General Information
Time 
MWF, 2  3 p.m. 
Location  Building 4, Room 163 
Professor 
Ben Brubaker (brubaker@math.mit.edu)
Office: 2267
Office Phone: 34079
Office Hours: We 34 p.m., Th 24 p.m., or by appointment. 
Textbook  Measure Theory and Probability, by M. Adams and V. Guillemin 
Grade Breakdown  2 Midterms  20 % each, Final  40%, Homework  20% 
Course Content  We will march through Adams and Guillemin, covering Lebesgue measure and integration theory and then moving on to Fourier analysis. Along the way, we'll see how the Lebesgue theory allows for rigorous development of probability (and where Riemann's theory of measure and integration falls short). If time permits, we'll discuss applications of both probability and Fourier analysis to modern analytic number theory. 
Prereqs 
18.100 is the listed prerequisite. As our starting point will be the theory of Riemann integration and how to develop a more general theory, some familiarity with abstract definitions and proofs will be useful. However, this doesn't rule out 18.100A as a precursor. 
Announcements & Dates
 Important Dates
 Wednesday, February 7: First Day of Class
 Monday, February 19: No Class (Presidents Day)
 Tuesday, February 20: Monday Schedule. Class held.
 Friday, March 9: Add deadline
 Wednesday, March 21: First Midterm
 March 2630: Spring Vacation
 Monday, April 16: No Class (Patriot's Day)
 Wednesday, May 2: Second Midterm
 Wednesday, May 16: Last Day of Class
 May 2125: Final Exam Week
OUR FINAL: Tuesday, May 22, 1:304:30 pm, Walker Gym
Further Reading
 M. Kac, Statistical Independence in Probability, Analysis, and Number Theory is a classic, and only 96 pages. Adams and Guillemin borrow from this book throughout the beginning sections of Chapter 1.
 For those interested in learning more about the development of measure theory, including a fuller discussion of Borel sets, check out Measure Theory by Paul Halmos.
 Wikipedia's entry on the Vitali Set, a construction of a nonmeasurable set.
 Royden's book Real Analysis gives a second look at many of the topics we're covering in the course, with special emphasis on the real numbers. The book contains many good exercises and, in fact, major theorems are left to the reader but contain significant hints.
 J.E. Littlewood is known both as a great mathematician and as a man who could turn a phrase. Here are some links to Littlewood's quotes:
 I'll try to augment this list with further curiosities as the course gets going.
Solutions and Handouts:
 Second Midterm, with solutions
 For the Second Midterm, the style will be similar to the first, with 45 questions including proofs of two of the following theorems:
 Product measure (defined by extending measure on "rectangles") agrees with measure defined by iterated integration of characteristic function on slices
 Fubini's Theorem (for integrable functions)
 Law of Large Numbers (using expectation values, random variables)
 L^p(X,u) is a Banach space for all p greater than or equal to 1.
 L^2(X,u) is a Hilbert space
 The Fourier coefficients of a complete orthonormal system converge in the Hilbert space norm.
 The Fourier inversion formula
 First Midterm, with solutions
 Instructions and Sample Problems for Midterm I
 Eventually, I may post further exam solutions and inclass handouts here. For partial solutions to the homework, see the homework page.
