| Week |
Topics |
Homework |
| 1: Feb 16-20 |
Review of real numbers, continuous functions, sequence of functions |
8, 11, 13, 18 p. 38 |
| 2: Feb 23-27 |
Elementary Set Theory : Open , Closed, Sigma algebra, Borel sets |
27, 28, 36 p 46
43, 47 p 50 |
| 3: Mar 01-05 and |
Lebesgue Measure : Outer measure, Measurable sets, Measurable functions |
|
| 4: Mar 7-12 |
Almost everywhere notion, Egoroff's theorem , Riemann Integrals : Darboux sums |
5,8 p 58
10, 11, 12 p 64 |
| 5: Mar 15-19 |
Riemann Characterization theorem , Integrable functions, defects of Riemann integration |
|
| 6: Mar 22-26 |
The Lebesgue Integrals : Integral of bounded functions, Bdd convergence theorem,
Integral of nonnegative function |
|
| 7: March 29-April 02 |
Fatou's lemma, MCT General Lebesgue Integration, Dominated conv. Theorem |
3, 6, 7, 8 p 89
10, 13, 14, 16 p93 |
| 8: April 05-09 |
Conergence in measure, Relations between different types of convergence |
|
| 9: April 19-23 |
Differentiation : Monotone functions, Bounded Variation functions, |
|
| 10: April 26-30 |
Bounded Variation functions,
Total variation Absolute continuity |
9, 10 p 104
11, 12, 20 p 111. |
| 11: May 03-07 |
The L p Spaces : Review of functional analysis, Definitions, Norms |
|
| 12: May 10-14 |
Completeness, Minkowski's inequality, Holder's inequality, Approximation and density |
|
| 13: May 17-21 |
Bounded linear functionals, Riesz Representation theorem |
8 p 123
10, 16, 17 p 127 |
| 14: May 24-28 |
The Abstract Measure and Integration : Measurable spaces, measure spaces |
|
| 15: June 31-04 |
Measurable functions, Integration, Signed measures, The Radon-Nikodym theorem |
TBA |