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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