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My graduate text on measure theory (based on these lecture notes) is now published by the AMS as part of the Graduate Studies in Mathematics series.  (See also my own blog page for this book, which among other things contains a draft copy of the book in PDF format.)

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.