Department of Mathematics
Request for course updating and revision
I. Number and title of course
MAT 418 - Introduction to Real Analysis IIII. Reasons for addition to present curricula
A. The contents of this course are prerequisites to graduate studies in mathematical analysis.III. Major objectives of the courseB. 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.IV. Topical outlineB. A thorough development of integral calculus and its generalization to Lebesgue and Stieltjes integration.
A. Partial DifferentiationV. Bibliography, texts and readings1. Implicit functionsB. Multiple Integrals2. Taylor's Theorem
3. Jacobians
1. Inner and outer areaC. Measurable sets2. Fubini's Theorem
3. Transformation of multiple integrals
1. Measure of bounded setsD. Measurable functions2. Inner and outer measure
3. Vitale's Theorem
1. Properties of measurable functionsE. Lebesgue Integration2. Sequences of measurable functions
3. Theorems of Weierstrass
1. Definition of LebesgueF. The Stieltjes Integral2. Fundamental properties of Lebesgue integral
3. Comparison of Lebesgue and Riemann integral
1. Functions of finite variation2. 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.VI. Presentation and evaluationBilodean, 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, examinationsVII. Prerequisite
MAT 417VIII. Credit
Three semester hoursIX. Statement of approval
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.X. Catalog description
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.