  A function $f\in \BUC(\R^d)$ is said to be weakly* almost periodic, denoted
$f\in\APs(\R^d)$, if there is $g\in\AP(\R^d)$, such that, $\oM(|f-g|)=0$, where
$\BUC(\R^d)$ and $\AP(\R^d)$ are, respectively, the space of bounded uniformly
continuous functions and the space of almost periodic functions, in $\R^d$, and
$\oM(h)$ denotes the mean value of $h$, if it exists. We give a very simple
direct proof of the stochastic homogenization property of the Dirichlet problem
for fully nonlinear uniformly elliptic equations of the form
$F(\om,\frac{x}{\ve},D^2u)=0$, $x\in U$, in a bounded domain $U\subset\R^d$, in
the case where for almost all $\om\in \Om$, the realization $F(\om,\cdot,M)$ is
a weakly* almost periodic function, for all $M\in\S^d$, where $\S^d$ is the
space of $d\X d$ symmetric matrices. Here $(\Om,\mu,\FF)$ is a probability
space with probability measure $\mu$ and $\s$-algebra $\FF$ of $\mu$-measurable
subsets of $\Om$. For each fixed $M\in\S^d$, $F(\om,y,M)$ is a stationary
process, that is, $F(\om,y,M)=\tilde F(T(y)\om,M):= F(T(y)\om,0,M)$, where
$T(y):\Om\to\Om$ is an ergodic group of measure preserving mappings such that
the mapping $(\om,y)\to T(y)\om$ is measurable. Also, $F(\om,y,M)$, $M\in\S^d$,
is uniformly elliptic, with ellipticity constants $0<\l<\gL$ independent of
$(\om,y)\in\Om\X\R^d$. The result presented here is a particular instance of
the general theorem of Caffarelli, Souganidis and Wang, in CPAM 2005. Our point
here is just to show a straightforward proof for this special case, which
serves as a motivation for that general theorem, whose proof involves much more
intricate arguments. We remark that any continuous stationary process verifies
the property that almost all realizations belong to an ergodic algebra, and
that $\APs(\R^d)$ is, so far, the greatest known ergodic algebra on $\R^d$.
