This set is due Friday, April 27.
The goal of these problems is to prove Carathéodory‘s theorem that “extracts” a measure from any outer measure. In particular, when applied to Lebesgue outer measure, this construction recovers Lebesgue measure.
This set is due Feb. 29 at the beginning of lecture. Let me know if more time is needed or anything like that. Problem 4 was incorrect as stated; I have fixed it now. Thanks to Tara Sheehan for bringing the problem to my attention.
Suppose that and has an antiderivative . Then
This set is due Feb. 8 at the beginning of lecture. Of course, let me know if more time is needed or anything like that.
Syllabus for Math 515: Advanced calculus AKA Analysis II.
Instructor: Andrés E. Caicedo.
Contact Information: See here.
Time: MWF 9:40-10:30 am.
Place: MG 124.
Office Hours: MF 11-12.
Mathematicians find it easier to understand and enjoy ideas which are clever rather than subtle. Measure theory is subtle rather than clever and so requires hard work to master.
Thomas W. Körner, Fourier Analysis, p. 572.
Contents: From the Course Description on the Department’s site:
Introduction to the fundamental elements of real analysis and a foundation for writing graduate level proofs. Topics may include: Banach spaces, Lebesgue measure and integration, metric and topological spaces.
We will emphasize measure theory, paying particular attention to the Lebesgue integral. Additional topics, depending on time, may include the Banach-Tarski paradox, and an introduction to Functional Analysis.
Grading: Based on homework. No late homework is allowed. Collaboration is encouraged, although you must turn in your own version of the solutions, and give credit to books/websites/… you consulted and people you talked/emailed/… to.
There will be no exams in this course. However, an important component of being proficient in mathematics is a certain amount of mental agility in recalling notions and basic arguments. I plan to assess these by requesting oral presentations of solutions to some of the homework problems throughout the term.
I will use this website to post additional information, and encourage you to use the comments feature. If you leave a comment, please use your full name, which will simplify my life filtering spam out.