  We provide equivalent conditions for Liouville property of actions of groups.
As an application, we show that there is a Liouville measure for the action of
the Thompson group $F$ on dyadic rationals. This result should be compared with
a recent result of Kaimanovich, where he shows that the action of the Thompson
group F on dyadic rationals is not Liouville for all finitely supported
measures. As another application we show that there is a Liouville measure for
lamplighter actions. This gives more examples of non-amenable Liouville
actions.
