Transportation of stationary random measures not charging small sets

Let (, rl) be a pair of jointly stationary, ergodic random measures of equal finite intensity. A balancing allocation is a translation-invariant (equivariant) random map T : IIBd -> IIBd, defined on the same probability space, such that the image measure of under T is rl. We show that as soon as does not charge small sets, i.e. does not give mass to ( d -1)-rectifiable sets, there is always a balancing allocation T which is measurably depending only on (, rl), i.e. T is a factor of (,rl).