Uncountable Hyperfiniteness and The Random Ratio Ergodic Theorem

We show that the orbit equivalence relation of a free action of a locally compact group is hyperfinite (à 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.