In this case M is a von Neumann regular ring: for on the closure of its image |A| has a measurable inverse B and then T = BV* defines a measurable operator with ATA = A.
Within this context, M is a von Neumann regular ring; in the closure of |A|'s image, there's a measurable inverse B, which allows T = BV* to define a measurable operator satisfying ATA = A.