  In the present paper, we study the distribution of the return points in the
fibers for a RDS (random dynamical systems) nonuniformly expanding preserving
an ergodic probability, we also show the abundance of nonlacunarity of
hyperbolic times that are obtained along the orbits through the fibers. We
conclude that any ergodic measure with positive Lyapunov exponents satisfies
the nonuniform specification property between fibers. As consequences, we prove
that any expanding measure is the limit of probability measure whose measures
of disintegration on the fibers are supported by a finite number of return
points and we prove that the average of the measures on the fibers
corresponding to a disintegration, along an orbit in the base dynamics is the
limit of Dirac measures supported in return orbits on the fibers.
