On some subsets of spaces equipped with transformation groups (Q1979060)

Let \(E\) be a nonempty set and \(G\) be a subgroup of the group of all bijective mappings on \(E\) and \(I\) be a \(G\)-invariant \(\sigma \)-ideal of subsets of \(E\). We say that \(X \subset E\) is \((G,I)\)-thick if there exists a family \(\{g_k \in G;\;k \in \mathbb N\}\) with \(E \setminus \{g_k(X);\;k \in \mathbb N\} \in I\). The set \(X \subset E\) is said to be \((G,I)\)-thin if \(X\) is not \((G,I)\)-thick. The author studies the relationships between these notions and the theory of \(G\)-quasiinvariant measures (\(\mu \) on \(E\) is \(G\)-quasiinvariant if the family of \(\mu \)-measurable sets is \(G\)-invariant and the same holds for the family of \(\mu \)-zero sets). One of the main results shows (under some conditions) the equivalence between the notion of \((G,I)\)-thickness of a set \(X \subset E\) and countably \((G,I)\)-equidecomposability of \(X\) and \(E\).