  The weak mean equicontinuous properties for a countable discrete amenable
group $G$ acting continuously on a compact metrizable space $X$ are studied. It
is shown that the weak mean equicontinuity of $(X \times X,G)$ is equivalent to
the mean equicontinuity of $(X,G)$. Moreover, when $(X,G)$ has full measure
center or $G$ is abelian, it is shown that $(X,G)$ is weak mean equicontinuous
if and only if all points in $X$ are uniquely ergodic points and the map $x \to
\mu_x^G$ is continuous, where $\mu_x^G$ is the unique ergodic measure on
$\{\ol{Orb(x)}, G\}$.
