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
Search by ISBN on Amazon: 0-471-31716-0
Prerequisites:
MAT 127B, 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.