by Paul R. Halmos

- Set inclusion
- Unions and intersections
- Limits, complements, and differences
- Rings and algebras
- Generated rings and s-rings
- Monotone classes
Chapter II: MEASURES AND OUTER MEASURES

- Measure on rings
- Measure on intervals
- Properties of measures
- Outer measures
- Measurable sets
Chapter III: EXTENSION OF MEASURES

- Properties of induced measures
- Extension, completion, and approximation
- Inner measures
- Lebesgue measure
- Non measurable sets
Chapter IV: MEASURABLE FUNCTIONS

- Measure spaces
- Measurable functions
- Combinations of measurable functions
- Sequences of measurable functions
- Pointwise convergence
- Convergence in measure
Chapter V: INTEGRATION

- Integrable simple functions
- Sequences of integrable simple functions
- Integrable functions
- Sequences of integrable functions
- Properties of integrals
Chapter VI: GENERAL SET FUNCTIONS

- Signed measures
- Hahn and Jordan decompositions
- Absolute continuity
- The Radon-Nikodym theorem
- Derivatives of signed measures
Chapter VII: PRODUCT SPACES

- Cartesian products
- Sections
- Product measures
- Fubini's theorem
- Finite dimensional product spaces
- Infinite dimensional product spaces
Chapter VIII: TRANSFORMATIONS AND FUNCTIONS

- Measurable transformations
- Measure rings
- The isomorphism theorem
- Function spaces
- Set functions and point functions
Chapter IX: PROBABILITY

- Heuristic introduction
- Independence
- Series of independent functions
- The law of large numbers
- Conditional probabilities and expectations
- Measures on product spaces
Chapter X: LOCALLY COMPACT SPACES

- Topological lemmas
- Borel sets and Baire sets
- Regular measures
- Generation of Borel measures
- Regular contents
- Classes of continuous functions
- Linear functionals
Chapter XI: HAAR MEASURE

- Full subgroups
- Existence
- Measurable groups
- Uniqueness
Chapter XII: MEASURE AND TOPOLOGY IN GROUPS

- Topology in terms of measure
- Weil topology
- Quotient groups
- The regularity of Haar measure

Chapter I: SETS AND CLASSES

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.