  The classic Birkhoff- von Neumann theorem states that the set of doubly
stochastic matrices is the convex hull of the permutation matrices. In this
paper, we study a generalisation of this theorem in the type $II_1$ setting.
Namely, we replace a doubly stochastic matrix with a collection of measure
preserving partial isomorphisms, of the unit interval, with similar properties.
We show that a weaker version of this theorem still holds.
