Department of Mathematics

Syllabus

**Math 01.511 – Real Analysis II**

**CATALOG DESCRIPTION:**

**Math 01.511 Real Analysis II
3 s.h.**

(Prerequisite Math 01.510)

This course is a continuation of Real Analysis I and covers Riemann-Stieltjes integration; elements of measure theory and Lebesgue integration. This course may not be offered annually.

**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.

**CONTENT:**

**1. The Riemann-Stieltjes Integral**

1.1 Definition and properties

1.2 Drawbacks of the Riemann-Stieltjes 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

**TEXTS:**

The following are indicative of texts suitable for this course:

1) Wilcox H. and Myers, D., AN INTRODUCTION TO LEBESGUE INTEGRATION
AND FOURIER SERIES,

Dover Publishing Company, NYC, 1994.

2) Goldberg, Richard, METHODS OF REAL ANALYSIS, 2^{nd }edition,
John Wiley & Sons, 1976