MATH-6200 Real Analysis

Scheduled: TF 12:00-1:50, Sage 3705; Credits: 4

Web-page: www.rpi.edu/~roytbv/REAL/real-04.html (also accessible from the math department page).

Traditionally, introductory courses in real analysis are devoted to a very careful study of basic concepts and constructions of measure theory and integration. If done in all the gory detail, it usually doesn’t leave any time to go into useful relation of real analysis to other parts of mathematics. One of the most spectacular applications of real analysis is probability theory (that is considered by some purists as a part of measure theory). In this course we will try to connect measure-theoretical concepts to probability and consider some meaningful applications from probability theory. No prior knowledge of probability is required.

Course Outline

• Elementary set theory and topology.
• Measures: sigma-algebras, rings and semirings, completion
• Lebesgue measure.
• Integration.
• Convergence theorems for integrals.
• Product measures. Fubini's theorem.
• Basic probability. Infinite products of probability spaces.
• L^P Spaces.
• Hilbert spaces, orthonormal bases.
• Relations between two measures: Radon–Nikodym theorem.
• Laws of large numbers.
• Distribution functions and probability densities.
• Characteristic functions (Fourier transform).
• Central limit theorem.

Suggested Texts

· R. M. Dudley, Real Analysis and Probability, Cambridge University Press 2002.

· A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis, Dover. (This is a beautiful book from one of the greatest mathematicians ever. The book won't be used in the class but I highly recommend it).

The text is rather dense. To help you navigate through it here is a Reading List; it will be updated periodically.

Homework Assignments:

Assignment #1, due January 27, 2004

Assignment #2 UPDATED VERSION, due February 10, 2004

Reading Assignment: Section 4.3.

Note that the Lebesgue function construction and a non-Borel set example that were discussed in class some time ago are presented in the book, Proposition 4.2.3, with some minor differences.

Assignment #3, due February 24, 2004

Assignment #4, due March 16, 2004

Assignment #5, due March 30, 2004

Trigonometric polynomials span L² : an outline

Corrected Version of Assignment #6, due April 13, 2004

Assignment #7, due April 30, 2004

Grading policy (tentative):

Homework, 50% of the course grade (there will be 6-7 assignments); midterms and a final, 50%

Instructor’s Information

Instructor's Web Page: http://www.rpi.edu/~roytbv/
Tentative Office Hours: Tu, Fr 4-5 or by appointment, ph. 276-6889
Email: roytbv at rpi dot edu

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