In this edition, a set of Supplementary Notes and Remarks has been added at the end, grouped according to chapter. Some of these call attention to subsequent developments, others add further explanation or additional remarks. Most of the remarks are accompanied by a briefly indicated proof, which is sometimes different from the one given in the reference cited. The list of references has been expanded to include many recent contributions, but it is still not intended to be exhaustive.

1．Measure and Category on the Line．，

 Countable sets,sets offirst category，nullsets,the theorems ofCantor．Baire,and BOreI

2．Liouville Numbers．

 Algebraic and transcendental numbers,measure and category of the set of

 Liouviile humbers

3．Lcbesgue 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 con．

 tinuum hypothesis

6．The Banach-Mazur Game

 Winning strategies,categoff 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 measurablc functions and of functimis having the property of Baire,uniform convergence on subsets

9．Metric and Topological Spaces

 Definitions,complete and topologically complete spaces,the Baire categorytheorem

10．Examples of Metric Spaces

 Uniform and integral metrics in the space of continuous functions,integrabl functions,pseudmetric spaces，the space of measurable sets

11．Nowhere Differentiable Functions

 Banach’S application of the category method

12．The Theorem ofAlexandroff．

 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 ORiemann 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 tFubinis theorem by means of a product transformation

16．The Banach Category Theorem．

 Open sets of first category or measure zero,MontgomeryS lemma，the thecrums of Marczewski and Sikorski．cardinals of measure zerodecomposition into a nullset and a set of first category

17．The Poincar6 Recurrence Theorem

 Measure and category of the set of points recurrent under a nondissipativ transformation，application to dynamical systems

18．Transitive Transformations

 Existence of transitive automorphisms of the square,the category method

19．The Sierpinski·Erd6s Duality Theorem．

 Similarities between the classes of sets of measure zero and of first category，th principie 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 lowe densities,the Lebesgue density topology

Supplementary Notes and Remarks．

References

Supplementary References．

Index