The singular value function, associated with a Maharam trace

Let M be a finite von Neumann algebra, let $S(M)$ be the $*$-algebra of measurable operators affiliated with M. Maharam traces Φ on a von Neumann algebra M with values in complex Dedekind complete vector lattices are considered. The singular value function of operators from $S(M)$, associated with such a trace Φ are determined. The main properties of these singular value functions, similar to classical singular value functions of measurable operators, are studied.