Abstract:Almost forty years ago, Connes, Feldman and Weiss proved that for measurable equivalence relations the notions of amenability and hyperfiniteness coincide. In this paper we define the uniform version of amenability and hyperfiniteness for measurable graphed equivalence relations of bounded vertex degrees and prove that these two notions coincide as well. Roughly speaking, a measured graph $\mathcal {G}$ is uniformly hyperfinite if for any ${\varepsilon }>0$ there exists $K\geq 1$ such that not only $\mathcal {G}$, but all of its subgraphs of positive measure are $({\varepsilon },K)$-hyperfinite. We also show that this condition is equivalent to weighted hyperfiniteness and a strong version of fractional hyperfiniteness, a notion recently introduced by Lovász. As a corollary, we obtain a characterization of exactness of finitely generated groups via uniform hyperfiniteness.

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中文翻译：

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均匀超有限性

摘要：将近 40 年前，Connes、Feldman 和 Weiss 证明，对于可测量的等价关系，服从性和超有限性的概念是一致的。在本文中，我们为有界顶点度的可测量图形等价关系定义了服从性和超有限性的统一版本，并证明这两个概念也一致。粗略地说，如果对于任何 ${\varepsilon }>0$ 存在 $K\geq 1$ 使得不仅 $\mathcal {G}$，而且所有的它的正测度子图是 $({\varepsilon },K)$-超有限。我们还表明，这种情况等价于加权超有限性和分数超有限性的强版本，这是 Lovász 最近引入的一个概念。作为推论，

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