  It has been shown by Le Jan that, given a memoryless-noise random dynamical
system together with an ergodic distribution for the associated Markov
transition probabilities, if the support of the ergodic distribution admits
locally asymptotically stable trajectories, then there is a random attracting
set consisting of finitely many points, whose basin of forward-time attraction
includes a random full measure open set. In this paper, we present necessary
and sufficient conditions for this attracting set to be a singleton; our result
does not require the state space to be compact, but holds on general Lusin
metric spaces.
