by A B Kharazishvili (Tbilisi State University, Republic of Georgia)

This book is devoted to some topics of the general theory of invariant and quasi-invariant measures. Such measures are usually defined on various σ-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 the viewpoint of the theory of invariant (quasi-invariant) measures.

• Some Properties of Transformation Groups
• Quasiinvariant and Invariant Measures
• Some Examples and Constructions
• Nonmeasurable Sets with Respect to Quasiinvariant and Invariant Measures
• Small Sets with Respect to Quasiinvariant Measures
• Almost Invariant Sets
• Some Invariant σ-Ideals and σ-Algebras
• Density Points and Invariant Extensions of Lebesgue Measure
• The Uniqueness of Lebesgue and Borel Measures
• Quasiinvariant Borel Measures on Standard Groups