  In the present paper we investigate $L_0$-valued states and Markov operators
on $ C^*$-algebras over $L_0$. In particular, we give representations for
$L_0$-valued state and Markov operators on $ C^*$ algebras over $L_0$,
respectively, as measurable bundles of states and Markov operators. Moreover,
we apply the obtained representations to study certain ergodic properties of $
C^*$-dynamical systems over $L_0$.
