  In this paper, we introduce topologically stable points, $\beta$-persistent
points, $\beta$-persistent property, $\beta$-persistent measures and almost
$\beta$-persistent measures for first countable Hausdorff group actions of
compact metric spaces. We prove that the set of all $\beta$-persistent points
is measurable and it is closed if the action is equicontinuous. We also prove
that the set of all $\beta$-persistent measures is a convex set and every
almost $\beta$-persistent measure is a $\beta$-persistent measure. Finally, we
prove that every equicontinuous pointwise topologically stable first countable
Hausdorff group action of a compact metric space is $\beta$-persistent. In
particular, every equicontinuous pointwise topologically stable flow is
$\beta$-persistent.
