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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. 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