REAL ANALYSIS

MA 380

Catalogue Description

This is a study of sets, mappings, sequences, connected, open and closed sets, continuity, uniform convergence, and metric spaces. This course offers an introduction to measure theory.

3 credits Prerequisite: MATH 295 Survey of Modern Math and MATH 191 Calculus 2 or equivalent

Goals

A. To serve as an introduction to mathematical analysis.

B. To prepare the student for possible graduate work in math in the future.

C. To introduce measure theory and the disadvantages of the Riemann Integral used in Calculus II.

Procedures

A. Lecture/Discussion

B. Readings and problems (mainly proofs) assigned.

Course Content

1. Set Theory

A. Sets and Functions

1. Basic Definitions

2. Operations on sets

3. Functions and mappings, images and pre-images

4. Decomposition of a set into classes. Equivalence Relations.

B. Equivalence of Sets. The Power of a Set.

1. Finite and infinite sets

2. Countable sets

3. Equivalence of sets

4. Uncountability of the real numbers

5. Power of a set

6. Cantor-Bernstein Theorem

C. Systems of Sets

1. Rings of sets

2. Semirings of sets

3. Ring generated by a semiring

4. Sigma-rings of sets

5. Algebra of sets

6. Sigma-Algebra of sets (Borel-algebra)

D. Partial Orderings

E. The Axiom of Choice

2. Metric Spaces

A. Basic Concepts

1. Definitions and examples

2. Continuous mappings and homomorphisms

3. Isometric spaces

B. Convergence. Open and Closed Sets.

1. Closure of a set. Limit points.

2. Convergence, Limits, Uniform Continuity.

3. Dense subsets. Separable spaces.

4. Closed sets.

5. Open sets.

6. Open and closed sets on the real line (example: The Cantor Set).

C. Complete Metric Spaces

3. Measure Theory

1. Measure on rings

2. Measure on intervals

3. Properties of measures

4. Outer measures

5. Measurable sets

6. Disadvantages of Riemann Integral in Calculus

7. Extensions of Measures, Lebesgue measure

8. Measurable Functions

9. Lebesgue Integral

Evaluation methods

1. Three (3) exams. There may be a take home exam due to the nature of the material.

2. Final -- 2 hours. Comprehensive exam covering the term.

3. Three (3) exams -------- 25% Each

Final exam ------------- 25% (50% if 1 of the 3 exams is dropped)

Bibliography

Required Text: Lewin, J., An Introduction to Math Analysis, 2nd Ed., McGraw Hill Publishing, 1993.

Cuppillari, W., The Nuts & Bolts of Proofs, Wadsworth Publishing Co., 1988.

Depree, John D. & Swartz, Charles W., Introduction to Real Analysis, John Wiley & Sons, 1988.

Halmos, Paul R., Measure Theory, New York, N.Y., Springer-Verlag, 1974.

Kingman, J.F. & Taylor, S.J., Introduction to Measure and Probability, New York, N.Y., Cambridge University Press, 1977.

Lange, Serge, Real Analysis, 2nd Ed., Addison-Wesley, Reading, Ma., 1983.

Royden, H.L., Real Analysis, 3rd Ed., MacMillan Publishing Co., New York, N.Y., 1988.

Rudin, W., Real and Complex Analysis, 3rd Ed., McGraw Hill Publishing, 1987.

Spiegel, Murray R., Schaum's Outline Series: Theory & Problems of Real Variables, McGraw Hill Publishing, 1969.

Torchinsky, Alberto, Real Variables, Addison-Wesley, 1988.