Generalized notion of a trace for von Neumann algebras

On a von Neumann algebra $M$, we consider traces with values in the algebra $L^0$ of measurable complex-valued functions. We show that every faithful normal $L^0$-valued trace on $M$ generates an $L^0$-valued metric on the algebra of measurable operators that are affiliated with $M$. Moreover, convergence in this metric coincides with local convergence in measure.