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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Lifting theory ： ウィキペディア英語版 Lifting theory Lifting Theory was first introduced by John von Neumann in his (1931) pioneering paper (answering a question raised by Alfréd Haar),〔von Neumann, J.: Algebraische Repräsentanten der Funktionen bis auf eine Menge von Maße Null. J. Crelle 165, 109-115 (1931)〕 followed later by Dorothy Maharam’s (1958) paper,〔Maharam, D.: On a theorem of von Neumann. Proc. Amer. Math. Soc. 9, 987-995 (1958)〕 and by A. Ionescu Tulcea and C. Ionescu Tulcea’s (1961) paper.〔A. Ionescu Tulcea and C. Ionescu Tulcea: On the lifting property, I., J. Math. Anal. App. 3, 537-546 (1961)〕 Lifting Theory was motivated to a large extent by its striking applications; for its development up to 1969, see the Ionescu Tulceas' work and the monograph,〔Alexandra Ionescu Tulcea and Cassius Ionescu Tulcea, ''Topics in the Theory of Lifting'', Ergebnisse der Mathematik, Vol. 48, Springer-Verlag, Berlin, Heidelberg, New York (1969)〕 now a standard reference in the field. Lifting Theory continued to develop after 1969, yielding significant new results and applications. A lifting on a measure space (''X'', Σ, μ) is a linear and multiplicative inverse :$T:L^\backslash infty(X,\backslash Sigma,\backslash mu)\backslash to\; \backslash mathcal\; L^\backslash infty(X,\backslash Sigma,\backslash mu)$ of the quotient map : $\backslash begin\backslash mathcal\; L^\backslash infty(X,\backslash Sigma,\backslash mu)\backslash to\; L^\backslash infty(X,\backslash Sigma,\backslash mu)\; \backslash \backslash$ f\mapsto ()\end In other words, a lifting picks from every equivalence class () of bounded measurable functions modulo negligible functions a representative— which is henceforth written ''T''(()) or ''T''() or simply ''Tf'' — in such a way that :$T(r()+s())(p)=rT()(p)\; +\; sT()(p),\; \backslash qquad\; \backslash forall\; p\backslash in\; X,\; r,s\backslash in\; \backslash mathbf\; R;$ :$T(()\backslash times())(p)=T()(p)\backslash times\; T()(p),\; \backslash qquad\; \backslash forall\; p\backslash in\; X;$ :$T()=1.$ Liftings are used to produce disintegrations of measures, for instance conditional probability distributions given continuous random variables, and fibrations of Lebesgue measure on the level sets of a function. 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「Lifting theory」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.