# Book:Paul R. Halmos/Measure Theory

## Paul R. Halmos: Measure Theory

Published $\text {1950}$, Springer

ISBN 0-387-90088-8.

This book is part of Springer's Graduate Texts in Mathematics series.

### Contents

Preface
Acknowledgments

0. Prerequesites
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 $\sigma$-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
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 content
55. Classes of continuous functions
56. Linear functionals
CHAPTER XI: HAAR MEASURE
57. Full subgroups
58. Existence
59. Measurable groups
60. Uniqueness
CHAPTER XII: MEASURES AND TOPOLOGY IN GROUPS
61. Topology in terms of measure
62. Weil topologgy
63. Quotient groups
64. The regularity of Haar measure

References
Bibliography
List of frequently used symbols
Index