测试与范畴学Measure and category

1.Measure and Category on the Line 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