  Let $\lambda$ be a probability measure on $\mathbb T^{n-1}$ where $n=2$ or 3.
Suppose $\lambda$ is invariant, ergodic and has positive entropy with respect
to the linear transformation defined by a hyperbolic matrix. We get a measure
$\mu $ on $SL_n(\mathbb Z)\backslash SL_n(\mathbb R)$ by putting $\lambda$ on
some unstable horospherical orbit of the right translation of
$a_t=\mathrm{diag}(e^t,..., e^t, e^{-(n-1)t})$ $(t>0)$. We prove that if the
average of $\mu$ with respect to the flow $a_t$ has a limit, then it must be a
scalar multiple of the probability Haar measure. As an application we show that
if the entropy of $\lambda$ is large, then Dirichlet's theorem is not
improvable $\lambda$ almost surely.
