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.
In this scenario, M is a von Neumann regular ring: within the closure of |A|'s range, there exists a measurable inverse B, and then T = BV* defines a measurable operator with ATA = A.