Book:Paul R. Halmos/Measure Theory

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