We prove a concentration inequality for invariant means on topological groups, namely for such adapted to a chain of amenable topological subgroups. The result is based on an application of Azuma’s martingale inequality and provides a method for establishing extreme amenability. Building on this technique, we exhibit new examples of extremely amenable groups arising from von Neumann’s continuous geometries. Along the way, we also answer a question by Pestov on dynamical concentration in direct products of amenable topological groups.

中文翻译：

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连续几何形状中链稳定剂的不变均值和动力学浓度

我们证明了拓扑群上不变均值的集中不等式，即适应于一系列顺应的拓扑子群的集中不等式。该结果基于 Azuma 鞅不等式的应用，并提供了一种建立极端顺应性的方法。在此技术的基础上，我们展示了由冯·诺依曼的连续几何产生的极其顺从的群的新例子。在此过程中，我们还回答了佩斯托夫关于顺应拓扑群的直积的动态集中的问题。