Reserve Text Information for

MATH 5453-001, Fall 2009

(Last updated on 17 August 2009)

I have requested that the following books be placed on overnight reserve in the Chemistry-Mathematics Library. Please note that some of these books were checked out at the time that the reserve request was submitted to the Library, so it may take a couple of weeks from the date posted above for the books to actually be available. These books represent some of my favorite references for real analysis and most of the material that I will present in class can be traced back to at least one of them.

Real Analysis (3/e), by H. L. Royden (QA 331.5 .R6 1988)

This, of course, is the recommended course text and is, in my opinion, one of the best general references on real analysis and measure theory.

Principles of Mathematical Analysis, by Walter Rudin; also known as "baby Rudin" (QA 300.R8 1976)

Technically, this is an "undergraduate" level text, but it has a good presentation of the prerequisite topics for this course, including nice treatments of the theory of metric spaces and the Riemann (Stieltjes) integral. The last chapter of the book touches on the Lebesgue integral and gives a somewhat different construction of Lebesgue measure than the one we will carry out in class.

Real and Complex Analysis, by Walter Rudin; also known as "big Rudin" (QA 300.R82 1987)

This is about the slickest presentation of graduate level real and complex analysis around. The first eight chapters comprise most of what I hope we can accomplish in both semesters of 5453-5463 (there are 20 chapters altogether!). However, my experience dictates that the presentation is a bit too slick for most students who are learning the material for the first time, so I have not adopted it formally as the course text. Rudin's construction of Lebesgue measure (in Chapter 2) is based on something called the Riesz Representation Theorem and is quite a bit different than the approach we will use. Nevertheless, this is a book that should owned by every doctoral-level student who is focusing on some area of analysis.

The Elements of Integration, by Robert G. Bartle (QA 312.B3)

This slim book contains a concise, but very readable, introduction to the theory of Lebesgue integration. Many of the proofs I will give in class are unabashedly lifted from this book. Were it not for its omission of some important topics (like absolutely continuous functions and functions of bounded variation), I would probably consider it for designation as the recommended course text.

Probability and Measure Theory (2/e), by Robert B. Ash (QA 273.A78 2000)

Measure theory is absolutely essential to the development of modern probability theory. This book not only contains an elegant development of measure theory, but also sets forth the foundations of measure-theoretic probability theory, which in turn is crucial for the rigorous mathematical treatment of such hot topics as stochastic processes and stochastic differential equations. As we wind our way through the course, I may try to give a brief glimpse of measure-theoretic probability theory, but otherwise you can view this book as "deep background."