  We show that for certain classes of actions of Z^d, d >= 2, by automorphisms
of the torus any measurable conjugacy has to be affine, hence measurable
conjugacy implies algebraic conjugacy; similarly any measurable factor is
algebraic, and algebraic and affine centralizers provide invariants of
measurable conjugacy. Using the algebraic machinery of dual modules and
information about class numbers of algebraic number fields we consruct various
examples of Z^d-actions by Bernoulli automorphisms whose measurable orbit
structure is rigid, including actions which are weakly isomorphic but not
isomorphic. We show that the structure of the centralizer for these actions may
or may not serve as a distinguishing measure-theoretic invariant.
