

Measure Theory
by Paul R. Halmos
Chapter I: SETS AND CLASSES
Chapter II: MEASURES AND OUTER MEASURES
Chapter III: EXTENSION OF MEASURES
Chapter IV: MEASURABLE FUNCTIONS
Chapter V: INTEGRATION
Chapter VI: GENERAL SET FUNCTIONS
Chapter VII: PRODUCT SPACES
Chapter VIII: TRANSFORMATIONS AND FUNCTIONS
Chapter IX: PROBABILITY
Chapter X: LOCALLY COMPACT SPACES
Chapter XI: HAAR MEASURE
Chapter XII: MEASURE AND TOPOLOGY IN GROUPS
Excerpts from the Preface:
My main purpose in this book is to present a unified treatment of that part of measure theory which in recent years has shown itself to be most useful for its applications in modern analysis. If I have accomplished my purpose, then the book should be found usable both as a text for students and as a source of reference for the more advanced mathematician.
At the end of almost every section there is a set of exercises which appear sometimes as questions but more usually as assertations that the reader is invited to prove. These exercises should be viewed as corollaries to and sidelights on the results more formally expounded. They constitute an integral part of the book; among them appear not only most of the examples and counter examples necessary for understanding the theory, but also definitions of new concepts and, occasionally, entire theories that not long ago were still subjects of research.
It might appear inconsistent that, in the text, many elementary notions are treated in great detail, while, in the exercises, some quite refined and profound matters (topological spaces, transfinite numbers, Banach spaces, etc.) are assumed to be known. The material is arranged, however, so that when a beginning student comes to an exercise which uses terms not defined in this book he may simply omit it without loss of continuity. The more advanced reader, on the other hand, might be pleased at the interplay between measure theory and other parts of mathematics which it is the purpose of such exercises to exhibit.