  In this paper, we provide a generalized version of the Voiculescu theorem for
normal operators by showing that, in a von Neumann algebra with separable
pre-dual and a faithful normal semifinite tracial weight $\tau$, a normal
operator is an arbitrarily small $(\max\{\|\cdot\|,
\Vert\cdot\Vert_{2}\})$-norm perturbation of a diagonal operator. Furthermore,
in a countably decomposable, properly infinite von Neumann algebra with a
faithful normal semifinite tracial weight, we prove that each self-adjoint
operator can be diagonalized modulo norm ideals satisfying a natural condition.
