The main goal of this book is to demonstrate the usefulness of set-theoretical methods in various questions of real analysis and classical measure theory. In this context, many statements and facts from analysis are treated as consequences of purely set-theoretical assertions which can successfully be applied to measures and Baire category. Topics covered include similarities and differences between measure and category; constructions of nonmeasurable sets and of sets without the Baire property; three aspects of the measure extension problem; the principle of condensation of singularities from the point of view of the Kuratowski-Ulam theorem; transformation groups and invariant (quasi-invariant) measures; the uniqueness property of an invariant measure; and ordinary differential equations with nonmeasurable right-hand sides. Audience: The material presented in the book is essentially self-contained and is accessible to a wide audience of mathematicians. It will appeal to specialists in set theory, mathematical analysis, measure theory and general topology. It can also be recommended as a textbook for postgraduate students who are interested in the applications of set-theoretical methods to the above-mentioned domains of mathematics.
Preface. 0. Introduction: Preliminary Facts. 1. Set-Valued Mappings. 2. Nonmeasurable Sets and Sets without the Baire Property. 3. Three Aspects of the Measure Extension Problem. 4. Some Properties of sigma-algebras and sigma-ideals. 5. Nonmeasurable Subgroups of the Real Line. 6. Additive Properties of Invariant sigma-Ideals on the Real Line. 7. Translations of Sets and Functions. 8. The Steinhaus Property of Invariant Measures. 9. Some Applications of the Property (N of Luzin. 10. The Principle of Condensation of Singularities. 11. The Uniqueness of Lebesgue and Borel Measures. 12. Some Subsets of Spaces Equipped with Transformation Groups. 13. Sierpinski's Partition and Its Applications. 14. Selectors Associated with Subgroups of the Real Line. 15. Set Theory and Ordinary Differential Equations. Bibliography. Subject Index.