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.

Subject Matter

• Measures
• Lebesgue Integral
• Locally Compact Spaces
• Probability
• Haar Measures

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

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 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