Real Analysis: Readings and Homework


Math 531 -
072
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