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...

This book is devoted to some results from the classical Point Set Theory and their applications to certain problems in mathematical analysis of the real line. Notice that various topics from this theory are presented in several books and surveys. From among the most important works devoted to Point Set Theory, let us first of all mention the excellent book by Oxtoby 83] in which a deep analogy between measure and category is discussed in detail. Further, an interesting general approach to problems concerning measure and category is developed in the well-known monograph by Morgan 79] where a...

This book is devoted to some topics of the general theory of invariant and quasi-invariant measures. Such measures are usually defined on various sigma-algebras of subsets of spaces equipped with transformation groups, and there are close relationships between purely algebraic properties of these groups and the corresponding properties of invariant (quasi-invariant) measures. The main goal of the book is to investigate several aspects of those relationships (primarily from the set-theoretical point of view). Also of interest are the properties of some natural classes of sets, important from...