The Lebesgue-Stieltjes Integral
The Lebesgue-Stieltjes Integral
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DOI：
10.1007/978-1-4612-1174-7_4
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
M. Carter;B. Brunt
中科院分区：
文献类型：
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作者：
M. Carter;B. Brunt
我们现在开始用公式表示我们将要研究的积分的定义。这是两个人的想法结合的结果。法国数学家亨利·勒贝格（Henri Lebesgue，1875-1941）在埃米尔·博雷尔（Emile Borel，1871-1956）关于集合测度的早期工作的基础上，成功地定义了一种积分（勒贝格积分），它适用于比黎曼积分更广泛的函数类，并且收敛定理要简单得多。荷兰数学家托马斯Stieltjes（1856年至1894年）负责的概念，整合一个功能方面的另一个功能。他的思想最初是作为黎曼积分的扩展而发展起来的，被称为黎曼-斯蒂尔杰斯积分。随后结合他的想法与措施理论的方法勒贝格导致了一个非常强大和灵活的概念整合。
We now proceed to formulate the definition of the integral that we are going to study. It results from combining the ideas of two people. The French mathematician Henri Lebesgue (1875-1941), building on earlier work by Emile Borel (1871-1956) on the measure of a set, succeeded in defining an integral (the Lebesgue integral) that applied to a wider class of functions than did the Riemann integral, and for which the convergence theorems were much simpler. The Dutch mathematician Thomas Stieltjes (1856-1894) was responsible for the notion of integrating one function with respect to another function. His ideas were originally developed as an extension of the Riemann integral, known as the Riemann-Stieltjes integral. The subsequent combination of his ideas with the measure-theoretic approach of Lebesgue has resulted in a very powerful and flexible concept of integration.