Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

MAT 206: Measure Theory
Approved: 2010-11-01, Jim Bremer

Units/Lecture:

Spring, alternate years; 4 units; lecture, extensive problem solving

Suggested Textbook: (actual textbook varies by instructor; check your instructor)
Real Analysis: Modern Techniques and Their Applications, Gerald Folland, John Wiley & Sons, 1999. ($133).
Search by ISBN on Amazon: 0-471-31716-0

Prerequisites:

MAT 125B, or consent of instructor

Course Description:

Introduction to measure theory. The study of lengths, surface areas, and volumes in general spaces, as related to integration theory.

Suggested Schedule:

Lectures Sections Topics/Comments
SECTION ONE - MEASURE SPACES
(6 LECTURES)


2 Lectures 1.1 - 1.3 σ-algebras and measure spaces
2 lectures 1.4 The construction of measure spaces from premeasures on algebras. Comment: The material on Cantor sets and functions in 1.5 could be introduced at this stage
2 lectures 1.5 Lebesgue-Stieltjes measures on R
SECTION TWO - INTEGRATION
(9 LECTURES)


1 lecture 2.1 Motivations and measurable functions. Comment: Compare the geometric underpining of Lebesgue integrals to that of Riemann integrals and illustrate the failings of Riemann integration
3 lectures 2.2 - 2.3 The Lebesgue integral and associated convergence theorems
1 lecture 2.4 Convergence in measure; Egoroff's theorem; and Lusin's theorem
2 lectures 2.5 Product measures and the Fubini-Tonelli theorem
2 lectures 2.6 - 2.7 Lebesgue measures on Rn. Comment: There is far too much material here for two lectures; it is suggested that much of the material (particularly Theorem 2.47 on nonlinear changes of variable) be presented without complete proofs.
SECTION THREE - THE FUNDAMENTAL THEOREM
(6 LECTURES)


1 lecture 3.1 and 3.3 Signed and complex measures
1 lecture 3.2 Radon-Nikodym theorem
2 lectures 3.4 Lebesgue differentiation theorem. Comment: There are alternative proofs of this theorem which are simpler than that presented in the book (e.g., it is an easy consequence of the Markov inequality).
2 lectures 3.5 Functions of bounded variation and the fundamental theorem
SECTION FOUR - Lp SPACES
(5 LECTURES)


2 lectures 6.1 - 6.2 Basic definitions; the Holder inequality; and Lp duality
1 lecture 6.4 Weak Lp spaces and inequalities
1 lecture 6.5 Interpolation of Lp spaces

Additional Notes:

Comment: The textbook is a very elegant introduction to measure theory and elementary analysis at the graduate level. It is a bit terse in places but is nonetheless always extremely clear.