# Measure Theory and Integration

Course Aim

Students successfully completing this course will be able to:

- State and apply the definitions and properties of Lebesgue measure and measurable sets;
- State and apply the definitions and properties of measurable functions, Lebesgue integrals, limit theorems of integrals, Fubini theorem, LP space;
- Use Latex effectively for mathematical writing;
- Demonstrate rigorous mathematical proof and writing skills;
- Communicate mathematics effectively and demonstrate rigorous math thinking which prepares for more advanced courses.

Course Description

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.

Course Contents

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

Assessment

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.

Prerequisites or Prior Knowledge

B36 “Introduction to Real Analysis” is recommended but not required. The following is expected prerequisite knowledge: basic set theory, mathematical logic, the fundamental property of real numbers; familiarity with limit definitions, and how to use these definitions in rigorous proofs of sequences, continuity and differentiation of real-valued functions; properties of a supremum (or least upper bound) and infimum (or greatest lower bound); basic topology including the definitions of open, closed, compact sets in the Euclidean space; basic definitions and properties of Riemann integrals. Please contact the instructor at the beginning of the course with questions.

Textbooks

Lebesgue Integration on Euclidean spaces, Frank Johns.

Reference Books

A User-Friendly Introduction to Lebesgue Measure and Integration, Gail Nelson.
Real Analysis, H.L. Royden.

ノート

Alternate years course: AY2025