An uncountable Mackey–Zimmer theorem

Abstract

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 ⋊ρ 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 ⋊ρ 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 Γ 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.

An uncountable Furstenberg–Zimmer structure theory

“…We start by recalling the uncountable Mackey-Zimmer theorem [32]. Let K be a compact Hausdorff group and (Y , ν) a PrbAlg-space.…”

“…THEOREM 5.2. (Uncountable Mackey-Zimmer [32]) Let Y = (Y , ν, S) be an ergodic PrbAlg -system and K be a compact Hausdorff group. Every ergodic PrbAlg -homogeneous extension X of Y by K/L for some compact subgroup L of K is isomorphic in PrbAlg to a PrbAlg -homogeneous skew-product Y ρ H/M for some compact subgroup H of K, some compact subgroup M of H, and some H-valued PrbAlg -cocycle ρ.…”

“…We establish the equivalence of (a)-(c) in smaller conditional L ∞ -modules similarly to the classical theory and also in larger conditional L 0 -modules (where L 0 denotes the ring of equivalence classes of complex measurable functions), and we show that these descriptions in the larger and the smaller modules are equivalent. In the case of ergodic systems, we prove in §5 that relatively compact extensions are isomorphic to isometric extensions (that is, probability algebra homogeneous skew-product systems as introduced in [32]).…”

An uncountable ergodic Roth theorem and applications

“…Some foundational aspects arising in the ergodic theory of uncountable groups and inseparable spaces was systematically investigated by the fourth author and Tao in [28,30,29] and by the fourth author in [27]. For example, in the area of multiple recurrence, the tool of disintegration of measures is used extensively.…”

A category-theoretic proof of the ergodic decomposition theorem

“…The goal is to capture a wider range of structures and phenomena in probability theory and related fields such as statistics and information theory. On one hand, there is interest in moving measure theory beyond the need of countability, for example in the work of Jamneshan and Tao [15], and in their work with others in expanding ergodic theory in that direction [12][13][14]16].…”