ChapterZero

Looks like my plan for the weekend is to spend some time in bed with Folland’s Real Analysis. I didn’t realize at the time I took the course what a godsend this book is– I thought it was too dense at the time– but now I see how it functions almost perfectly as a reference book for the analysis I need to know to handle advanced probability treatises.

The fact that he immediately takes the abstract approach to measures means that– besides the fact that, unlike in many books which serve the same function, pages aren’t wasted on the development of exclusively the Lebesgue measure/integral, which is then generalized and repeated for abstract measures/integrals– he completely covers probability spaces as a (albeit unmentioned) subset of his discussion. His functional analysis chapter is also pretty comprehensive in relation to the functional analysis I’ve seen used in probability treatises, even though he doesn’t cover the important topics of self-adjoint or compact operators. And, I’m particularly grateful for the chapter on point-set topology, because to my shame I forgot most of the general topology I knew, and that turns out to be very important (duh, I guess).

At this stage though, the most attractive feature of his book is the short chapter which provides short, rigorous proofs of the Central Limit Theorem, 0-1 Law, Laws of Large Numbers, and other basic probabilistic results.