Here are some notes, mostly from Tao's Introduction to Measure Theory. A few exercises are thrown in, mostly sketches. I focused on material not found or emphasized in Analysis II. I'll update the pdf regularly, if anyone stopping by is interested in reading them I will be typesetting the occasional solution to exercises in this book. If anyone wants to do so together, or to discuss some exercises in general at any point, that would be nice!
Why should $F_{\sigma}$ and $G_{\delta}$ be unions and intersections of closed and open respectively? In class Professor Zhou said it was sensible that $G_{\delta}$ sets are open since we often work with finite open covers over compact sets. I think one reason $F_{\sigma}$ ought to be closed is that if it weren't, we couldn't find descriptive inner measure sets for measure zero sets. Any measure zero set (I think) has to be closed and couldn't contain such an inner approximation. Closed sets can be 'smaller' than open sets. The measure zero sets I can think of in $R^n$ are those of countable many points, boundary points (Cantor set), and affine transformations of graphs in $n-1$ or fewer variables. Are there others?
I find myself wanting to say something along the lines of “This set is measurable because it's homeo/diffeo/something-morphic to $X$, which is measurable”, or “it's an $n-m$ manifold in $R^n$ and has measure 0”, or something else intuitive. A function $f: \mathbb{R} \rightarrow \mathbb{R}$ preserves measurability if it satisfies the $\textit{Luzin property}$; it sends Lebesgue measure zero sets to Lebesgue measure zero sets. It seems natural that this should also be true for $R^n$. Then again, a lot of the functions we're considering, we're considering because they aren't nice in these ways, otherwise they should be Riemann integrable. I think that smooth functions should behave nicely. This is probably explained more coherently in our books a few pages ahead of where I am now. For some reason my understanding of necessary conditions and these sorts of things feels pretty confused/poorly organized in my mind, but I think it should be fairly simple. Looking forward to cleaning up.
2/14: General approaches for thinking about statements when $f$ is measurable: Find a simple function property/construction that helps. If working with integrals, step into sup first. See if desired statement is more natural with the property you thought would be helpful. If so, prove statement in this context. Step out of sup, be mindful of limits and see if general statement follows.
Riemann and Lebesgue Integrals (informal discussion, intuition): The definitions of the Jordan and Lebesgue integrals, as well as some examples can be found in the notes pdf on this page. The required definition of simple functions and more are there as well. Roughly, Jordan measure is to the Riemann integral as Lebesgue measure is to the Lebesgue integral. Elementary sets are to the Riemann integral as measurable sets are to the Lebesgue integral. Riemann integrable functions are Lebesgue integrable. Lebesgue theory extends the Riemann theory. Every Jordan measurable set is Lebesgue measurable, and every Riemann integrable function is a Lebesgue measurable function.
As Terence Tao said, the Lebesgue integral can handle “noise” or “error”. The difference between the two and the motivation for the Lebesgue theory is demonstrated nicely by $[0,1]\Q^2$, the “bullet-riddled-square”. The square and the set of bullets $[0,1]^2\cup\Q^2$ have Jordan inner measure zero, and Jordan outer measure 1. The indicator function of the bullets over the square is not Riemann integrable, but it is Lebesgue integrable, as the “noise” $Q^2$ poses no problems.
Littlewood's Three Principals: (i) Every (measurable) set is nearly a finite sum of intervals;
(ii) Every (absolutely integrable) function is nearly continuous;
(iii) Every (pointwise) convergent subsequence of functions is nearly uniformly convergent.
These three principals are an intuitive rephrasing of some results that we have learned. We can find such a sum in (i) using a G-delta cover, and expressing at open set as a countable collection of disjoint (up to some error), or almost disjoint boxes.
Littlewood's second theorem is a consequence of Egorov's theorem. We can take some closed set on which our function is continuous, and an epsilon set on which it is not. This is also essentially Lusin's theorem, mentioned above.
The third principle is essentially Egorov's theorem.
Most recent copy of:
