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The theory of decreasing sequences of measurable partitions. (English. Russian original) Zbl 0853.28009
St. Petersbg. Math. J. 6, No. 4, 705-761 (1995); translation from Algebra Anal. 6, No. 4, 1-68 (1994).
The theory of decreasing sequences of measurable partitions of full separable spaces with full probability measures is considered. The simplest such sequence is the sequence of “pasts” of Bernoulli scheme. Sequences isomorphic to it are called standard. One of the main results of the paper is: Every ergodic (having trivial intersection) homogeneous sequence has a standard subsequence (theorem abut lacunar isomorphism). It means in particular that every dyadic sequence has a standard subsequence. A criterion for standardness is given. Examples of nonstandard sequences are constructed. A theorem on the existence of a continuum of pairwise nonisomorphic nonstandard sequences is proven. Asymptotic invariants of sequences are constructed with the method worked out by the author and called “method of tower”. The entropy theory is used as well. The author mentions that his method may be useful in other mathematical researches.

28D05 Measure-preserving transformations