The existence of invariant probability measures for a group of transformations
Let \(G\) be a group of measurable transformations of a \(\sigma\)-finite measure space \((X,{\mathcal A},m)\). A set \(A\in{\mathcal A}\) is said to be weakly wandering under \(G\) if there exists a sequence \(g_ n\) of elements of \(G\) such that the sets \(g_ n A\) are all pairwise disjoint. The author proves that the following conditions are equivalent: (i) There exists a \(G\)-invariant probability measure \(\mu\) defined on \(\mathcal A\) such that the measure \(m\) is absolutely continuous with respect to \(\mu\). (ii) There does not exist any set of positive measure which is weakly wandering under \(G\). This result gives a proof of a conjecture of Chuaqui. Under additional assumptions, versions of this result were known before; in particular when \(G\) is cyclic (due to Hopf), and when \(G\) is an arbitrary group of nonsingular transformations (Hajian and Itô).
groups of measure-preserving transformations
scientific article; zbMATH DE number 446879
scientific article