Transportation of diffuse random measures on R^d

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