In the author's PhD thesis (2019) universal envelopes were introduced as a tool for studying continuously obtainable information on discontinuous functions. To any function $f \colon X \to Y$ between $\operatorname{qcb}_0$-spaces one can assign so-called envelope which, in well-defined sense, encodes all function. A consists of two continuous functions $F L$ and $\xi_L Y with values $\Sigma$-sp...

Almost forty years ago, Connes, Feldman and Weiss proved that for measurable equivalence relations the notions of amenability hyperfiniteness coincide. In this paper we define uniform version graphed bounded vertex degrees prove these two coincide as well. Roughly speaking, a measured graph $\mathcal {G}$ is uniformly hyperfinite if any ${\varepsilon }>0$ there exists $K\geq 1$ such not only {G...