  Let $X$ be a compact metrizable space, and let $\Delta$ be a closed set of
Borel probability measures on $X$. We study the small boundary property of the
pair $(X, \Delta)$. In particular, it is shown that $(X, \Delta)$ has the small
boundary property if it has a restricted version of property Gamma.
  As an application, it is shown that, if $A$ is the crossed product C*-algebra
$\mathrm{C}(X)\rtimes\mathbb Z^d$, where $(X, \mathbb Z^d)$ is a free minimal
topological dynamical system, or if $A$ is an AH algebra with diagonal maps,
then, $A$ is $\mathcal Z$-stable if the set of extreme tracial states is
compact, regardless of its dimension.
