The Course is one of the core courses for the above majors.The following topic ered in thisou measure and measurable functions:exterior measure,Lebesgu measure and properties of measurable sets,measurable functions and thei properties,Egorov theorem. 2)Lebesgue integral,convergence theorems and Fubini theorem:How to define Lebesgue integral starting from simple functions,Riemann integral and Lebesgue integr ab 课程简介（英Lebesgue integrable functions,the proof and applications of Fubini theorem 文) the space of Lebesgue integral functions and its properties. (Description)3)Differentiability of functions of bounded variations,absolutely continuous functions:definitions and properties. 4)Abstract measure spaces and Radon-Nikodyn derivatives:definitions and several important examples. This course is an important prerequisite not only for math major,but also field such as theoretical foundation of machine learning,financial engineering, tochastic. 课程目标与内容(Course objectives and contents) 通过这门课程的学习，学生要掌握如下的知识点，掌握测度空间及其上的积分 理论。这是数学，概率统计，经济学，金融学和数据科学的重要理论基础和描 述工具。 通过这门课程的学习，学生要学会如何做严格的证明推理，学习认真求实，慎 密推理的思维习惯和工作态度。 ()Lebesgue测度和可测函数：外测度，Lebesgue测度的定义及Lebesgue可测 集的基本性质，可测函数的定义和基本性质，Egorov定理，可测函数与连续函 “课程目标 数的关系。(A3,A4,B1,B2,C1,C3) (Course Object)(2)Lebesgue积分及收敛性定理，Fubini定理：如何从简单函数出发定义Lebesgue 积分，Lebesgue积分与Riemann积分的关系，几个重要的积分收敛性定理 Lebesgue可积函数的绝对连续性，Fubini定理的证明与应用.(A3,A4,B1,B2,C1,C3) 3)有界变差函数的定义及主要性质，绝对连续函数的定义及性质。 A3,A4,B1,B2,C1,C3) (4)抽象测度空间的定义，及其上的可测函数，积分的定义，Radon--Nikodyn号 数：定义和几个重要的例子。(A3,A4,B1,B2,B3,C1,C3,C5) *课程简介（英 文） （Description） The Course is one of the core courses for the above majors. The following topics are covered in this course: 1) Lebesgue measure and measurable functions: exterior measure, Lebesgue measure and properties of measurable sets, measurable functions and their properties, Egorov theorem . 2) Lebesgue integral, convergence theorems and Fubini theorem: How to define Lebesgue integral starting from simple functions, Riemann integral and Lebesgue integral, important convergence theorems, absolute continuity of Lebesgue integrable functions, the proof and applications of Fubini theorem, the space of Lebesgue integral functions and its properties. 3) Differentiability of functions of bounded variations, absolutely continuous functions: definitions and properties. 4) Abstract measure spaces and Radon-Nikodyn derivatives: definitions and several important examples. This course is an important prerequisite not only for math major, but also fields such as theoretical foundation of machine learning, financial engineering, stochastic control etc. 课程目标与内容（Course objectives and contents） *课程目标 (Course Object) 通过这门课程的学习，学生要掌握如下的知识点，掌握测度空间及其上的积分 理论。这是数学，概率统计，经济学，金融学和数据科学的重要理论基础和描 述工具。 通过这门课程的学习，学生要学会如何做严格的证明推理，学习认真求实，慎 密推理的思维习惯和工作态度。 (1) Lebesgue 测度和可测函数：外测度，Lebesgue 测度的定义及 Lebesgue 可测 集的基本性质，可测函数的定义和基本性质，Egorov 定理，可测函数与连续函 数的关系。(A3,A4,B1,B2,C1,C3) (2) Lebesgue 积分及收敛性定理，Fubini 定理：如何从简单函数出发定义 Lebesgue 积分，Lebesgue 积分与 Riemann 积分的关系，几个重要的积分收敛性定理， Lebesgue 可积函数的绝对连续性，Fubini 定理的证明与应用。(A3，A4,B1,B2,C1,C3) (3) 有 界 变 差 函 数 的 定 义 及 主 要 性 质 ， 绝 对 连 续 函 数 的 定 义 及 性 质 。 (A3,A4,B1,B2,C1,C3) (4) 抽象测度空间的定义，及其上的可测函数，积分的定义，Radon-Nikodyn 导 数：定义和几个重要的例子。(A3,A4,B1,B2,B3,C1,C3,C5)