Strong ergodicity around countable products of countable equivalence relations

This paper deals with countable products of countable Borel equivalence relations and equivalence relations "just above" those in the Borel reducibility hierarchy. We show that if $E$ is strongly ergodic with respect to $\mu$ then $E^\mathbb{N}$ is strongly ergodic with respect to $\mu^\mathbb{N}$. We answer questions of Clemens and Coskey regarding their recently defined $\Gamma$-jump operations, in particular showing that the $\mathbb{Z}^2$-jump of $E_\infty$ is strictly above the $\mathbb{Z}$-jump of $E_\infty$. We study a notion of equivalence relations which can be classified by infinite sequences of "definably countable sets". In particular, we define an interesting example of such equivalence relation which is strictly above $E_\infty^\mathbb{N}$, strictly below $=^+$, and is incomparable with the $\Gamma$-jumps of countable equivalence relations. We establish a characterization of strong ergodicity between Borel equivalence relations in terms of symmetric models. The proofs then rely on a fine analysis of the very weak choice principles "every sequence of $E$-classes admits a choice sequence", for various countable Borel equivalence relations $E$.