Ergodic Subequivalence Relations Induced by a Bernoulli Action

Let $\mathrm{\Gamma }$ be a countable group and denote by $\text{Cal}S$ the equivalence relation induced by the Bernoulli action $\Gamma\curvearrowright [0,1]^{\Gamma}$,where$[0,1]^{\Gamma}$ is endowed with the product Lebesgue measure. We prove that for any subequivalence relation $\text{Cal}R$ of $\text{Cal}S$, there exists a partition $\{X_i{\}}_{i\ge 0}$ of $[0,1{]}^{\mathrm{\Gamma }}$ with $\Cal R$-invariantmeasurablesetssuchthat$\Cal R_{|X_0}$ishyperfiniteand$\Cal R_{|X_i}$isstronglyergodic(henceergodic),forevery$i\geq 1$.