Kolmogorov's example (a survey of actions of infinite-dimensional groups with an invariant probability measure)

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.