Lectures: MWF 2:00PM-2:50PM at Thack 704
Office hours (subject to change): M,Th 11:00-11:50 or by appointment, held at Thackeray Hall 610.
Content and Prerequisites:
This course is an introduction to Real Analysis and Measure and Integration Theory.
Textbook: Measure and Integral: An Introduction to Real Analysis, by Richard L. Wheeden and Antoni Zygmund is recommened.
Other reading resources: Functional Analysis, Sobolev Spaces and Partial Differential Equations (Universitext) by H. Brezis
Measure Theory and Fine Properties of Functions by C.L. Evans and R.F. Gariepy
Real and Complex Analysis (International Series in Pure and Applied Mathematics) by W. Rudin
Real Analysis by Royden
Functional Analysis by Rudin
A tentative syllabus:
BV functions on real line, Riemann-Stieltjes Integral, Lebesgue measure and measurable functions, Lebesgue Integral, passing to the limit under the integral, Lebesgue integral vs. Riemann integral, Abstract measures, Caratheodory's theorem, Product measures and the Fubini-Tonelli theorem, Signed Measures, Hahn Decomposition, Vitali covering Theorem, Absolutely continuous measures, Radon-Nikodym Theorem, The Egoroff and Luzin theorems, Baire category. If time permits we will also cover Banach Spaces, The Big Theorems of Functional Analysis (Open Mapping Theorem/Closed Graph Theorem, Uniform Boundedness Principle and Hahn-Banach Theorem), Riesz Representation Theorem, Hilbert Spaces.
Grading: Homework (10%), 2 Midterms (25% each), Final Exam (40%).
Homework: They will appear weakly on this website and will be collected on Mondays in class. They will NOT be returned to you so make a copy for yourself before handing in your homework.
Midterm Policy: There will be no make up midterm exams. If you miss the midterm exam for a *documented* serious medical reason or similar tragedy, your grade on it will be prorated on your other exam and homework grade. No Incompletes will be accorded except in extreme circumstances. As for auditing the course, you need to talk with me about it at the latest by the end of September.
Midterm exams: They will be held in class on Friday Oct. 4 and Friday Nov. 8. They will mostly consist of theorems of class and problems of the homework. The cut-off will be announced here.
Final exam: Will be held on the last day of classes during the
undergraduate exam week. The exact date will be announced.