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An uncountable Mackey-Zimmer theorem

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Tao, Terence

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The Mackey–Zimmer theorem classifies ergodic group extensions X of a measure-preserving system Y by a compact group K, by showing that such extensions are isomorphic to a group skew-product X?Y??H for some closed subgroup H of K. An analogous theorem is also available for ergodic homogeneous extensions X of Y, namely that they are isomorphic to a homogeneous skew-product Y??H/M. These theorems have many uses in ergodic theory, for instance playing a key role in the Host–Kra structural theory of characteristic factors of measure-preserving systems.The existing proofs of the Mackey–Zimmer theorem require various “countability”, “separability”, or “metrizability” hypotheses on the group ? that acts on the system, the base space Y, and the group K used to perform the extension. In this paper we generalize the Mackey–Zimmer theorem to “uncountable” settings in which these hypotheses are omitted, at the cost of making the notion of a measure-preserving system and a group extension more abstract. However, this abstraction is partially counteracted by the use of a “canonical model” for abstract measure-preserving systems developed in a companion paper. In subsequent work we will apply this theorem to also obtain uncountable versions of the Host–Kra structural theory.

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Institute of Mathematics of the Polish Academy of Sciences

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Mathematics

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Studia Mathematica

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10.4064/sm201125-1-5

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