Measure and Category : A Survey of the Analogies between Topological and Measure Spaces / by John C. Oxtoby.

Format:

Book

Author/Creator:

Oxtoby, John C., author.

Contributor:

SpringerLink (Online service)

Series:

Graduate texts in mathematics 0072-5285 ; 2.

Graduate Texts in Mathematics, 0072-5285 ; 2

Language:

English

Subjects (All):

Science.

Science, general.

Local Subjects:

Science, general.

Physical Description:

1 online resource.

Contained In:

Springer eBooks

Place of Publication:

New York, NY : Springer US, 1971.

System Details:

text file PDF

Summary:

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.

Contents:

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.

Other Format:

Printed edition:

ISBN:

9781461599647

Access Restriction:

Restricted for use by site license.