用泛函观点看LEBESGUE积分

从泛函分析观点来看Lebesgue积分，使得Lebesgue积分可以用泛函分析最简单最基本的方法独立导出．基本做法是将Riemann对于区间0，1]上的连续函数的积分看成连续函数空间C0，1]上的连续线性泛函，再将它“自然”延拓到C0，1]在积分范数意义下的完备化空间，而这个完备化空间正是Lebesgue可积函数空间L10，1]．

A Functional View of Lebesgue Integration

This note is devoted to describe the classical Lebesgue integration from a functional point of view. Let C0, 1] be the linear space of all real-valued continuous functions endowed with the norm ｜｜x｜｜ = f0^1 ｜x（t）｜dt and let X = C0, 1] be its completion. We define a linear functional xR^＊ on C0, 1] by {xR^＊, x） = fg x（t）dt in Riemann＇s sense, and let x＊ be the natural extension of xR^＊ from C0, 1] to its completion C0, 1]. With a sketch but self-contained proof, we show the Lebesgue integration is just the natural extension of xR^＊ to C0, 1], that is, C0, 1] = L10, 1] and （x＊, xI =- f0^1 x（t）dt in Lebesgue＇s sence.