  We prove existence of (at most denumerable many) absolutely continuous
invariant probability measures for random one-dimensional dynamical systems
with asymptotic expansion. If the rate of expansion (Lyapunov exponents) is
bounded away from zero, we obtain finitely many ergodic absolutely continuous
invariant probability measures, describing the asymptotics of almost every
point.
  We also prove a similar result for higher-dimensional random non-uniformly
expanding dynamical systems. The results are consequences of the construction
of such measures for skew-products with essentially arbitrary base dynamics and
asymptotic expansion along the fibers. In both cases our method deals with
either critical or singular points for the random maps.
