Preface

In February of 2007, I converted my “What’s new” web page of research

updates into a blog at terrytao.wordpress.com. 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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