Let $\theta$ be a partition of a Lebesgue space $(X,m)$ induced by a Borel equivalence relation $\sim$ and let $G$ be the associated full group consisting of all automorphisms $T$ such that $x \sim Tx$ for almost all $x$. Finally let $\xi$ be the measurable partition of $X$ into ergodic components of $G$.
If the $\sigma$-field generated by $\theta$ is the trivial $\sigma$-field $\{\varnothing, X\}$ modulo null sets, is it known whether the following sentence is true: "$\xi$ is atomic with all elements having the same measure or $\xi$ is isomorphic to the continuous partition of $[0,1]$" ?
