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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.
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