Weak mixing for nonsingular Bernoulli actions of countable amenable groups

Abstract

Let G G be an amenable discrete countable infinite group, let A A be a finite set, and let ( μ g ) g ∈ G (\mu _g)_{g\in G} be a family of probability measures on A A such that inf g ∈ G min a ∈ A μ g ( a ) > 0 \inf _{g\in G}\min _{a\in A}\mu _g(a)>0 . It is shown (among other results) that if the Bernoulli shiftwise action of G G on the infinite product space ⨂ g ∈ G ( A , μ g ) \bigotimes _{g\in G}(A,\mu _g) is nonsingular and conservative, then it is weakly mixing. This answers in the positive a question by Z. Kosloff, who proved recently that the conservative Bernoulli Z d \mathbb {Z}^d -actions are ergodic. As a byproduct, we prove a weak version of the pointwise ratio ergodic theorem for nonsingular actions of G G .