  Conley in \cite{Con} constructed a complete Lyapunov function for a flow on
compact metric space which is constant on orbits in the chain recurrent set and
is strictly decreasing on orbits outside the chain recurrent set. This
indicates that the dynamical complexity focuses on the chain recurrent set and
the dynamical behavior outside the chain recurrent set is quite simple. In this
paper, a similar result is obtained for random dynamical systems under the
assumption that the base space $(\Omega,\mathcal F,\mathbb P)$ is a separable
metric space endowed with a probability measure. By constructing a complete
Lyapunov function, which is constant on orbits in the random chain recurrent
set and is strictly decreasing on orbits outside the random chain recurrent
set, the random case of Conley's fundamental theorem of dynamical systems is
obtained. Furthermore, this result for random dynamical systems is generalized
to noncompact state spaces.
