We study the weighted generalized empirical distribution $H_n$ which is of U-statistic structure and obtain that the rate of the almost sure convergence of $sup_{o< u< 1}\{|H_n(u)-u|/\sqrt{u(1-u)}\}$ is $o(n^{-1/2}(\log \;n)^{1/2})$, where $H_n(\cdot)$ is defined in section 1. The finding somehow fills a gap in the asymptotic theory of the generalized empirical processes.