We've done our last problem set, but I recommend the following problems from Section 6.2 (and am happy to discuss them with you): Folland exercises 20(a), 21, and 22.
I've added two problems from Chapter 6.1 to the homework. This is the last assignment. I'll give some suggested problems from Chapter 6.2 as well, but they won't be collected.
Some commentary on last week's homework on Problem #3: It is possible to give an elementary solution that doesn't use any of this class's theory. But this is not a good solution. The good solution is to define a measure μ placing mass \( 2^{-n} \) on each point \( q_n \), where \( q_1,q_2,\ldots \) is an enumeration of the rationals. (You don't need to use \( 2^{-n} \), just something summable.) Then define \( F(x) \) to be the right-continuous, increasing function associated with μ, i.e., \( F(x) = \mu((-\infty,x]) \). Then F is continuous at all points except the rationals by Folland exercise 1.28 from Homework #3. Really this exercise could have been assigned back then, but the idea of doing it now is that we should all internalize the connection between positive measures and right-continuous increasing functions (and between signed measures and functions in NBV).
We will have one last homework assignment due Tuesday, May 7th. I've posted the assignment now but will be adding a few problems from Section 6.1 after class on May 2nd.
As we discussed in class today, the final needs to be held on the last day of class (Tuesday, May 14) because of the scheduling of the qualifying exams. I've updated the calendar now.
I've uploaded Homework 10. Sorry for the delay.
Update: All videos now uploaded. See you on Thursday. If you want to meet with me tomorrow to discuss homework, etc., please send me an email to set it up.
Update: I've added one more lecture video. I expect to add one more.
Here are the first two lecture videos in place of Tuesday's class. I'll post the rest by Tuesday.
I've posted Homework 9.