Quasi-invariant measures for continuous group actions

Abstract

The class of ergodic, invariant probability Borel measure for the shift action of a countable group is a G_δ set in the compact, metrizable space of probability Borel measures. We study in this paper the descriptive complexity of the class of ergodic, quasi-invariant probability Borel measures and show that for any infinite countable group Γ it is Π⁰₃-hard, for the group Z it is Π⁰₃-complete, while for the free group F_∞ with infinite, countably many generators it is Π⁰_α-complete, for some ordinal α with 3 ≤ α ≤ ω +2. The exact value of this ordinal is unknown.