Explore foundational concepts of modern measure theory that underpin advanced mathematical topics such as functional analysis, partial differential equations, and Fourier analysis. Through lectures and exercises, investigate fundamental concepts of Lebesgue measure and integration theory and apply the definitions and properties of Lebesgue measure and measurable sets. Discussion includes measurable functions, Lebesgue integrals, limit theorems of integrals, the Fubini theorem, and LP space. Using Latex for mathematical writing, hone mathematical proof and writing skills to communicate mathematics effectively and develop rigorous math thinking to prepare for more advanced courses.

This is an alternating years course.

Week 1. Preliminary Review

Week 2. Review on convergence of function sequences

Week 3. Lebesgue outer measure

Week 4. Sigma-algebra and measurable sets

Week 5. Construction of Lebesgue measure

Week 6. Properties of Lebesgue measure

Week 7. Exam 1

Week 8. Measurable functions

Week 9. Integrals

Week 10.Integral limit theorems Part I

Week 11.Integral limit theorems Part II

Week 12.LP Space

Week 13.Exam 2

Exam 1 25 %

Exam 2 25 %

Homework 50 % (2-3 exercises every week posted on slack group)

Each exam consists of 5 questions plus one bonus question. At least 3 questions are based on the homework and exercise discussed in class. Some questions may also ask for explicit statements of definitions and theorems.

Lebesgue Integration on Euclidean spaces, Frank Johns.

A User-Friendly Introduction to Lebesgue Measure and Integration, Gail Nelson.

Real Analysis, H.L. Royden.

Alternate years course: AY2025