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As I have done in the last three years, I am spending some time at the beginning of this year converting some of my posts on this blog into book format. This time round, the situation is a bit different because the majority of mathematical posts last year came from three courses I have taught: random matrices, higher-order Fourier analysis, and measure theory. These topics are sufficiently unrelated to each other, and to the other mathematical posts from 2010, that I am thinking of having as many as four distinct books this time around, though my plans are not yet definite in this regard.
In any event, I have started the process by converting the measure theory notes to book form, a draft copy of which is now available here. I have also started up a stub of a book page for this text, though it has little content at present beyond that link. I will be continuing to work on it in parallel with the rest of the conversion process. As always, any comments and corrections are very welcome.
This is going to be a somewhat experimental post. In class, I mentioned that when solving the type of homework problems encountered in a graduate real analysis course, there are really only about a dozen or so basic tricks and techniques that are used over and over again. But I had not thought to actually try to make these tricks explicit, so I am going to try to compile here a list of some of these techniques here. But this list is going to be far from exhaustive; perhaps if other recent students of real analysis would like to share their own methods, then I encourage you to do so in the comments (even – or especially – if the techniques are somewhat vague and general in nature).
(See also the Tricki for some general mathematical problem solving tips. Once this page matures somewhat, I might migrate it to the Tricki.)
Note: the tricks occur here in no particular order, reflecting the stream-of-consciousness way in which they were arrived at. Indeed, this list will be extended on occasion whenever I find another trick that can be added to this list.