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Measure and Category: A Survey of the Analogies between Topological and Measure Spaces
by John C. Oxtoby

  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-delta subset, topologically complete subspaces

  13. Transforming Linear Sets into Nullsets
    The space of automorphisms of an interval, effects 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 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-Erdoes 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
Measure and Category:
A Survey of the Analogies
Between Topological
and Measure Spaces
Measure and Category:
A Survey of the Analogies
Between Topological
and Measure Spaces
John C. Oxtoby John C. Oxtoby

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