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Meaure Theory
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Measure Theory
by 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.






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Measure Theory
by Paul R. Halmos




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