Uncountable Hyperfiniteness and The Random Ratio Ergodic Theorem

TL;DR

This paper extends classical ergodic theorems to uncountable settings, characterizing hyperfiniteness of group actions and establishing new results on amenability and ergodic properties of locally compact groups.

Contribution

It introduces uncountable versions of the Ornstein-Weiss and Danilenko's Random Ratio Ergodic Theorems, linking hyperfiniteness with hypercompactness and amenability.

Findings

Orbit equivalence relation is hyperfinite iff hypercompact

Uncountable Ornstein-Weiss Theorem established

Uncountable Danilenko's Random Ratio Ergodic Theorem proved

Abstract

We show that the orbit equivalence relation of a free action of a locally compact group is hyperfinite (\`a la Connes-Feldman-Weiss) precisely when it is 'hypercompact'. This implies an uncountable version of the Ornstein-Weiss Theorem and that every locally compact group admitting a hypercompact probability preserving free action is amenable. We also establish an uncountable version of Danilenko's Random Ratio Ergodic Theorem. From this we deduce the 'Hopf dichotomy' for many nonsingular Bernoulli actions.