18.125 Measure and Integration |
Introduction to Lesbesgue Integration Theory, and Elements of Functional Analysis
This is the core graduate course in measure theory and integration, focusing largely on Euclidean spaces. It is a prerequisite for 18.175 (the core graduate course in probability theory), which will be taught in Spring 2009 by Dr. Kemp). Undergraduate math majors interested in this material may wish to consider 18.103 instead, which focuses more on the real line and Fourier integrals. 18.103 will also be considered a prerequisite for 18.175 in Spring 2009. We will follow the course textbook "A Concise Introduction to the Theory of Integration" by D. Stroock, as well as recommended texts "Real and Complex Analysis" by W. Rudin and "Real Analysis and Probability" by R. Ash. Topics include: Riemann-Stieltjes integration, measures, the Caratheodory extension theorem, Lebesgue measure on Euclidean space, abstract integration theory, change of variables and the Radon-Nikodym theorem, introduction to Hilbert spaces and Banach spaces (concentrating on L^p spaces), and (time-permitting) elements of Fourier analysis. There will be 10 homework sets. They will be available below, and due on the dates specified. There will be a midterm exam on Thursday, April 3, in class. There will also be a final exam on Thursday, May 22, 1:30pm - 4:30pm in 4-153. This is the core graduate course in measure theory and integration, focusing largely on Euclidean spaces. It is a prerequisite for 18.175 (the core graduate course in probability theory), which will be taught in Spring 2009 by Dr. Kemp). Undergraduate math majors interested in this material may wish to consider 18.103 instead, which focuses more on the real line and Fourier integrals. 18.103 will also be considered a prerequisite for 18.175 in Spring 2009. We will follow the course textbook "A Concise Introduction to the Theory of Integration" by D. Stroock, as well as recommended texts "Real and Complex Analysis" by W. Rudin and "Real Analysis and Probability" by R. Ash. Topics include: Riemann-Stieltjes integration, measures, the Caratheodory extension theorem, Lebesgue measure on Euclidean space, abstract integration theory, change of variables and the Radon-Nikodym theorem, introduction to Hilbert spaces and Banach spaces (concentrating on L^p spaces), and (time-permitting) elements of Fourier analysis. There will be 10 homework sets. They will be available below, and due on the dates specified. There will be a midterm exam on Thursday, April 3, in class. There will also be a final exam on Thursday, May 22, 1:30pm - 4:30pm in 4-153. Homework 1: due on Friday, February 15, by 4:00pm. Homework 1. Solution. Homework 2: due on Friday, February 22, by 4:00pm. Homework 2. Solution. Homework 3: due on Friday, February 29, by 4:00pm. Homework 3. Solution. Homework 4: due on Friday, March 7, by 4:00pm. Homework 4. Solution. Homework 5: due on Friday, March 14, by 4:00pm. Homework 5. Solution. Homework 6: due on Friday, March 21, by 4:00pm. Homework 6. Solution. Midterm: due on Tuesday, April 8, by in lecture. Midterm. Solution. Homework 7: due on Wednesday, April 23, by 4:00pm. Homework 7. Solution. Homework 8: due on Friday, May 2, by 4:00pm. Homework 8. Homework 9: due on Friday, May 16, by 4:00pm. Homework 9. |