**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