Math 73/103
Measure Theory and Complex Analysis
Last updated November 14, 2017 21:06:47 EST

Announcements:

• Our first class meeting is Monday, September 11th

Homework Assigments

 Monday September 11: Read: Review the Riemann integral: $\mathcal R\int_a^b f(x)\,dx$. HW: Work problems 1, 2 and 3 from the First homework assignment. (The assignment was last modified: 12:42 pm, August 22, 2017). If you want the source for the first homework assignment, it is here. You'll need my personal exam class: dpwexam-new.cls. Let me know if there additional missing macro files. Update: Wednesday, September 13: Read: We're more or less in the very beginning of Chapter 1 of Rudin. Homework: Now you're ready for problems 4, 5 and 6. Optional: Here's an optional little worksheet on Borel sets. Let me know if you want to talk about it. Some solutions: Hw1: solutions. borel-sets: solutions. Saturday, September 23: Homework: Here is the second assignment. You should start on problems 1--4. E-Mail: I tried to use canvas to circulate this assignment on Saturday morning. If you didn't see the email, you need to tweak your canvas settings. Here is the LaTeX source for homework #2. Sunday October 1: Homework #2 You should complete the second homework assignment by Friday. (Really, it should be Wednesday, but I forgot to say so earlier.) Not to be turned in: Let $f\in \mathcal{L}^1(X)$ and define $\nu: \mathcal{M}\to \mathbf{C}$ by $$\nu(E)=\int_E f(x)\,d\mu(x).$$ Show that $\nu$ is a complex measure on $(X\mathcal M)$. Show directly (without invoking Hahn/Jordan decompositions) that there are finite (positive) measures $\mu_i$ such that $$\nu(E)= \mu_1(E) -\mu_2(E) +i \bigl(\mu_3(E) - \mu_4(E)\bigr)$$ with $\mu_1\perp \mu_2$ and $\mu_3\perp \mu_4$. Some Date in early October: Homework: Here is the third assignment. I'm a bit late with this, so all problems are in play. I don't think that the source will be available this time. Monday, October 16th: Homework: Here is the fourth assignment. You should start on problems 2--8 right away. This is technically "backgroud material", but it will be new to some of you. It is certainly good background for the written quals. October 21: Homework: Here are some solutions for the third assignment. Monday, October 30: Last Homework fifth homework. Some of the Rudin problems are trickly. Feel free to ask questions! November 1: Homework: Here are some solutions for the fourth assignment. Tuesday 14 November: FINAL EXAM final exam. Sorry for the delay in posting!