Uniform hyperfiniteness

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}$, but all its subgraphs positive measure are $({\varepsilon },K)$-hyperfinite. We also show condition equivalent to weighted strong fractional hyperfiniteness, notion recently introduced by Lovász. As corollary, obtain characterization exactness finitely generated groups via hyperfiniteness.