目錄
1.Measure and Category on the Line
Countable sets, sets offirst category, nullsets, 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, 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 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 nullset 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 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
Index
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