| Lebesgue积分的新定义 |
| |
| 引用本文： | 曹怀信.Lebesgue积分的新定义[J].安康学院学报,2010,22(6):1-4. |
| |
| 作者姓名： | 曹怀信 |
| |
| 作者单位： | 陕西师范大学数学与信息科学学院; |
| |
| 摘 要： | 对于可测集ERn上的非负可测函数f,证明了f的下方图形G（f,E）是Rn＋1中的Lebesgue可测集;进而,定义f的Lebesgue积分为G（f,E）的Lebesgue测度mG（f,E）;对于E上的一般可测函数f,定义其在E上的Lebesgue积分为mG（f＋,E）-mG（f-,E）,只要它们之一有限。利用测度的性质,证明了这种新的定义与传统定义是等价的。这种新定义使得Lebesgue积分具有非常明显的几何意义,且使得Levi渐升列定理及关于积分域的可数可加性定理等重要结论都成为测度与极限换序定理的简单推论。
|
| 关 键 词： | 可测函数 下方图形 测度 Lebesgue积分 |
A New Definition of Lebesgue Integral |
| |
| CAO Huaixin.A New Definition of Lebesgue Integral[J].Journal of Ankang Teachers College,2010,22(6):1-4. |
| |
| Authors: | CAO Huaixin |
| |
| Institution: | CAO Huaixin(College of Mathematics and Information Science,Shaanxi Normal University,Xi'an 710062,Shaanxi,China) |
| |
| Abstract: | For a nonnegative measurable function f on a Lebesgue measurable set ERn,it is proved that the lower-graph G(f,E) of f is a measurable set in Rn+1 and then the Lebesgue integral of f on E is defined as the Lebsegue measure mG(f,E)of G(f,E).For a measurable function f on E,the Lebesgue integral of f on E is defined as mG(f+,E)-mG(f-,E) whenever at least one of them is finite.By using the properties of Lebesgue measure,it is proved that this new definition is equivalent to the usual one.This new definiton has an explicit geometric illustration and makes the Levi’s theorem and countable additivity theorem hold automatically. |
| |
| Keywords: | measurable function lower graph measure Lebesgue integral |
| 本文献已被 CNKI 维普 万方数据 等数据库收录！ |
|
