Measure and Category: A Survey of the Analogies between Topological and Measure Spaces (Graduate Texts in Mathematics (2))

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. 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. 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 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. Front Matter....Pages I-VIII Measure and Category on the Line....Pages 1-5 Liouville Numbers....Pages 6-9 Lebesgue Measure in r -Space....Pages 10-18 The Property of Baire....Pages 19-21 Non-Measurable Sets....Pages 22-26 The Banach-Mazur Game....Pages 27-30 Functions of First Class....Pages 31-35 The Theorems of Lusin and Egoroff....Pages 36-38 Metric and Topological Spaces....Pages 39-41 Examples of Metric Spaces....Pages 42-44 Nowhere Differentiable Functions....Pages 45-46 The Theorem of Alexandroff....Pages 47-48 Transforming Linear Sets into Nullsets....Pages 49-51 Fubini’s Theorem....Pages 52-55 The Kuratowski-Ulam Theorem....Pages 56-61 The Banach Category Theorem....Pages 62-64 The Poincaré Recurrence Theorem....Pages 65-69 Transitive Transformations....Pages 70-73 The Sierpinski-Erdös Duality Theorem....Pages 74-77 Examples of Duality....Pages 78-81 The Extended Principle of Duality....Pages 82-85 Category Measure Spaces....Pages 86-91 Back Matter....Pages 92-96 [From the Preface] 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. [by] John C. Oxtoby. Bibliography: P. 92-93.