用泛函观点看LEBESGUE积分(英文)  

A Functional View of Lebesgue Integration

作　　者：程立新[1] 

机构地区：[1]厦门大学数学科学学院,厦门361005

出　　处：《应用泛函分析学报》2011年第4期349-350,391,共3页Acta Analysis Functionalis Applicata

基　　金：Supported partially by the National Natural Science Foundation of China(11071201)

摘　　要：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C[0,1]上的连续线性泛函,再将它"自然"延拓到C[0,1]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L_1[0,1].This note is devoted to describe the classical Lebesgue integration from a functional point of view. Let C[0, 1] be the linear space of all real-valued continuous functions endowed with the norm ｜｜x｜｜ = f0^1 ｜x（t）｜dt and let X = C[0, 1] be its completion. We define a linear functional xR^＊ on C[0, 1] by {xR^＊, x） = fg x（t）dt in Riemann＇s sense, and let x＊ be the natural extension of xR^＊ from C[0, 1] to its completion C[0, 1]. With a sketch but self-contained proof, we show the Lebesgue integration is just the natural extension of xR^＊ to C[0, 1], that is, C[0, 1] = L1[0, 1] and （x＊, xI =- f0^1 x（t）dt in Lebesgue＇s sence.

关 键 词：LEBESGUE积分 LEBESGUE测度 线性泛函 

分 类 号：O177[理学—数学]