  Investigating the direct integral decomposition of von Neumann algebras of
bounded module operators on self-dual Hilbert W*-moduli an equivalence
principle is obtained which connects the theory of direct disintegration of von
Neumann algebras on separable Hilbert spaces and the theory of von Neumann
representations on self-dual Hilbert {\bf A}-moduli with countably generated
{\bf A}-pre-dual Hilbert {\bf A}-module over commutative separable W*-algebras
{\bf A}. Examples show posibilities and bounds to find more general relations
between these two theories, (cf. R. Schaflitzel's results). As an application
we prove a Weyl--Berg--Murphy type theorem: For each given commutative
W*-algebra {\bf A} with a special approximation property (*) every normal
bounded {\bf A}-linear operator on a self-dual Hilbert {\bf A}-module with
countably generated {\bf A}-pre-dual Hilbert {\bf A}-module is decomposable
into the sum of a diagonalizable normal and of a ''compact'' bounded {\bf
A}-linear operator on that module.
