A few comments on last week's homework:
Problem #1 was straightforward and people did well on it, but in arguing that Lipschitz continuity implied that \( |F'|\leq M \) a.e., several people wrote that \[ | F'(x) | = \lim_{h\to 0}\frac{| F(x+h) - F(x)|}{h}\leq \lim_{h\to 0} \frac{M|h|}{|h|}=M. \] This argument only proves that if the limit exists, then it's bounded by M. So it's only valid if you've already shown that F is absolutely continuous and then pointed out that hence it is in \( BV([a,b]) \) for every \( a < b \) and hence it is a.e. differentiable on every interval and hence everywhere. (It's also fine just to say that since it's absolutely continuous, it's differentiable a.e.)
On problem #2, one of the key points in the problem is a fact that we use a lot in Chapter 3.5 and which probably ought to be its own lemma: the Lebesgue-Radon-Nikodym decomposition of a measure \( \mu_F \) with respect to Lebesgue measure has the form \( F'\,dm + \lambda \). That is, the portion of \( \mu_F \) that's absolutely continuous with respect to Lebesgue measure has density \( F' \). This fact is an immediate consequence of the differentiation theorem at the end of Chapter 3.4. You do not need to reprove it if you use it on the final or a qualifying exam (or in a paper). It's a standard fact that people will know.
To do problem #2, first you need to explain why you can reduce to the case that all \( F_j \) are in NBV, namely that you can shift them and make them right continuous while only changing their derivatives on sets of measure zero, and then their new sum will still be finite and its derivative will differ from that of F by only a set of measure zero. Now we'll be able to talk about measures \( \mu_F \) and \(\mu_{F_j}\), and \( \mu_F \) is the sum of the \( \mu_{F_j} \). The Lebesgue-Radon-Nikodym decomposition of \( \mu_{F_j} \) is \( F'_j\,dm +\lambda_j\), and the absolutely continuous part of the decomposition of \( \mu_F \) is \( F'\,dm +\lambda\). You can then confirm that \( \sum F'_j\,dm +\sum \lambda_j \) is a valid Lebesgue-Radon-Nikodym decomposition of \( \mu_F \), and by uniqueness of the decomposition we obtain \( F'=\sum F'_j \) a.e.
On problem #5, a lot of people struggled with the details of proving separability of \( L^p(\mathbb{R}^n,m) \), though everyone seemed to get the main idea. I didn't think about the details ahead of time, but in general I approve of your using the results about n-dimensional Lebesgue measure from Chapter 2.6. The key fact from here for this problem is that any measurable set with finite measure in \( \mathbb{R}^n \) can be approximated by a finite union of sets that are products of intervals. (Incidentally, many people on this problem used the word rectangle to mean such a set. But rectangles in a σ-algebra \( \mathcal{M}\otimes\mathcal{N} \) are just defined as sets \( M\times N \) for arbitrary \( M\in\mathcal{M} \) and \( N\in\mathcal{N} \), not necessarily intervals.) Here is a sketch of the proof of this: We start with the fact that \( m(E) \) is equal to the infimum of \( \sum m(E_j) \) over all covers \( \cup E_j \) where \( E_j \) is a rectangle, i.e., has the form \( A_1\times\cdots\times A_n \). Thus we can find a finite union of sets of this form with arbitrarily small symmetric difference with E. So it suffices to approximate each rectangle \( A_1\times\cdots\times A_n \) by a finite union of rectangles with intervals for sides. And you can do this by approximating each set \( A_i\subset\mathbb{R} \) by a finite union of intervals.
From Theorem 2.40c, we need to construct our dense subset of functions in \( L^p \). We know that integrable simple functions are dense in \( L^p \). Now take an arbitrary such function, and show that we can arbitrarily approximate it in \( L^p \) by an integrable simple function with rational cofficients. Now take such a function, and show we can arbitrarily approximate such a function in \( L^p \) by a simple function one where the sets are rectangles with intervals as sides; it suffices to show that for any \( E\subset\RR^n \), you can approximate \( \chi_E \) in \( L^p \) by \( \chi_{E'} \) for such a set \( E' \), which follows nearly immediately from Theorem 2.40c. Now show you can approximate \( \chi_{E'} \) for any such set \( E' \) by \( \chi_{E''} \) where \(E''\) is a rectangle whose sides are intervals with rational endpoints. It is a lot simpler to do this one step at a time as described than all at once, because if you can arbitrarily approximate an object in one set by an object in another, which can be arbitrarily approximated by an object in another set, and so on, then you can arbitrarily approximate anything in the first set by anything in the last.
By the way, in the \( L^{\infty} \) part of the problem, many people gave solutions that at various points referred to two different collections of balls with the same notation, namely balls of points in \( \mathbb{R}^n \) with the standard Euclidean metric, and balls of functions in \( L^{\infty} \) with the \( L^{\infty} \) metric.
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.