Radon测度与积分+文献综述
毕业论文关键词 抽象测度与积分 局部紧Hausdorff空间 Riesz表示定理 Radon积分
毕业设计说明书外文摘要
Title Radon Measure and Integration
Abstract
In my undergraduate thesis, based on the measure theory,firstly define the integration of the generalized nonnegative measurable function on the basis of the integration of the nonnegative measurable simple function ,then the integration of the real measurable function can be defined as the difference between the integrations of its positive and negative parts.Finally,the the integration of complex measurable function can easily be denoted by the integrations of its real and imaginary part.Next,prove the Riesz representation theorem on ( is the locally compact Hausdorff space),and define the positive linear functional as the Radon positive integration and the measure derived from the positive linear functional as Radon positive measure.When the Radon positive integration on is Riemann integration,the corresponding Radon positive measure is the familiar Lebesgue measure. As application,show that the Radon positive integration on indeed is the Stieltjes integration produced by increasing function.In the end,extend the Radon positive integration on to the Radon integration,and prove that the Radon integration on is equivalent to the Stieltjes integration produced by the bounded variation of function.
Keywords Abstract Measure and Integration Locally Compact Hausdorff Space Riesz Representation Theorem Radon Integration
目 次
1 引言 1
2 预备知识 2
2.1 度量空间 2
2.2 拓扑空间 3
2.3 5
3 测度理论 7
3.1 可测函数 7
3.2 测度空间 8
3. 3 积分 11
3. 4 可测函数的连续性 16
4 Riesz表示定理 18
4. 1 基本知识 18
4. 2 Riesz表示定理 18
4. 3 上Lebesgue测度 23
5 Radon积分理论 26
5. 1 Radon正积分 27
5. 2 Radon积分 27
结论 30
致谢 31
参考文献32
1 引言
测度与积分是对称的，既可由测度定义积分，也可由积分定义测度。但测度概念的出现要比积分概念晚的多。积分源于Newton和Leibniz,其中Leibniz给出积分独立于微分并作为Riemann和的思想，并由几代数学家初步完善。其后积分概念从两个方向得到扩充：第一次扩充的结果是Stieltjes积分，推广了Riemann-Darboux的积分概念；沿完全不同的另一条思想路线扩充的结果是Lebesgue测度与积分，继承自Jordan、Borel等人的测度理论。与之前的积分在处理级数等方面相比，Lebesgue积分显示出极大的优越性。最后Radon积分统一了Stieltjes积分和Lebesgue积分，实际上它被称为Lebesgue-Stieltjes积分[1]。这一推广不仅使积分范围更广，或者统一了 中点集上的不同积分概念，而且还扩展至函数空间 （其中 为局部紧Hausdorff空间）。
上一篇：电信套餐优惠定价问题研究
下一篇：直纹面及其应用+文献综述