Oxtoby Measure and Category

A Survey of the Analogies between Topological and Measure Spaces
Erscheinungsjahr 2012
ISBN: 978-1-4615-9964-7
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 2, eBook

Reihe: Graduate Texts in Mathematics

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.

Zielgruppe

Research

Autoren/Hrsg.

Oxtoby, J. C.

Inhaltsverzeichnis

1. Measure and Category on the Line.- 2. Liouville Numbers.- 3. Lebesgue Measure in r-Space.- 4. The Property of Baire.- 5. Non-Measurable Sets.- 6. The Banach-Mazur Game.- 7. Functions of First Class.- 8. The Theorems of Lusin and Egoroff.- 9. Metric and Topological Spaces.- 10. Examples of Metric Spaces.- 11. Nowhere Differentiate Functions.- 12. The Theorem of Alexandroff.- 13. Transforming Linear Sets into Nullsets.- 14. Fubini’s Theorem.- 15. The Kuratowski-Ulam Theorem.- 16. The Banach Category Theorem.- 17. The Poincaré Recurrence Theorem.- 18. Transitive Transformations.- 19. The Sierpinski-Erdös Duahty Theorem.- 20. Examples of Duahty.- 21. The Extended Principle of Duality.- 22. Category Measure Spaces.- References.