We prove that, if is a second-countable topological group with a compatible right-invariant metric and is a sequence of compactly supported Borel probability measures on converging to invariance with respect to the mass transportation distance over and such that concentrates to a fully supported, compact –space , then is homeomorphic to a –invariant subspace of the Samuel compactification of . In particular, this confirms a conjecture by Pestov and generalizes a well-known result by Gromov and Milman on the extreme amenability of topological groups. Furthermore, we exhibit a connection between the average orbit diameter of a metrizable flow of an arbitrary amenable topological group and the limit of Gromov’s observable diameters along any net of Borel probability measures UEB–converging to invariance over the group.
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