types_of_functions.
- Measurable_function comment "In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration.".
- Measurable_function label "Fonction mesurable".
- Measurable_function label "Función medible".
- Measurable_function label "Funkcja mierzalna".
- Measurable_function label "Funzione misurabile".
- Measurable_function label "Função mensurável".
- Measurable_function label "Measurable function".
- Measurable_function label "Meetbare functie".
- Measurable_function label "Messbare Funktion".
- Measurable_function label "Измеримая функция".
- Measurable_function label "可测函数".
- Measurable_function label "可測関数".
- Measurable_function sameAs Měřitelná_funkce.
- Measurable_function sameAs Messbare_Funktion.
- Measurable_function sameAs Función_medible.
- Measurable_function sameAs Fonction_mesurable.
- Measurable_function sameAs Funzione_misurabile.
- Measurable_function sameAs 可測関数.
- Measurable_function sameAs 가측함수.
- Measurable_function sameAs Meetbare_functie.
- Measurable_function sameAs Funkcja_mierzalna.
- Measurable_function sameAs Função_mensurável.
- Measurable_function sameAs m.0c0hb.
- Measurable_function sameAs Q516776.
- Measurable_function sameAs Q516776.
- Measurable_function wasDerivedFrom Measurable_function?oldid=605938615.
- Measurable_function isPrimaryTopicOf Measurable_function.