《实变函数》是一门数学系重要的专业基础课程,这是一门以证明为主 的课程。它的主要任务是通过各教学环节,使学生掌握Lebesgue 测度, Lebesgue 积分及收敛性定理,有界变差函数和绝对连续函数的基本性质, 和抽象测度理论基础。本课程不仅对数学专业的学生很重要,对学习应 用数学,概率统计和金融等专业也很重要,为学生进一步学习泛函分析, 偏微分方程,概率统计和随机过程等方向打下坚实的基础。 *课程简介(Description) corecourses abovemajors. allmath majors. followingtopics Lebesguemeasure measurablefunctions: exterior measure, Lebesgue measure measurablesets, measurable functions properties,Egorov theorem Lebesgueintegral, convergence theorems Fubinitheorem: How defineLebesgue integral starting from simple functions, Riemann integral Lebesgueintegral, important convergence theorems, absolute continuity Lebesgueintegrable functions, Fubinitheorem, Lebesgueintegral functions itsproperties. boundedvariations, absolutely continuous functions, averagingproperty integrablefunctions: definitions Abstractmeasure spaces Radon-Nikodynderivatives: definitions severalimportant examples. 课程教学大纲(course syllabus) *学习目标(Learning Outcomes) 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