| Class | Date | Sections | Brief Description |
|---|---|---|---|
| 1 | Thursday, 8/24 | - | Introduction, Overview, Analysis Review, Metric Spaces |
| 2 | Tuesday, 8/29 | Notes: 1.1 - 1.3 | Metric Spaces, Interior and Closure Points, Open and Closed Sets |
| 3 | Thursday, 8/31 | Notes: 1.4 - 1.5 | Compact Sets, Images and Preimages |
| 4 | Tuesday, 9/5 | Notes: 1.5 Nelson: 0.1 - 0.3 |
Continuous Functions Between Metric Spaces, Review of the Riemann Integral |
| 5 | Thursday, 9/7 | Nelson: 1.1 SteinShak: 1.1 - 1.2 |
Lebesgue Outer Measure, Basic Properties |
| 6 | Tuesday, 9/12 | Nelson: 1.1 - 1.2 SteinShak: 1.2 |
Outer Measure of Intervals, Additivity on Disjoint Compact Sets |
| 7 | Thursday, 9/14 | Nelson: 1.2 SteinShak: 1.3 |
Measurable Sets |
| 8 | Tuesday, 9/19 | Nelson: 1.2 - 1.3 SteinShak: 1.3 |
Properties of Measurable Sets, Sigma Algebras |
| 9 | Thursday, 9/21 | Nelson: 1.3 SteinShak: 1.3 |
Nonmeasurable Sets, Baire Category Theorem |
| 10 | Tuesday, 9/26 | Nelson: 2.1 SteinShak: 1.4 |
Riemann Integration and Points of Continuity, Measurable Functions |
| 11 | Thursday, 9/28 | Nelson: 2.1 SteinShak: 1.4 |
Borel Sets, Measurable Functions, Simple Functions |
| 12 | Tuesday, 10/3 | Nelson: 2.2 | The Lebesgue Integral of a Bounded Function on an Interval |
| 13 | Thursday, 10/5 | Nelson: 2.2 | Integrability and Measurability, Bounded Convergence Theorem |
| 14 | Tuesday, 10/10 | Nelson: 2.3 | The Lebesgue Integral of a Nonnegative Function, Fatou's Lemma |
| 15 | Thursday, 10/12 | Nelson: 2.3 - 2.4 | Monotone Convergence Theorem, Lebesgue Dominated Convergence Theorem |
| - | - | - | Fall Break |
| 16 | Tuesday, 10/24 | Nelson: 4.1, 4.4 SteinShak: 6.1 |
Measure Spaces, Outer Measures |
| 17 | Thursday, 10/26 | Nelson: 4.4 SteinShak: 6.1, 7.1 |
Measures from Outer Measures, Hausdorff Measures |
| 18 | Tuesday, 10/31 | Nelson: 4.2 - 4.3 SteinShak: 6.2 |
Integration on a General Measure Space |
| 19 | Thursday, 11/2 | Nelson: 3.1 SteinShak: 2.2 |
The Space L^1[a,b] |
| 20 | Tuesday, 11/7 | Nelson: 3.1 SteinShak: 2.2 |
Completeness of L^1[a,b] |
| 21 | Thursday, 11/9 | Nelson: 3.4 SteinShak: 4.2 |
Inner Products, Orthogonality, Projections |
| 22 | Tuesday, 11/14 | Nelson: 3.4 SteinShak: 4.2 |
Cauchy-Schwarz Inequality, Hilbert Spaces |
| 23 | Thursday, 11/16 | SteinShak: 4.2, 4.4 | Projections in Hilbert Spaces, Bases, Fourier Series |
| 24 | Tuesday, 11/21 | - | Group Project Time |
| - | - | - | Thanksgiving Break |
| 25 | Tuesday, 11/28 | - | Measure-Preserving Transformations, Poincare Recurrence |
| 26 | Thursday, 11/30 | - | Ergodic Transformations, Birkhoff Ergodic Theorem |
| 27 | Tuesday, 12/5 | - | Project Presentations |
| 28 | Thursday, 12/7 | - | Project Presentations |