The multiplicative ergodic theorem is valid under an integrability condition on the linearized flow with respect to an invariant measure. We investigate the case were the flow is generated by . . . a s:ochas:ic diKeien:ia! ecjna:ion and give a criterion in t e r m ~f !he :' ecto: fie!& ar,d the (generally non-adapted) invariant measure assuring the validity of the integrability condition.