  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.
