Amenable groups, stationary measures and partitions with independent iterates

Suppose a discrete amenable group G acts freely on a probability space (X, B, μ) and {g _i } is any mixing sequence of group elements, that is μ(g _i ^-1 A ∩ B) → μ(A)μ(B) for all A, B ε B. Then given any finite partition P and ε > 0 there is a subsequence {h _j } of {g _i } and a partition P′ differing from P on a set of measure less than ε such that the partitions {gP: g ε IP′ {h _j }} are jointly independent, where IP′{h _j } denotes the set { ^eG} ∩ { h _jk h_jk-1 ⋯h_j1 : j1 < j2 < ⋯ < jk} consisting of the identity of G together with all finite products of the {h _j } taken with indices in decreasing order.