# Measure Theoryby Paul R. Halmos

Chapter I: SETS AND CLASSES

1. Set inclusion
2. Unions and intersections
3. Limits, complements, and differences
4. Rings and algebras
5. Generated rings and s-rings
6. Monotone classes

Chapter II: MEASURES AND OUTER MEASURES

7. Measure on rings
8. Measure on intervals
9. Properties of measures
10. Outer measures
11. Measurable sets

Chapter III: EXTENSION OF MEASURES

12. Properties of induced measures
13. Extension, completion, and approximation
14. Inner measures
15. Lebesgue measure
16. Non measurable sets

Chapter IV: MEASURABLE FUNCTIONS

17. Measure spaces
18. Measurable functions
19. Combinations of measurable functions
20. Sequences of measurable functions
21. Pointwise convergence
22. Convergence in measure

Chapter V: INTEGRATION

23. Integrable simple functions
24. Sequences of integrable simple functions
25. Integrable functions
26. Sequences of integrable functions
27. Properties of integrals

Chapter VI: GENERAL SET FUNCTIONS

28. Signed measures
29. Hahn and Jordan decompositions
30. Absolute continuity
31. The Radon-Nikodym theorem
32. Derivatives of signed measures

Chapter VII: PRODUCT SPACES

33. Cartesian products
34. Sections
35. Product measures
36. Fubini's theorem
37. Finite dimensional product spaces
38. Infinite dimensional product spaces

Chapter VIII: TRANSFORMATIONS AND FUNCTIONS

39. Measurable transformations
40. Measure rings
41. The isomorphism theorem
42. Function spaces
43. Set functions and point functions

Chapter IX: PROBABILITY

44. Heuristic introduction
45. Independence
46. Series of independent functions
47. The law of large numbers
48. Conditional probabilities and expectations
49. Measures on product spaces

Chapter X: LOCALLY COMPACT SPACES

50. Topological lemmas
51. Borel sets and Baire sets
52. Regular measures
53. Generation of Borel measures
54. Regular contents
55. Classes of continuous functions
56. Linear functionals

Chapter XI: HAAR MEASURE

57. Full subgroups
58. Existence
59. Measurable groups
60. Uniqueness

Chapter XII: MEASURE AND TOPOLOGY IN GROUPS

61. Topology in terms of measure
62. Weil topology
63. Quotient groups
64. The regularity of Haar measure

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.