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: 20101101, 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: 0471317160
Search by ISBN on Amazon: 0471317160
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  σalgebras 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  LebesgueStieltjes 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 FubiniTonelli theorem 
2 lectures  2.6  2.7  Lebesgue measures on R^{n}. 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  RadonNikodym 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  L^{p} SPACES (5 LECTURES) 


2 lectures  6.1  6.2  Basic definitions; the Holder inequality; and L^{p} duality 
1 lecture  6.4  Weak L^{p} spaces and inequalities 
1 lecture  6.5  Interpolation of L^{p} 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.