Geodesic
Kolmogorov's example (a survey of actions of infinite-dimensional groups with an invariant probability measure)
Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 2, pp. 386-391

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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 non-locally-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.
Mots-clés : invariant set, permutation group
Keywords: transitive action, ergodic components, simplex of invariant measures. 
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     author = {A. M. Vershik},
     title = {Kolmogorov's example (a survey of actions of infinite-dimensional groups with an invariant probability measure)},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
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     publisher = {mathdoc},
     volume = {48},
     number = {2},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2003_48_2_a9/}
}
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A. M. Vershik. Kolmogorov's example (a survey of actions of infinite-dimensional groups with an invariant probability measure). Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 2, pp. 386-391. http://geodesic.mathdoc.fr/item/TVP_2003_48_2_a9/