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In mathematics, the Henstock–Kurzweil integral or gauge integral (also known as the (narrow) Denjoy integral (pronounced (:dɑ̃ˈʒwa)), Luzin integral or Perron integral, but not to be confused with the more general wide Denjoy integral) is one of a number of definitions of the integral of a function. It is a generalization of the Riemann integral, and in some situations is more general than the Lebesgue integral. This integral was first defined by Arnaud Denjoy (1912). Denjoy was interested in a definition that would allow one to integrate functions like : This function has a singularity at 0, and is not Lebesgue integrable. However, it seems natural to calculate its integral except over the interval () and then let ε, δ → 0. Trying to create a general theory, Denjoy used transfinite induction over the possible types of singularities, which made the definition quite complicated. Other definitions were given by Nikolai Luzin (using variations on the notions of absolute continuity), and by Oskar Perron, who was interested in continuous major and minor functions. It took a while to understand that the Perron and Denjoy integrals are actually identical. Later, in 1957, the Czech mathematician Jaroslav Kurzweil discovered a new definition of this integral elegantly similar in nature to Riemann's original definition which he named the gauge integral; the theory was developed by Ralph Henstock. Due to these two important mathematicians, it is now commonly known as the Henstock–Kurzweil integral. The simplicity of Kurzweil's definition made some educators advocate that this integral should replace the Riemann integral in introductory calculus courses,〔(【引用サイトリンク】url=http://www.math.vanderbilt.edu/~schectex/ccc/gauge/letter/ )〕 but this idea has not gained traction. ==Definition== Henstock's definition is as follows: Given a tagged partition ''P'' of (''b'' ), say : and a positive function : which we call a ''gauge'', we say ''P ''is -fine if : For a tagged partition ''P'' and a function : we define the Riemann sum to be : Given a function : we now define a number ''I'' to be the Henstock–Kurzweil integral of ''f'' if for every ε > 0 there exists a gauge such that whenever ''P'' is -fine, we have : If such an ''I'' exists, we say that ''f'' is Henstock–Kurzweil integrable on (''b'' ). Cousin's theorem states that for every gauge , such a -fine partition ''P'' does exist, so this condition cannot be satisfied vacuously. The Riemann integral can be regarded as the special case where we only allow constant gauges. 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「Henstock–Kurzweil integral」の詳細全文を読む スポンサード リンク
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