By
Alexander Kharazishvili, Tbilisi State University, Tbilisi, Republic of Georgia.
Description
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.
Included in series
North-Holland Mathematics Studies
Audience:
Pure mathematicians and post-graduate students. Especially, those ones whoseresearch interests lie in set theory, real analysis,measure theory, general topology , geometry ofEuclidean spaces, group theory.
