The existence of invariant \(\sigma\)-finite measures for a group of transformations (Q690064)

Let \(G\) be a group of measurable transformations of a \(\sigma\)-finite measure space \((X,{\mathcal A},m)\). The author proves that the following two properties are equivalent: (i) There exists a \(G\)-invariant \(\sigma\)-finite measure \(\mu\) on \(\mathcal A\) such that the measure \(m\) is absolutely continuous with respect to \(\mu\). (ii) \(X\) is \(\sigma-G\)-bounded with respect to \(m\) and the \(\sigma\)-ideal \(I_ G(X)\) is \(\sigma\)-saturated in \(\mathcal A\). The result generalizes earlier work of E. Hopf, Kawada, Hajian and Itô, Chuaqui and K. Arnold.