Let {X(n), n=0,1,2,...} denote a Markov chain on a general state space and let f be a nonnegative function. The purpose of this paper is to present conditions which will imply that f(X(n)) tends to 0 a.s., as n tends to infinity. As an application we obtain a result on "synchronisation for random dynamical systems". At the end of the paper we also present a result on  "convergence in distribution" for random dynamical system on complete, separable, metric spaces, a result, which is a generalisation of  a similar result for random dynamical systems on compact, metric spaces.