Transportation of diffuse random measures on Rd

We consider two jointly stationary and ergodic random measures xi and n on Rd with equal finite intensities, assuming xi to be diffuse (non-atomic). An allocation is a random mapping taking Rd to Rd ? {infinity} in a translation invariant way. We construct allocations transporting the diffuse xi to arbitrary n, under the mild condition of existence of an 'auxiliary' point process which is needed only in the case when n is diffuse. When that condition does not hold we show by a counterexample that an allocation transporting xi to n need not exist.