Measure and category

Abstract

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 6 10 4. The Property of Baire .................. 19 Its analogy to measurability, properties of regular open sets 5. Non-Measurable Sets .................. 22 Vitali sets, Bernstein sets, Ulam's theorem, inaccessible cardinals, the con- tinuum hypothesis 6. The Banach-Mazur Game ................ 27 Winning strategies, category and local category, indeterminate games 7. Functions of First Class ................ 31 Oscillation, the limit of a sequence of continuous functions, Riemann inte- grability 8. The Theorems of Lusin and Egoroff ............ 36 Continuity of measurable functions and of functions having the property of Baire, uniform convergence on subsets 9. Metric and Topological Spaces .............. 39 Definitions, complete and topologically complete spaces, the Baire category theorem 10. Examples of Metric Spaces ............... 42 Uniform and integral metrics in the space of continuous functions, integrable functions, pseudo-metric spaces, the space of measurable sets 11. Nowhere Differentiable Functions ............ 45 Banach's application of the category method 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. The Theorem of Alexandroff ............... Remetrization of a G subset, topologically complete subspaces 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 Fubini's Theorem ................... Measurability and measure of sections of plane measurable sets 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 The Banach Category Theorem .............. Open sets of first category or measure zero, Montgomery's lemma, the theo- rems of Marczewski and Sikorski, cardinals of measure zero, decomposition into a nullset and a set of first category The Poincar6 Recurrence Theorem ............ Measure and category of the set of points recurrent under a nondissipative transformation, application to dynamical systems Transitive Transformations ............... Existence of transitive automorphisms of the square, the category method The Sierpinski-Erdt3s Duality Theorem .......... Similarities between the classes of sets of measure zero and of first category, the principle of duality Examples of Duality .................. Properties of Lusin sets and their duals, sets almost invariant under transforma- tions that preserve nullsets or category The Extended Principle of Duality ........... A counter example, product measures and product spaces, the zero-one law and its category analogue Category Measure Spaces ................ Spaces in which measure and category agree, topologies generated by lower densities, the Lebesgue density topology