# Math 617

Instructor: Dr. Clayton Shonkwiler
Time: Monday, Tuesday, Wednesday, Friday 1:00–1:50
Location: Weber 008
Office: Weber 216
Office Hours: Monday 2:00–3:00, Wednesday 2:00–3:00 and 4:00–5:00, and by appointment
Syllabus

Lecture Notes

Homework

## Overview

Measure theory provides the theoretical underpinnings of modern definitions of the integral and serves as the foundation for current approaches to functional analysis and distribution theory – and hence in particular to solving partial differential equations – as well as to probability theory, fractals, and dynamical systems.

The main goal of the course is to develop the basic theory: definitions and examples of $$\sigma$$-algebras and measures, the definition of measurable functions and of the Lebesgue integral, and the Lebesgue–Radon–Nikodym theorem. That groundwork will then allow us to develop two applications of the theory: to Fourier analysis (by way of functional analysis), and to probability theory. If time permits, we will conclude with some connections to random walks and polymer models.

A background in classical real analysis (i.e., MATH 517 material) and some familiarity with the basic concepts of point-set topology and vector spaces are essential prerequisites.

Some other texts that may be useful supplements:

• Real Analysis: Measure Theory, Integration, and Hilbert Spaces, by Elias M. Stein and Rami Shakarchi
• Real and Complex Analysis, by Walter Rudin
• Measure and Integral: An Introduction to Real Analysis, by Richard L. Wheeden and Antoni Zygmund
• An Introduction to Lebesgue Integration and Fourier Series, by Howard J. Wilcox and David L. Myers
• An Introduction to Measure Theory, by Terence Tao