The key ingredient in establishing this result is the study of
the limiting distributions of certain translates of a
homogeneous measure. We show that if in addition $G$ is
generated by unipotent elements then there exists $a\in F$ such
that the following holds: Let $U\subset F$ be the subgroup
generated by all unipotent elements of $F$, $x\in L/\Lambda$,
and $\lambda$ and $\mu$ denote the Haar probability measures on
the homogeneous spaces $\cl{Ux}$ and $\cl{Gx}$, respectively
(cf.~Ratner's theorem). Then $a^n\lambda\to\mu$ weakly as
$n\to\infty$.
We also give an algebraic characterization of algebraic
subgroups $F<\SL_n(\R)$ for which all orbit closures are finite
volume almost homogeneous spaces, namely {\it iff\/} the
smallest observable subgroup of $\SL_n(\R)$ containing $F$ has
no nontrivial characters defined over $\R$.