Measure and Category

1. Measure and Category on the Line.- Countable sets, sets of first category, nullsets, the theorems of Cantor, Baire and Borel.- 2. Liouville Numbers.- Algebraic and transcendental numbers, measure and category of the set of Liouville numbers.- 3. Lebesgue 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, category 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 measurable functions and of functions having the property of Baire, uniform convergence on subsets.- 9. Metric and Topological Spaces.- Definitions, complete and topologically complete spaces, the Baire category theorem.- 10. Examples of Metric Spaces.- Uniform and integral metrics in the space of continuous functions, integrable functions, pseudo-metric 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 G? 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 Fubini’s 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 Poincaré 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-Erdös Duality Theorem.- Similarities between the classes of sets of measure zero and of first category, the principle 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.