Vershik, A. M. Kolmogorov’s example (A survey of actions of infinite-dimensional groups with an invariant probability measure). (English. Russian original) Zbl 1055.37006 Theory Probab. Appl. 48, No. 2, 373-378 (2003); translation from Teor. Veroyatn. Primen. 48, No. 2, 386-391 (2003). Summary: In the late 1940s, A. N. Kolmogorov suggested a remarkably simple example of a transitive, but not ergodic, action of the group of all permutations of positive integers. It turned out that such examples arise, as a rule, in the theory of actions of nonlocally compact groups, and for locally compact groups this phenomenon cannot happen. Kolmogorov’s example also helps to give a correct definition of the decomposition into ergodic components and orbit partition for actions of general groups. Cited in 2 Documents MSC: 37A15 General groups of measure-preserving transformations and dynamical systems 28D15 General groups of measure-preserving transformations 37A40 Nonsingular (and infinite-measure preserving) transformations 37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems Keywords:invariant set; permutation group; transitive action; ergodic components; simplex of invariant measures PDFBibTeX XMLCite \textit{A. M. Vershik}, Theory Probab. Appl. 48, No. 2, 373--378 (2003; Zbl 1055.37006); translation from Teor. Veroyatn. Primen. 48, No. 2, 386--391 (2003) Full Text: DOI