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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 Khinchin integral ： ウィキペディア英語版 Khinchin integral In mathematics, the Khinchin integral (sometimes spelled Khintchine integral), also known as the Denjoy–Khinchin integral, generalized Denjoy integral or wide Denjoy integral, is one of a number of definitions of the integral of a function. It is a generalization of the Riemann and Lebesgue integrals. It is named after Aleksandr Khinchin and Arnaud Denjoy, but is not to be confused with the (narrow) Denjoy integral. ==Motivation== If ''g'' : ''I'' → R is a Lebesgue-integrable function on some interval ''I'' = (), and if :$f(x)\; =\; \backslash int\_a^x\; g(t)\backslash ,dt$ is its Lebesgue indefinite integral, then the following assertions are true: #''f'' is absolutely continuous (see below) #''f'' is differentiable almost everywhere #Its derivative coincides almost everywhere with ''g''(''x''). (In fact, ''all'' absolutely continuous functions are obtained in this manner.) The Lebesgue integral could be defined as follows: ''g'' is Lebesgue-integrable on ''I'' iff there exists a function ''f'' that is absolutely continuous whose derivative coincides with ''g'' almost everywhere. However, even if ''f'' : ''I'' → R is differentiable ''everywhere'', and ''g'' is its derivative, it does not follow that ''f'' is (up to a constant) the Lebesgue indefinite integral of ''g'', simply because ''g'' can fail to be Lebesgue-integrable, i.e., ''f'' can fail to be absolutely continuous. An example of this is given by the derivative ''g'' of the (differentiable but not absolutely continuous) function ''f''(''x'')=''x''²·sin(1/''x''²) (the function ''g'' is not Lebesgue-integrable around 0). The Denjoy integral corrects this lack by ensuring that the derivative of any function ''f'' that is everywhere differentiable (or even differentiable everywhere except for at most countably many points) is integrable, and its integral reconstructs ''f'' up to a constant; the Khinchin integral is even more general in that it can integrate the ''approximate'' derivative of an approximately differentiable function (see below for definitions). To do this, one first finds a condition that is weaker than absolute continuity but is satisfied by any approximately differentiable function. This is the concept of ''generalized'' absolute continuity; generalized absolutely continuous functions will be exactly those functions which are indefinite Khinchin integrals. 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「Khinchin integral」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.