  We consider a minimal equicontinuous action of a finitely generated group $G$
on a Cantor set $X$ with invariant probability measure $\mu$, and stabilizers
of points for such an action. We give sufficient conditions under which there
exists a subgroup $H$ of $G$ such that the set of points in $X$ whose
stabilizers are conjugate to $H$ has full measure. The conditions are that the
action is locally quasi-analytic and locally non-degenerate. An action is
locally quasi-analytic if its elements have unique extensions on subsets of
uniform diameter. The condition that the action is locally non-degenerate is
introduced in this paper. We apply our results to study the properties of
invariant random subgroups induced by minimal equicontinuous actions on Cantor
sets and to certain almost one-to-one extensions of equicontinuous actions.
