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In mathematics, the concepts of essential supremum and essential infimum are related to the notions of supremum and infimum, but adapted to measure theory and functional analysis, where one often deals with statements that are not valid for ''all'' elements in a set, but rather ''almost everywhere'', i.e., except on a set of measure zero. ==Definition== Let ''f'' : ''X'' → R be a real valued function defined on a set ''X''. A real number ''a'' is called an ''upper bound'' for ''f'' if ''f''(''x'') ≤ ''a'' for all ''x'' in ''X'', i.e., if the set : is empty. Let : be the set of upper bounds of ''f''. Then the supremum of ''f'' is defined by : if the set of upper bounds is nonempty, and sup ''f'' = +∞ otherwise. Now assume in addition that (''X'', Σ, ''μ'') is a measure space and, for simplicity, assume that the function ''f'' is measurable. A number ''a'' is called an ''essential upper bound'' of ''f'' if the measurable set ''f''−1(''a'', ∞) is a set of measure zero, i.e., if ''f''(''x'') ≤ ''a'' for ''almost all'' ''x'' in ''X''. Let : be the set of essential upper bounds. Then the essential supremum is defined similarly as :, and ess sup ''f'' = +∞ otherwise. Exactly in the same way one defines the essential infimum as the supremum of the ''essential lower bounds'', that is, : if the set of essential lower bounds is nonempty, and as −∞ otherwise. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Essential supremum and essential infimum」の詳細全文を読む スポンサード リンク
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