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Measure and Category

A Survey of the Analogies between Topological and Measure Spaces

  • John C. Oxtoby

Part of the Graduate Texts in Mathematics book series (GTM, volume 2)

Table of contents

  1. Front Matter
    Pages I-VIII
  2. John C. Oxtoby
    Pages 1-5
  3. John C. Oxtoby
    Pages 6-9
  4. John C. Oxtoby
    Pages 10-18
  5. John C. Oxtoby
    Pages 19-21
  6. John C. Oxtoby
    Pages 22-26
  7. John C. Oxtoby
    Pages 27-30
  8. John C. Oxtoby
    Pages 31-35
  9. John C. Oxtoby
    Pages 36-38
  10. John C. Oxtoby
    Pages 39-41
  11. John C. Oxtoby
    Pages 42-44
  12. John C. Oxtoby
    Pages 45-46
  13. John C. Oxtoby
    Pages 47-48
  14. John C. Oxtoby
    Pages 49-51
  15. John C. Oxtoby
    Pages 52-55
  16. John C. Oxtoby
    Pages 56-61
  17. John C. Oxtoby
    Pages 62-64
  18. John C. Oxtoby
    Pages 65-69
  19. John C. Oxtoby
    Pages 70-73
  20. John C. Oxtoby
    Pages 74-77
  21. John C. Oxtoby
    Pages 78-81
  22. John C. Oxtoby
    Pages 82-85
  23. John C. Oxtoby
    Pages 86-91
  24. Back Matter
    Pages 92-96

About this book

Introduction

This book has two main themes: the Baire category theorem as a method for proving existence, and the "duality" between measure and category. The category method is illustrated by a variety of typical applications, and the analogy between measure and category is explored in all of its ramifications. To this end, the elements of metric topology are reviewed and the principal properties of Lebesgue measure are derived. It turns out that Lebesgue integration is not essential for present purposes, the Riemann integral is sufficient. Concepts of general measure theory and topology are introduced, but not just for the sake of generality. Needless to say, the term "category" refers always to Baire category; it has nothing to do with the term as it is used in homological algebra. A knowledge of calculus is presupposed, and some familiarity with the algebra of sets. The questions discussed are ones that lend themselves naturally to set-theoretical formulation. The book is intended as an introduction to this kind of analysis. It could be used to supplement a standard course in real analysis, as the basis for a seminar, or for inde­ pendent study. It is primarily expository, but a few refinements of known results are included, notably Theorem 15.6 and Proposition 20A. The references are not intended to be complete. Frequently a secondary source is cited, where additional references may be found.

Keywords

calculus

Authors and affiliations

  • John C. Oxtoby
    • 1
  1. 1.Bryn Mawr CollegeBryn MawrUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4615-9964-7
  • Copyright Information Springer-Verlag New York 1971
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-0-387-05349-3
  • Online ISBN 978-1-4615-9964-7
  • Series Print ISSN 0072-5285
  • Buy this book on publisher's site