  The article presents a new perspective on the isomorphism problem for
non-ergodic measure-preserving dynamical systems with discrete spectrum which
is based on the connection between ergodic theory and topological dynamics
constituted by topological models. By first solving the isomorphism problem for
a certain class of topological dynamical systems, it is shown that the
measure-preserving case can in fact be deduced from the topological one via the
construction of topological models. As a byproduct, a new characterization of
mean ergodicity for topological dynamical systems is obtained.
