內容簡介

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.
 

目錄

1．Measure and Category on the Line．，
　Countable sets,sets offirst category，nullsets,the theorems ofCantor．Baire,and BOreI
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 con．
　tinuum 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 ofAlexandroff．
　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 ORiemann 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 tFubinis theorem by means of a product transformation
16．The Banach Category Theorem．
　Open sets of first category or measure zero,MontgomeryS lemma，the thecrums of Marczewski and Sikorski．cardinals of measure zerodecomposition into a nullset and a set of first category
17．The Poincar6 Recurrence Theorem
　Measure and category of the set of points recurrent under a nondissipativ transformation，application to dynamical systems
18．Transitive Transformations
　Existence of transitive automorphisms of the square,the category method
19．The Sierpinski•Erd6s Duality Theorem．
　Similarities between the classes of sets of measure zero and of first category，th principie 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 lowe densities,the Lebesgue density topology
Supplementary Notes and Remarks．
References
Supplementary References．
Index
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