Measure and Category: A Survey of the Analogies between Topological and Measure SpacesSpringer, 06.12.2012 - 108 Seiten 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. |
Inhalt
| 1 | |
The Property of Baire | 19 |
The Theorems of Lusin and Egoroff | 36 |
The Theorem of Alexandroff | 47 |
The Banach Category Theorem | 62 |
Examples of Duality | 78 |
| 92 | |
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Measure and Category: A Survey of the Analogies between Topological and ... John C. Oxtoby Eingeschränkte Leseprobe - 2013 |
Häufige Begriffe und Wortgruppen
A₁ algebraic assuming the continuum automorphism Baire space belongs Borel measure Borel set cardinal category measure category method choose class of sets closed intervals closed set closed set F closed subset complement contains continuous functions continuum hypothesis convergent countable set countable union covering sequence defined dense set density duality endpoints exists F₁ finite measure finite number following theorem G is open G₁ homeomorphism I₁ implies intersection Lebesgue measure Lemma Let f linear set Liouville number measurable set measure and category measure space measure zero metric space non-empty open set non-measurable nullset o-algebra one-to-one mapping open intervals open set G outer measure P₁ Poincaré positive integer positive number positive outer measure Proof property of Baire Proposition rational numbers real number regular open set second category sequence of intervals set of measure set of points shows Theorem 1.6 topological space α α