The following is a tentative syllabus for the course. This page will be regularly updated as we go forward.
| Week | Lectures | Topics | Remarks |
|---|---|---|---|
| 1 | 9/13 | Definition of a metric space and examples | |
| 9/15 | Metric topology | ||
| 2 | 9/20 | Convergence and completeness with examples | |
| 9/22 | Completing a metric space | ||
| 3 | 9/27 | Compact metric spaces and uniform convergence | |
| 9/29 | Arzelà - Ascoli Theorem | ||
| 4 | 10/4 | Baire Category Theorem | Time permitting |
| 10/6 | Review of the Riemann Integral and its shortcomings. $\sigma$-algebras and Borel sets | ||
| 5 | 10/11 | Measurable functions | |
| 10/13 | Measures | ||
| 6 | 10/18 | Integration | |
| 10/20 | Dominated Convergence Theorem | ||
| 7 | 10/25 | Outer measures and Lebesgue measure on the real line | |
| 10/27 | Extension of premeasures | ||
| 8 | 11/1 | Product measures and integration on products | |
| 11/3 | $L^1$-spaces and Fubuini's Theorem | ||
| 9 | 11/8 | Signed measures, the Hahn decomposition and the Jordan decomposition | |
| 11/10 | Modes of convergence | ||
| 10 | 11/15 | The Radon-Nikodym Theorem | |
| 11/17 | Riesz Representation Theorem | Alternative: Haar measure |