In February of 2007, I converted my “What’s new” web page of research
updates into a blog at This blog has since grown
and evolved to cover a wide variety of mathematical topics, ranging from my
own research updates, to lectures and guest posts by other mathematicians,
to open problems, to class lecture notes, to expository articles at both basic
and advanced levels.
With the encouragement of my blog readers, and also of the AMS, I
published many of the mathematical articles from the first two years of the
blog as [Ta2008] and [Ta2009], which will henceforth be referred to as
Structure and Randomness and Poincar´ e’s Legacies Vols. I, II. This gave
me the opportunity to improve and update these articles to a publishable
(and citeable) standard, and also to record some of the substantive feedback
I had received on these articles by the readers of the blog.
The current text contains many (though not all) of the posts for the third
year (2009) of the blog, focusing primarily on those posts of a mathematical
nature which were not contributed primarily by other authors, and which
are not published elsewhere. It has been split into two volumes.
The current volume consists of lecture notes from my graduate real anal-
ysis courses that I taught at UCLA (Chapter 1), together with some related
material in Chapter 2. These notes cover the second part of the gradu-
ate real analysis sequence here, and therefore assume some familiarity with
general measure theory (in particular, the construction of Lebesgue mea-
sure and the Lebesgue integral, and more generally the material reviewed
in Section 1.1), as well as undergraduate real analysis (e.g., various notions
of limits and convergence). The notes then cover more advanced topics in
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