Ergodic Subequivalence Relations Induced by a Bernoulli Action

Let Γ be a countable group and denote by S{mathcal{S}} the equivalence relation induced by the Bernoulli action Gcurvearrowright [0, 1]^G{Gammacurvearrowright [0, 1]^{Gamma}}, where [0, 1]^Γ is endowed with the product Lebesgue measure. We prove that, for any subequivalence relation R{mathcal{R}} of S{mathcal{S}}, there exists a partition {X _i } _i≥0 of [0, 1]^Γ into R{mathcal{R}}-invariant measurable sets such that R_|X0{mathcal{R}_{vert X_{0}}} is hyperfinite and R_|Xi{mathcal{R}_{vert X_{i}}} is strongly ergodic (hence ergodic and non-hyperfinite), for every i ≥ 1.