  In this paper, for a discontinuous skew-product transformation with the
integrable observation function, we obtain uniform ergodic theorem and
semi-uniform ergodic theorem. The main assumptions are that discontinuity sets
of transformation and observation function are neglected in some
measure-theoretical sense. The theorems extend the classical results which have
been established for continuous dynamical systems or continuous observation
functions. Meanwhile, on the torus $\mathbb{T}^{d}$ with special rotation, we
prove the pointwise convergence of multiple ergodic average $\disp \f 1 N
\sum_{n=0}^{N-1} f_{1}(R_{\alpha}^{n}x)f_{2}(R_{\alpha}^{2n}x)$ in
$\mathbb{T}^{d}$.
