  We study harmonic functions and Poisson boundaries for Borel probability
measures on general (i.e., not necessarily locally compact) topological groups,
and we prove that a second-countable topological group is amenable if and only
if it admits a fully supported, regular Borel probability measure with trivial
Poisson boundary. This generalizes work of Kaimanovich--Vershik and Rosenblatt,
confirms a general topological version of Furstenberg's conjecture, and entails
a characterization of the amenability of isometry groups in terms of the
Liouville property for induced actions. Moreover, our result has non-trivial
consequences concerning Liouville actions of discrete groups on countable sets
