A remark on graphed equivalence relations with harmonic measures (Q2738872)

Let \((X,{\mathcal B})\) be a standard Borel space, \(\mu\) a measure on \((X,{\mathcal B})\), \({\mathcal R}\) an equivalence relation on \(X\), and let \(\Phi= \{\varphi_i\}\) be a countable set of Borel isomorphisms \(\varphi_i: A_i\to B_i\) between \(A_i\) and \(B_i\) in \({\mathcal B}\). The authors recall several notions in measure and graph theory such as erasings (élagage), graphings (graphage) of \({\mathcal R}\), \(\Phi\)-harmonicity of \(\mu\), Euler characteristic of \(\Phi\) and amenability of \({\mathcal R}\) for \(\mu\). Let \(({\mathcal R},X,\Phi,\mu)\) be a graphed equivalence relation with \(\Phi\)-harmonic measure \(\mu\). The authors give, among other things, two criteria for \({\mathcal R}\) to be amenable for \(\mu\).