# 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

**Email Address:** clayton@math.colostate.edu

**Syllabus**

## 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