Recently I'm reading the paper Ramsey–Milman phenomenon, Urysohn metric spaces, and extremely amenable groups by Pestov. When it comes to the definition of an extremely amenable topological group, it claims (without proof) that
(the extreme amenability) is equivalent to the existence of a left invariant multiplicative mean on the space of all bounded right uniformly continuous functions on the group.
I'm not aware of any references on this result.
Question: Is there any reference on extreme amenability of topological groups and invariant means?
Thanks for any comments or answers!