A110
Course Coordinator:
Xiaodan Zhou
Measure Theory and Integration
Description:

The concept of a measure is a natural generalization of length, area, volume and other common notions, such as mass and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to be widely used beyond mathematics.

Though the concept of measure dates to ancient Greece, it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. In particular, the work by French mathematician Henri Lebesgue plays a foundational role in modern measure theory. Lebesgue’s formulation of measure easily extends to quite an abstract setting and forms a basic language of probability theory and Lebesgue integral is robust under various limiting operations, overcoming the drawbacks of Riemann integrals. This is important, for instance, in the study of Fourier series, Fourier transforms, and other topics

The study of measure and integration lays the foundation for more advanced mathematical topics including functional analysis, partial differential equations, Fourier analysis, etc. In this course, we will visit many fundamental concepts of Lebesgue measure and integration theory through the lecture and exercise.

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 Type:
Elective
Credits:
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.
Text Book:
Lebesgue Integration on Euclidean spaces, Frank Johns.
Reference Book:
A User-Friendly Introduction to Lebesgue Measure and Integration, Gail Nelson.
Real Analysis, H.L. Royden.
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.