Measure and Category:
A Survey of the Analogies between Topological and Measure Spaces
by John C. Oxtoby
- Measure and Category on the Line
Countable sets, sets of first category, nullsets, the theorems of Cantor, Baire, and Borel
- Liouville Numbers
Algebraic and transcendental numbers, measure and category of the set of Liouville numbers
- Lebesgue Measure in r-Space
Definitions and principal properties, measurable sets, the Lebesgue density theorem
- The Property of Baire
Its analogy to measurability, properties of regular open sets
- Non-Measurable Sets
Vitali sets, Bernstein sets, Ulam's theorem, inaccessible cardinals, the continuum hypothesis
- The Banach-Mazur Game
Winning strategies, category and local category, indeterminate games
- Functions of First Class
Oscillation, the limit of a sequence of continuous functions, Riemann integrability
- The Theorems of Lusin and Egoroff
Continuity of measurable functions and of functions having the property of Baire,
uniform convergence on subsets
- Metric and Topological Spaces
Definitions, complete and topologically complete spaces, the Baire category theorem
- Examples of Metric Spaces
Uniform and integral metrics in the space of continuous functions, integrable functions,
pseudo-metric spaces, the space of measurable sets
- Nowhere Differentiable Functions
Banach's application of the category method
- The Theorem of Alexandroff
Remetrization of a G-delta subset, topologically complete subspaces
- 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
- 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 theorems of Marczewski and Sikorski,
cardinals of measure zero, decomposition into a nullset and a set of first category
- The Poincare 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-Erdoes 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 transformations
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