2016 Fall: Advanced Real Analysis
English Version|中文版
Basic Information:- Instructor: Professor Hao Yin, haoyin@ustc.edu.cn
- Time: 19:30-21:05, each Tuesday and Thursday.
- Classroom:@5201, East Campus.
- Textbook:
- Lawrence C. Evans, Ronald F. Gariepy: Measure Theory and Fine Properties of Functions, Revised Version, CRC Press, 2015. Chapter 1-4, 5.1-5.3.
- Lecture Notes(in Chinese): Link (Ch 1-3, 4.7-4.9)
- Grade: 70%Final Exam+30%min{100, Final Exam+Homework*3.33333}.
Final Exam:
- Time/Location: Jan. 11 th, 2017. 14:30-16:30, @5203, East Campus.
- Previous Final Exam: Link (2015-2016 Fall). This year the textbook is Stein's Real Analysis(Ch 6) and Functional Analysis (Ch 1-3)
- 2016 Final Exam: Link (2016-2017 Fall)
- Please contact with the instructor if you want to know more about your final grade.
- Problem 1:
- Due: Oct. 15th.
Prove that every monotone continuous function on R has finite derivative a.e. by using the theory of Radon measure.
- Solution: Click here.
- Problem 2:
- Due: Nov. 15th.
- Description: When we were proving Riesz Representation Theorem in the class, we first proved each non-negative linear functional on C_c(R^n;R) owns a Radon measure and satisfies the integral representation. Then we decomposed a general linear functional into its positive and negative part and 2 radon measures μ^+,μ^- are respectively corresponded to each part.
Prove or disprove μ^+ and μ^- are mutually singular.
- Solution: Click here
- Problem 3:
- Due: Dec. 20th.
- Description: We define s-dimensional spherical Hausdorff measure S^s by restricting the covering to be closed ball covering and define the corresponding upper, lower density similarly to Hausdorff measure.
Prove or disprove: For each subset E of R^n which is S^s-measurable and with finite S^s measure, its upper density equal to 1 for a.e. x in E.
More precise description is here.
- Solution: Click here