# Measure and Category: A Survey of the Analogies Between Topological and Measure Spaces: 002

## Oxtoby, John C.

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In this edition, a set of Supplementary Notes and Remarks has been added at the end, grouped according to chapter. Some of these call attention to subsequent developments, others add further explanation or additional remarks. Most of the remarks are accompanied by a briefly indicated proof, which is sometimes different from the one given in the reference cited. The list of references has been expanded to include many recent contributions, but it is still not intended to be exhaustive. John C. Oxtoby Bryn Mawr, April 1980 Preface to the First Edition 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.

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Contenuti:

1. Measure and Category on the Line.- Countable sets, sets of first category, nullsets, the theorems of Cantor, Baire and Borel.- 2. Liouville Numbers.- Algebraic and transcendental numbers, measure and category of the set of Liouville numbers.- 3. Lebesgue Measure in r-Space.- Definitions and principal properties, measurable sets, the Lebesgue density theorem.- 4. The Property of Baire.- Its analogy to measurability, properties of regular open sets.- 5. Non-Measurable Sets.- Vitali sets, Bernstein sets, Ulam’s theorem, inaccessible cardinals, the continuum hypothesis.- 6. The Banach-Mazur Game.- Winning strategies, category and local category, indeterminate games.- 7. Functions of First Class.- Oscillation, the limit of a sequence of continuous functions, Riemann integrability.- 8. The Theorems of Lusin and Egoroff.- Continuity of measurable functions and of functions having the property of Baire, uniform convergence on subsets.- 9. Metric and Topological Spaces.- Definitions, complete and topologically complete spaces, the Baire category theorem.- 10. Examples of Metric Spaces.- Uniform and integral metrics in the space of continuous functions, integrable functions, pseudo-metric spaces, the space of measurable sets.- 11. Nowhere Differentiable Functions.- Banach’s application of the category method.- 12. The Theorem of Alexandroff.- Remetrization of a G? subset, topologically complete subspaces.- 13. Transforming Linear Sets into Nullsets.- The space of automorphisms of an interval, effect of monotone substitution on Riemann integrability, nullsets equivalent to sets of first category.- 14. Fubini’s Theorem.- Measurability and measure of sections of plane measurable sets.- 15. The Kuratowski-Ulam Theorem.- Sections of plane sets having the property of Baire, product sets, reducibility to Fubini’s theorem by means of a product transformation.- 16. The Banach Category Theorem.- Open sets of first category or measure zero, Montgomery’s lemma, the theorems of Marczewski and Sikorski, cardinals of measure zero, decomposition into a nullset and a set of first category.- 17. The Poincaré Recurrence Theorem.- Measure and category of the set of points recurrent under a nondissipative transformation, application to dynamical systems.- 18. Transitive Transformations.- Existence of transitive automorphisms of the square, the category method.- 19. The Sierpinski-Erdös Duality Theorem.- Similarities between the classes of sets of measure zero and of first category, the principle of duality.- 20. Examples of Duality.- Properties of Lusin sets and their duals, sets almost invariant under transformations that preserve nullsets or category.- 21. The Extended Principle of Duality.- A counter example, product measures and product spaces, the zero-one law and its category analogue.- 22. Category Measure Spaces.- Spaces in which measure and category agree, topologies generated by lower densities, the Lebesgue density topology.- Supplementary Notes and Remarks.- References.- Supplementary References.

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## 1.Measure and Category: A Survey of the Analogies between Topological and Measure Spaces (Graduate Texts in Mathematics)

Editore: Springer (1996)
ISBN 10: 0387905081 ISBN 13: 9780387905082
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Descrizione libro Springer, 1996. Hardcover. Condizione libro: Neu. 2. Auflage. immediately available. Codice libro della libreria 000465

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## 2.Measure and Category: A Survey of the Analogies Between Topological and Measure Spaces, 2nd edition

Editore: Secaucus, New Jersey, U.S.A.: Springer Verlag (1980)
ISBN 10: 0387905081 ISBN 13: 9780387905082
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Descrizione libro Secaucus, New Jersey, U.S.A.: Springer Verlag, 1980. Soft cover. Condizione libro: New. Condizione sovraccoperta: New. 2nd Edition. **INTERNATIONAL EDITION** Read carefully before purchase: This book is the international edition in mint condition with the different ISBN and book cover design, the major content is printed in full English as same as the original North American edition. The book printed in black and white, generally send in twenty-four hours after the order confirmed. All shipments contain tracking numbers. Great professional textbook selling experience and expedite shipping service. Codice libro della libreria ABE-8034539661

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## 3.Measure and Category. A Survey of the Analogies between Topological and Measure Spaces

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Descrizione libro Springer, 1980. Hardback. Condizione libro: NEW. 9780387905082 This listing is a new book, a title currently in-print which we order directly and immediately from the publisher. Codice libro della libreria HTANDREE0275135

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## 4.Measure and Category: A Survey of the Analogies Between Topological and Measure Spaces: 002 (Graduate Texts in Mathematics)

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Descrizione libro Springer, 1996. Condizione libro: New. This item is printed on demand for shipment within 3 working days. Codice libro della libreria LP9780387905082

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## 5.Measure and Category: A Survey of the Analogies Between Topological and Measure Spaces

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Descrizione libro Springer, 2016. Paperback. Condizione libro: New. PRINT ON DEMAND Book; New; Publication Year 2016; Not Signed; Fast Shipping from the UK. No. book. Codice libro della libreria ria9780387905082_lsuk

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## 6.Measure and Category

Editore: Springer-Verlag New York Inc. (1980)
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Descrizione libro Springer-Verlag New York Inc., 1980. HRD. Condizione libro: New. New Book.Shipped from US within 10 to 14 business days.THIS BOOK IS PRINTED ON DEMAND. Established seller since 2000. Codice libro della libreria IP-9780387905082

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## 7.Measure and Category

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Descrizione libro Springer-Verlag New York Inc., 1980. HRD. Condizione libro: New. New Book. Delivered from our US warehouse in 10 to 14 business days. THIS BOOK IS PRINTED ON DEMAND.Established seller since 2000. Codice libro della libreria IP-9780387905082

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## 8.Measure and Category: A Survey of the Analogies between Topological and Measure Spaces: 002 (Graduate Texts in Mathematics)

Editore: Springer 1997-01-01 (1997)
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Descrizione libro Springer 1997-01-01, 1997. Condizione libro: New. Brand new book, sourced directly from publisher. Dispatch time is 24-48 hours from our warehouse. Book will be sent in robust, secure packaging to ensure it reaches you securely. Codice libro della libreria NU-ING-00701970

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## 9.Measure and Category: A Survey of the Analogies between Topological and Measure Spaces (Graduate Texts in Mathematics)

Editore: Springer (1996)
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Descrizione libro Springer, 1996. Hardcover. Condizione libro: New. Codice libro della libreria B26S4-74

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## 10.Measure and Category: A Survey of the Analogies Between Topological and Measure Spaces (Hardback)

Editore: Springer-Verlag New York Inc., United States (1996)
ISBN 10: 0387905081 ISBN 13: 9780387905082
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Descrizione libro Springer-Verlag New York Inc., United States, 1996. Hardback. Condizione libro: New. 2nd ed. 1980. 4th printing 1996. 236 x 155 mm. Language: English . Brand New Book ***** Print on Demand *****. In this edition, a set of Supplementary Notes and Remarks has been added at the end, grouped according to chapter. Some of these call attention to subsequent developments, others add further explanation or additional remarks. Most of the remarks are accompanied by a briefly indicated proof, which is sometimes different from the one given in the reference cited. The list of references has been expanded to include many recent contributions, but it is still not intended to be exhaustive. John C. Oxtoby Bryn Mawr, April 1980 Preface to the First Edition 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. Codice libro della libreria APC9780387905082

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