**Time and place:**MWF at 10 in MS 5233**Instructor:**Thomas M. Liggett**Office hours:**MWF 2-3 in MS 7919**Text:**Measure Theory and Probability Theory by K. B. Athreya and S. N. Lahiri- There will be a
**discussion section**with a TA (Yen Do) on Thursdays at 10, beginning October 4. The first lecture is on Friday, September 28. Yen's office hours will be Tu 11-12 and 1-2 in MS 6142. **Prerequisites:**(a) One (two is even better) quarter of real analysis (Mathematics 245). As a review of measure theory, you might want to read the following parts of the text before classes begin: pages 1-25, 39-44, 48-71, 147-164. (b) Some exposure to undergraduate probability (e.g., Mathematics 170AB) is desirable, but not required. This is formally less important than the real analysis requirement, but helps to motivate the material. No undergraduate probability theorems or concepts will be used, however, so students who have not had undergraduate probability should be able to follow the course with no difficulty.**Topics:**Introduction to (measure theoretic) probability theory, including concepts of convergence, independence, weak and strong laws of large numbers, random series, characteristic functions, and central limit theorems -- roughly chapters 6-11 of the text.**The future:**This course leads to 275B (random walks, martingales, ergodic theorems), followed by 275C (Brownian motion; introduction to continuous time Markov processes), 275D (stochastic calculus with applications to mathematical finance), and 275E (stochastic particle systems). Courses 275ABC are normally offered each year. Courses 275DE are generally offered every two or three years.**Probability theory**is a branch of pure mathematics that has strong ties with many areas of application. In terms of technique, it is part of analysis. Its motivation comes from areas such as statistics, economics, business, engineering, biology and physics. It is an ideal meeting ground for the pure and the applied. Not only is it used in formulating and analyzing models in the areas of application listed above, but it also plays important roles in areas of pure mathematics such as analysis, partial differential equations, and combinatorics.**Homework**will be assigned each Friday and due the following Thursday in the discussion section (except that the assignment that would be due on Thanksgiving day will be due in lecture November 21). It will be returned to you the following week in discussion section.**Grades**will be based on homework to be assigned from the text and the final exam (December 13, 3-6pm). (There will be no midterm.) You may enroll for a letter grade or on an S/U basis. If you enroll for a letter grade, you are expected to take the final exam; if S/U, then the final exam is optional.- Final exam

**Due Oct. 4**. Page 32 #1.4,1.16,1.19 and page 71 #2.1,2.6 (take T_1 smaller than T_2; what happens if you don't?),2.11,2.15.**Due Oct. 11**. Page 34 #1.23, Page 75 #2.14 and page 212 #6.1,6.3,6.4,6.5 (note that the phi in part (c) is the one from part (a), not the one from part (b)),6.6,6.22,6.23.**Due Oct. 18**. Page 228 #7.3,7.4(b,c,e,f),7.6(a),7.8(b),7.20,7.21,7.22.**Due Oct. 25**. Page 231 #7.10,7.12(b) (there is an error in the statement of this problem; you should prove a correct form, and find a counterexample to the statement as it is),7.13,7.15(\epsilon_i are iid); page 279 #8.3,8.7,8.9,8.10.**Due Nov. 1**. Page 229 #7.5,7.9,7.16,7.18; page 279 #8.5,8.8.**Due Nov. 8**. Page 279 #8.2(assume independence;ignore the "nonnegative" part of the hint),8.11,8.13(a),8.15(note that in the hint, 8.14 should be 8.2),8.31.**Due Nov. 15**. Page 283 #8.28(b,c),8.30; page 309 #9.1,9.2,9.3,9.7,9.8(a).**Due Nov. 21 (Wednesday!)**. Page 310 #9.9,9.15,9.17,9.18,9.22,9.24, 9.26(b,c),9.29,9.30.**Due Nov. 29**Page 314 #9.32,9.34,9.35(a); page 337 #10.1,10.6,10.10.**Due Dec. 6**Page 337 #10.3,10.12,10.13,10.14,10.17,10.19.