作者：John C. Oxtoby[著]
页数：106页
出版社：世界图书出版公司北京公司
出版日期：2009
ISBN：9787510004391
电子书格式：pdf/epub/txt
内容简介
This book has two main themes： the Baire cate8ory 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 8cncrality， Needless to say， the term “category” refers always to Bairc category； it has nothing to do with the term as it is used in homological algebra. The book is a revised and expanded version of notes originally prepared for a course of lectures given at Haverford College during the spring of 1957 under the auspices of the William Pyle Philips Fund. These， in turn， were based on the Earle Raymond Hedrick Lectures presented at the Summer Meeting of the Mathematical Association of America at Seattle， Washington， in August， 1956.
作者简介
John C. Oxtoby，美国布尔茅尔学院(Bryn Mawr College)数学系教授。上世纪40年代美国最著名的数学家，在测度论、遍历性理论和拓扑学方面做出了突出的贡献。
目录
Countable sets, sets offirst category, sets, the theorems of Cantor, Baire, and Borel
2.Liouville Numbers
Algebraic and transcendental numbers, measure and category of the set of Liouviile humbers
3.Lcbesgue 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, categoff 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 measurablc functions and of functimis having the property of Baire, uniform convergence on subsets
9.Metric and Topological Spaces
Definitions, complete and topologically complete spaces, the Baire categorytheorem
10.Examples of Metric Spaces
Uniform and integral metrics in the space of continuous functions, integrabl functions, pseudmetric 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 Gδ subset, topologically complete subspaces
13.Transforming Linear Sets into Nullsets
The space of automorphisms of an interval, effect of monotone substitution on Riemann integrability, sets 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 Fubinis 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 set and a set of first category
17.The Poincare 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-Erdos Duality Theorem
Similarities between the classes of sets of measure zero and of first category, the principie of duality
20.Examples of Duality
Properties of Lusin sets and their duals, sets almost invariant under transformations that preserve sets 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
Index
