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. 源`自*六)维[论*文'网www.lwfree.cn
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