Krieger's finite generator theorem for actions of countable groups I

Abstract

For an ergodic probability-measure-preserving action $G\curvearrowright (X,\mu )$ of a countable group $G$, we define the Rokhlin entropy $h_G^{\mathrm{Rok}}(X,\mu )$ to be the infimum of the Shannon entropies of countable generating partitions. It is known that for free ergodic actions of amenable groups this notion coincides with classical Kolmogorov--Sinai entropy. It is thus natural to view Rokhlin entropy as a close analogue to classical entropy. Under this analogy we prove that Krieger's finite generator theorem holds for all countably infinite groups. Specifically, if $h_G^{\mathrm{Rok}}(X,\mu )<\log (k)$ then there exists a generating partition consisting of $k$ sets.