An Introduction to Measure Theory

This text is based on the lecture notes for a one-quarter graduate course in real analysis. The course and the book focus on the basics of measure and integration theory, both in Euclidean spaces and in abstract measure spaces. The author mentions that this text is intended as a prequel to his 2010 book An Epsilon of Room I, which is an introduction to the analysis of Hilbert and Banach spaces. These two books can serve as material for a complete graduate course in real analysis.

The author used as an inspiration the book by E. Stein and R. Shakarchi Real Analysis. Measure Theory, Integration and Hilbert Spaces. Princeton Lectures in Analysis, III. In particular, the first half of the text (Chapter 1) is about measure theory in Euclidean spaces R^d, and the abstract aspects of measure theory are deferred to the second half (Chapter 2). This is because this approach strengthens the students’ intuition in the first part of the course, while providing motivation for more abstract facts, such as Caratheodory’s general construction of a measure from an outer measure.

Most of the material of the text is self contained and addressed to the students with only an undergraduate knowledge of real analysis. Some exercises require also some knowledge of point-set topology or set theory.

There are many exercises; in fact, many of the results and examples are presented through exercises. The intention of the author is that the reader perform a significant portion of these exercises while going through the book; the students taking a course based on this book and working through the exercises will have the added benefit of being well-prepared for the examinations.

The first half of the book (Chapter 1) includes all the basics of Measure Theory and makes-up the material for a full course on the subject. The second half (Chapter 2) contains Related Articles (problem solving strategies, probability spaces, the Rademacher differentiation theorem, infinite product spaces and the Kolmogorov extension theorem)—all of which could be considered as “optional material”.

The best part is that the author has a wonderful blog containing plenty of material, as well as a special part of the blog for the course he taught on Measure Theory.

For instructors, mathematicians and especially graduate students, this text, as well as the blog and the other books which appeared or are in preparation by Tao constitute a treasure trove of material by one of the best mathematicians of our time. It is a pleasure and a special gift to have these resources.

Mihaela Poplicher is an associate professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is Mihaela.Poplicher@uc.edu.