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- Essential_supremum_and_essential_infimum abstract "In mathematics, the concepts of essential supremum and essential infimum are related to the notions of supremum and infimum, but the former are more relevant in measure theory, where one often deals with statements that are not valid everywhere, that is for all elements in a set, but rather almost everywhere, that is, except on a set of measure zero.Let (X, Σ, μ) be a measure space, and let f : X → R be a function defined on X and with real values, which is not necessarily measurable. A real number a is called an upper bound for f if f(x) ≤ a for all x in X, that is, if the setis empty. In contrast, a is called an essential upper bound if the set is contained in a set of measure zero, that is to say, if f(x) ≤ a for almost all x in X. Then, in the same way as the supremum of f is defined to be the smallest upper bound, the essential supremum is defined as the smallest essential upper bound.More formally, the essential supremum of f, ess sup f, is defined by if the set of essential upper bounds is not empty, and ess sup f = +∞ otherwise. Exactly in the same way one defines the essential infimum as the largest essential lower bound, that is,if the set of essential lower bounds is not empty, and as −∞ otherwise.".
- Essential_supremum_and_essential_infimum wikiPageID "3535995".
- Essential_supremum_and_essential_infimum wikiPageRevisionID "586032137".
- Essential_supremum_and_essential_infimum hasPhotoCollection Essential_supremum_and_essential_infimum.
- Essential_supremum_and_essential_infimum id "2044".
- Essential_supremum_and_essential_infimum title "Essential supremum".
- Essential_supremum_and_essential_infimum subject Category:Integral_calculus.
- Essential_supremum_and_essential_infimum subject Category:Measure_theory.
- Essential_supremum_and_essential_infimum comment "In mathematics, the concepts of essential supremum and essential infimum are related to the notions of supremum and infimum, but the former are more relevant in measure theory, where one often deals with statements that are not valid everywhere, that is for all elements in a set, but rather almost everywhere, that is, except on a set of measure zero.Let (X, Σ, μ) be a measure space, and let f : X → R be a function defined on X and with real values, which is not necessarily measurable.".
- Essential_supremum_and_essential_infimum label "Espace L∞".
- Essential_supremum_and_essential_infimum label "Essential supremum and essential infimum".
- Essential_supremum_and_essential_infimum label "Supremo essencial".
- Essential_supremum_and_essential_infimum label "Wesentliches Supremum".
- Essential_supremum_and_essential_infimum label "Существенный супремум".
- Essential_supremum_and_essential_infimum label "本質的上限と本質的下限".
- Essential_supremum_and_essential_infimum label "本质上确界和本质下确界".
- Essential_supremum_and_essential_infimum sameAs Wesentliches_Supremum.
- Essential_supremum_and_essential_infimum sameAs Espace_L∞.
- Essential_supremum_and_essential_infimum sameAs 本質的上限と本質的下限.
- Essential_supremum_and_essential_infimum sameAs Supremo_essencial.
- Essential_supremum_and_essential_infimum sameAs m.09k25n.
- Essential_supremum_and_essential_infimum sameAs Q1969054.
- Essential_supremum_and_essential_infimum sameAs Q1969054.
- Essential_supremum_and_essential_infimum wasDerivedFrom Essential_supremum_and_essential_infimum?oldid=586032137.
- Essential_supremum_and_essential_infimum isPrimaryTopicOf Essential_supremum_and_essential_infimum.