I. Number and title of course

MAT 418 - Introduction to Real Analysis II

A. The contents of this course are prerequisites to graduate studies in mathematical analysis.

B. A continuation of MAT 417, this course covers topics needed for advanced work in applied mathematics, probability theory and advanced statistics.

A. An introduction to important definitions and theorems in real function theory which are not seen in the usual calculus courses.

B. A thorough development of integral calculus and its generalization to Lebesgue and Stieltjes integration.

A. Partial Differentiation

1. Implicit functions

2. Taylor's Theorem

3. Jacobians

B. Multiple Integrals

1. Inner and outer area

2. Fubini's Theorem

3. Transformation of multiple integrals

C. Measurable sets

1. Measure of bounded sets

2. Inner and outer measure

3. Vitale's Theorem

D. Measurable functions

1. Properties of measurable functions

2. Sequences of measurable functions

3. Theorems of Weierstrass

E. Lebesgue Integration

1. Definition of Lebesgue

2. Fundamental properties of Lebesgue integral

3. Comparison of Lebesgue and Riemann integral

F. The Stieltjes Integral

1. Functions of finite variation

2. Definition of Stieltjes integral

3. Passage to limit under the Stieltjes integral sign

Bartle, G.& Sherbert, R., Introduction to Real Analysis, John Wiley & Sons, New York, NY, 1990.

Bilodean, G. & Thie, R., An Introduction to Analysis, McGraw-Hill, New York, NY, 1995.

Bruckner, A. et al., Real Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1997.

Caplan, Wilfred, Advanced Calculus, Addison-Wesley, Reading, MA, 1984.

Fitzpatrick, P., Advanced Calculus: A Course in Mathematical Analysis, Brooks/ Cole, Belmont, CA, 1996.

Fulks, D. Advanced Calculus, 4th Ed., John Wiley & Sons, New York, NY, 1988.

Gariepy, R. & Ziemer, W., Modern Real Analysis, Brooks/ Cole, Belmont, CA, 1995.

Gaughan, E., Introduction to Analysis, 5th Ed., Brooks/ Cole, Belmont, CA, 1998.

Gordon, R., Real Analysis: A First Course, Addison Wesley, Reading, MA, 1997.

Kaplan, W., Advanced Calculus, 4th Ed., Addison Wesley, Reading, MA, 1992.

Kirkwood, J., An Introduction to Analysis, 2nd Ed., Brooks/ Cole, Belmont, CA, 1995.

Lay, S., Analysis with an Introduction to Proof, Prentice-Hall, Englewood Cliffs, NJ, 1990.

Royden, H.L., Real Analysis, 3rd Ed., Macmillan, New York, NY, 1988.

Stoll, M., Introduction to Real Analysis, Addison Wesley, Reading, MA, 1997.

Lectures, assignments, discussion, examinations

MAT 417

Three semester hours

This course outline updating and revision was examined in accord with recommended procedures and was approved by the Department of Mathematics. The course was initially approved on April 26, 1965.

MAT 418 - Introduction to Real Analysis II

a continuation of MAT 417; partial differentiation, multiple integrals, measurable sets, measurable functions, Lebesgue integration, the Stieltjes integral.

XI. Statement of qualifications of faculty members who will teach this course.

All members of the mathematics graduate faculty who have completed a Masters Degree in mathematics and who have had at least six semester hours of graduate analysis.