| Class Number |
Date |
Reading |
Brief Description |
| 1 |
Tuesday, 1/24 |
Chapter 1, Appendix A |
Introduction, Overview, Analysis Review |
| 2 |
Thursday, 1/26 |
Appendix B |
Base b Representations, Metric Spaces, Open, Closed, and Compact Sets |
| 3 |
Tuesday, 1/31 |
2.1 |
Open Subsets of R, Lebesgue Outer Measure |
| 4 |
Thursday, 2/2 |
2.2 - 2.3 |
The Cantor Set, Properties of Outer Measure |
| 5 |
Tuesday, 2/7 |
2.3 |
Measurable Sets |
| 6 |
Thursday, 2/9 |
2.4 |
Properties of Measurable Sets, Nonmeasurable Sets |
| 7 |
Tuesday, 2/14 |
2.5 - 2.6 |
Sigma Algebras, Measure Spaces, Borel Sets |
| 8 |
Thursday, 2/16 |
2.6 |
Borel Sets, Baire Category Theorem on R |
| 9 |
Tuesday, 2/21 |
2.7 - 2.8 |
Sufficient Semi-Rings, Caratheodory's Measurability Condition |
| 10 |
Thursday, 2/23 |
3.1 - 3.2 |
Ergodic and Mixing Notions, Measurable Functions,
Measure-Preserving Transformations |
| 11 |
Tuesday, 2/28 |
3.2 - 3.3 |
Rotation Transformations, The Doubling Map, Baire Category Theorem |
| 12 |
Thursday, 3/2 |
3.3 - 3.5 |
Topologically Transitive Maps, Sufficient Semi-Rings and
Measure-Preserving Transformations, Recurrence |
| 13 |
Tuesday, 3/7 |
3.5 - 3.6 |
Poincare Recurrence, Invariant Sets |
| 14 |
Thursday, 3/9 |
3.6 - 3.7 |
Invariant Sets, Ergodic Transformations |
| 15 |
Tuesday, 3/14 |
3.7, 3.12 |
Examples of Ergodic Transformations, Symbolic Spaces |
| 16 |
Thursday, 3/16 |
4.1 |
The Riemann Integral, Lebesgue's Characterization |
| - |
- |
- |
Spring Break |
| 17 |
Tuesday, 4/4 |
4.2 |
Measurable Functions |
| 18 |
Thursday, 4/6 |
4.3 |
Simple Functions, The Lebesgue Integral of a Bounded Function with
Finite Support |
| 19 |
Tuesday, 4/11 |
- |
Properties of the Lebesgue Integral, Bounded Convergence Theorem |
| 20 |
Thursday, 4/13 |
4.4 |
The Lebesgue Integral of a Nonnegative Function, Fatou's Lemma,
Monotone Convergence Theorem |
| 21 |
Tuesday, 4/18 |
4.6 - 4.7 |
Lebesgue Integrable Functions, Lebesgue Dominated Convergence
Theorem, Norms on Vector Spaces |
| 22 |
Thursday, 4/20 |
4.2, 4.7 |
Banach Spaces, L^p Spaces, Measurable and Ergodic Transformations Revisisted |
| 23 |
Tuesday, 4/25 |
5.1 |
Birkhoff Ergodic Theorem |
| 24 |
Thursday, 4/27 |
5.1 - 5.3 |
Consequences of the Birkhoff Ergodic Theorem |
| 25 |
Tuesday, 5/2 |
- |
Inner Products, Orthogonality, Projections |
| 26 |
Thursday, 5/4 |
5.4 |
Hilbert Spaces |
| 27 |
Tuesday, 5/9 |
5.4, 6.1 |
Mean Ergodic Theorem, Cesaro Mean, Mixing |
| 28 |
Thursday, 5/11 |
- |
Random Sequences |