Means on equivalence relations (Q926424)

A mean on a countable set \(A\) is a positive linear functional \(\phi:\ell^{\infty}(A)\to\mathbb C\) sending the constant sequence \((1)\) to \(1\). An assignment of means on a countable Borel equivalence relation \(E\) on a Polish space \(X\) is a map that associates a mean on each equivalence class to each equivalence class. The equivalence relation is called smooth if there is a Borel section for it (that is, a Borel subset that contains exactly one representative of each class). The main result here is that \(E\) admits a Borel assignment of means if and only if \(E\) is smooth. This is derived as a consequence of a more technical characterization of the existence of a Baire assignment of means.