Nonmeasurable Sets and Functions

A.B. Kharazishvili (Eds.)0444516263, 9780444516268, 9781423741848

The book is devoted to various constructions of sets which are nonmeasurable with respect to invariant (more generally, quasi-invariant) measures. Our starting point is the classical Vitali theorem stating the existence of subsets of the real line which are not measurable in the Lebesgue sense. This theorem stimulated the development of the following interesting topics in mathematics: 1. Paradoxical decompositions of sets in finite-dimensional Euclidean spaces; 2. The theory of non-real-valued-measurable cardinals; 3. The theory of invariant (quasi-invariant) extensions of invariant (quasi-invariant) measures. These topics are under consideration in the book. The role of nonmeasurable sets (functions) in point set theory and real analysis is underlined and various classes of such sets (functions) are investigated . Among them there are: Vitali sets, Bernstein sets, Sierpinski sets, nontrivial solutions of the Cauchy functional equation, absolutely nonmeasurable sets in uncountable groups, absolutely nonmeasurable additive functions, thick uniform subsets of the plane, small nonmeasurable sets, absolutely negligible sets, etc. The importance of properties of nonmeasurable sets for various aspects of the measure extension problem is shown. It is also demonstrated that there are close relationships between the existence of nonmeasurable sets and some deep questions of axiomatic set theory, infinite combinatorics, set-theoretical topology, general theory of commutative groups. Many open attractive problems are formulated concerning nonmeasurable sets and functions. · highlights the importance of nonmeasurable sets (functions) for general measure extension problem. · Deep connections of the topic with set theory, real analysis, infinite combinatorics, group theory and geometry of Euclidean spaces shown and underlined. · self-contained and accessible for a wide audience of potential readers. · Each chapter ends with exercises which provide valuable additional information about nonmeasurable sets and functions. · Numerous open problems and questions.

---

Table of contents :
Content:
Preface
Pages vii-xi
A.B. Kharazishvili

Chapter 1 The vitali theorem Original Research Article
Pages 1-16

Chapter 2 The bernstein construction Original Research Article
Pages 17-34

Chapter 3 Nonmeasurable sets associated with hamel bases Original Research Article
Pages 35-55

Chapter 4 The fubini theorem and nonmeasurable sets Original Research Article
Pages 56-78

Chapter 5 Small nonmeasurable sets Original Research Article
Pages 79-101

Chapter 6 Strange subsets of the euclidean plane Original Research Article
Pages 102-120

Chapter 7 Some special constructions of nonmeasurable sets Original Research Article
Pages 121-144

Chapter 8 The generalized vitali construction Original Research Article
Pages 145-162

Chapter 9 Selectors associated with countable subgroups Original Research Article
Pages 163-178

Chapter 10 Selectors associated with uncountable subgroups Original Research Article
Pages 179-194

Chapter 11 Absolutely nonmeasurable sets in groups Original Research Article
Pages 195-219

Chapter 12 Ideals producing nonmeasurable unions of sets Original Research Article
Pages 220-235

Chapter 13 Measurability properties of subgroups of a given group Original Research Article
Pages 236-258

Chapter 14 Groups of rotations and nonmeasurable sets Original Research Article
Pages 259-275

Chapter 15 Nonmeasurable sets associated with filters Original Research Article
Pages 276-293

Appendix 1 Logical aspects of the existence of nonmeasurable sets
Pages 294-307

Appendix 2 Some facts from the theory of commutative groups
Pages 308-316

Bibliography
Pages 317-333

Subject index
Pages 334-337