GROUP ACTIONS WITH TOPOLOGICALLY STABLE MEASURES

We prove that if an action $T$ of a finitely generated group $G$ on a compact metric space $X$ is measure expansive and has the measure shadowing property then it is measure topologically stable. This represents a measurable version of the main result in [4]. Moreover we prove that if $G$ is a finitely generated virtually nilpotent group and there exists $g \in G$ such that $T_g$ is expansive and has the invariant measure shadowing property then $T$ is invariant measure topologically stable. Finally we show that minimal actions approximated by periodic ones have no topologically stable measures.