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 $\Phi$ 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 $\Phi$ are determined. The main properties of these singular value functions, similar to classical singular value functions of measurable operators, are studied.