This article is concerned with measure equivalence and uniform measure equivalence of locally compact, second countable groups. We show that two unimodular, locally compact, second countable groups are measure equivalent if and only if they admit free, ergodic, probability measure preserving actions whose cross section equivalence relations are stably orbit equivalent. Using this we prove that in the presence of amenability any two such groups are measure equivalent and that both amenability and property (T) are preserved under measure equivalence, extending results of Connes–Feldman–Weiss and Furman. Furthermore, we introduce a notion of uniform measure equivalence for unimodular, locally compact, second countable groups, and prove that under the additional assumption of amenability this notion coincides with coarse equivalence, generalizing results of Shalom and Sauer. Throughout the article we rigorously treat measure theoretic issues arising in the setting of non-discrete groups.

中文翻译：

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测量单模局部紧群的等价和粗糙等价

本文涉及局部紧凑的第二可数组的等价度量和统一等价度量。我们证明了，当且仅当两个单模局部紧凑的第二可数组的横截面当量关系稳定地处于轨道等效状态时，它们才允许度量自由度，遍历度，概率度量。利用这一点，我们证明了在存在可接受性的情况下，任意两个这样的组在度量上是等效的，并且在可度量性对等下保留了可承受性和属性（T），从而扩展了Connes–Feldman–Weiss和Furman的结果。此外，我们为单模，局部紧致，第二可数组引入了统一度量等价的概念，并证明在可适应性的附加假设下，该概念与粗等价一致，Shalom和Sauer的推广结果。在整个文章中，我们严格地处理非离散组设置中出现的度量理论问题。