An ergodic theorem for quantum diagonal measures

We prove the quantum version of an ergodic result of H. Furstenberg relative to noninvariant measures. The natural setting will be the case of the "quantum diagonal measure" relative to the product measure. Even if in all the interesting situations such diagonal measures are neither invariant nor normal with respect to the corresponding product ones, we still provide an ergodic theorem for them, generalizing the classical case. As a natural application, we are able to prove the entangled ergodic theorem in some interesting situations out of the known ones, that is when the unitary is not almost periodic, or when the involved operators are not compact.