Real analysis 2

Math 482   Real analysis 2   (3Hours credit)

Course Format: Two Hours of lecture and one hour of exercises per week, and additional 5 Hours of Discussion/self-study and office hours, at discretion of the instructor.

Prerequisite: Math381

Download Specification Course MATH 482

Download Report Course MATH 482

Description: Definition of Riemann integral- Darbo theorem and Riemann sums - Properties and the principle theorem in calculus. Sequence Series of functions- Poi twice convergence and uniform convergence- Algebra and (sigma algebra)- Finite additive and countable additive- Main extension theorem and outer measure- Measurable sets - Measure - Lebesgue measure and its properties- Simple functions- Measurable functions- Lebesgue integral- Theorems of convergence- The relation between Lebesgue and Riemann integral.

Objective: Study of main concepts of Real analysis as follows:

1. Studying Definition of Riemann integral- Darboux theorem and Riemann sums and operations on them.

2. Studying the Properties and the principle theorem in calculus. Series of functions and their properties.

3. Solving system of Pointwice convergence and uniform convergence- Algebra and algebra (sigma algebra).

4. Have the knowledge of the Finite additivity and countable additivity- Main extension theorem and outer measure.

5. Have the knowledge of Measure - Lebesgue measure and its properties- Simple functions.

6. Have the knowledge of Theorems of convergence and their properties.

7. Studying determinants and operations on relation between Lebesgue and Riemann integral.

Outcomes:

1. Define the fundamental in linear real analysis as:

Definition of Riemann integral- Darboux theorem and Riemann sums - Properties and the principle theorem in calculus. Series of functions- Pointwice convergence and uniform convergence- Algebra and – algebra (sigma algebra)- Finite additivity and countable additivity- Main extension theorem and outer measure- Measurable sets - Measure - Lebesgue measure and its properties- Simple  functions- Measurable functions- Lebesgue integral- Theorems of convergence- The relation between Lebesgue and Riemann integral.

2. State the physical problems by mathematical method

3. Enable students to analyses the mathematical problems

4. The student should illustrate how take up responsibility

References:

1.      Introduction to Real Analysis. William F. Trench. Hyperlinked Pearson Education: 2012-0-13-045786-8.

2.      Real Analysis (4th Edition) H. Royden, P. Fitzpatrick Macmillan Publishing Co., Inc. New York: 2010 10:01314374x-13: 978-0131437470