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In mathematics, the Bochner integral, named for Salomon Bochner, extends the definition of Lebesgue integral to functions that take values in a Banach space, as the limit of integrals of simple functions. ==Definition== Let (''X'', Σ, μ) be a measure space and ''B'' a Banach space. The Bochner integral is defined in much the same way as the Lebesgue integral. First, a simple function is any finite sum of the form : where the ''E''''i'' are disjoint members of the σ-algebra Σ, the ''b''''i'' are distinct elements of ''B'', and χE is the characteristic function of ''E''. If ''μ''(''E''''i'') is finite whenever ''b''''i'' ≠ 0, then the simple function is integrable, and the integral is then defined by : exactly as it is for the ordinary Lebesgue integral. A measurable function ƒ : ''X'' → ''B'' is Bochner integrable if there exists a sequence of integrable simple functions ''s''''n'' such that : where the integral on the left-hand side is an ordinary Lebesgue integral. In this case, the Bochner integral is defined by : It can be shown that a function is Bochner integrable if and only if it lies in the Bochner space . 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「Bochner integral」の詳細全文を読む スポンサード リンク
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