Some properties of transformation groups; quasi-invariant and invariant measures; some examples and constructions; nonmeasurable sets with respect to quasi-invariant and invariant measures; small sets with respect to quasi-invariant measures; almost invariant sets; some invariant sigma-ideals and sigma-algebras; density points and invariant extensions of Lebesgue measure; the uniqueness of Lebesgue and Borel measures; quasi-invariant Borel measures on standard groups.

This work 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 the viewpoint of the theory of invariant (quasi-invariant) measures.