Syllabus for MATH01 511 – Real Analysis II

Dr. Abdul Hassen

E-mail          Hassen@rowan.edu               Phone         (856)256 4500 ext 3888

 

CATALOG DESCRIPTION:         This course is a continuation of Real Analysis I and covers Riemann-Stieltjes integration; elements of measure theory and Lebesgue integration.

OBJECTIVES:         The purpose is to provide students an opportunity to study the drawbacks of the Riemann-Stieltjes integral and introduce them to measure theory and the Lebesgue integral.

PREREQUISITE:    Real Analysis I (Math 01 510)

TEXT:                        Wilcox H. and Myers, D., AN INTRODUCTION TO LEBESGUE INTEEGRATION AND FOURIER SERIES,

Dover Publishing Company, NYC, 1994.

REFERENCE:         1)         Goldberg, Richard, METHODS OF REAL ANALYSIS, 2^nd edition, John Wiley & Sons, 1976

                                    2)         The text used for Real Analysis I (Math01 510)

CONTENT:   We will cover the following topics from the text. (Lecture notes are available at WebCT)

0.                  Preliminaries

0.1              Review of Real Analysis I  Part I(Sequences and Series, Limit and Continuity)

0.2              Review of Real Analysis I Part II (Integration and Derivative)

1.                  The Riemann  Integral

1.1              Definition and properties

1.2       Drawbacks of the Riemann Integral

 

2.         Measurable Sets

2.1       The outer measure and measurable sets

2.2       Properties of measurable sets such as countable additivity

2.3       Borel sets and the Cantor set.

2.4       Lebesgue measure for bounded and unbounded sets            

 

3.         Measurable Functions

3.1       Definition of measurable functions

3.2       Preservation of measurability for functions

3.3       Simple functions

4.         The Lebesgue integral

4.1       The Lebesgue Integral for bounded measurable functions

4.2       Simple functions

4.3       Integrability of bounded measurable functions

4.4       Elementary properties of the integral

4.5       The Lebesgue Integral for unbounded functions

 

5.         Convergence and the Lebesgue Integral

5.1       Convergence theorems

5.2       A necessary and sufficient condition for Riemann Integrability

5.3       Ergoff’s and Lusin’s theorems

 

GRADING POLICY:    Students will be graded based on mid term and final exams (60% of the total grade), projects (20% of total grade), and assignments (20% of the total grade). The dates for the tests will be announced in class at least a week in advance.

Numerical grades will be converted to letter grades by the following scale.

A (-)= 90 to 100, B(-,+)= 80 to 89, C(-,+)= 70 to 79, D(-,+)= 60 to 69, F= 0 to 59

HOMEWORK      These will be given in class and the due dates will be announced. I suggest that you get a separate notebook for homework assignments