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We consider two jointly stationary and ergodic random measures $\xi$ and$\eta$ on $\mathbb{R}^d$ with equal finite intensities, assuming $\xi$ to bediffuse. An allocation is a random mapping taking $\mathbb{R}^d$ to$\mathbb{R}^d\cup\{\infty\}$ in a translation invariant way. We constructallocations transporting the diffuse $\xi$ to arbitrary $\eta$, under the mildcondition of existence of an `auxiliary' point process which is needed only inthe case when $\eta$ is diffuse. When that condition does not hold we show by acounterexample that an allocation transporting $\xi$ to $\eta$ need not exist.

Transportation of diffuse random measures on $\mathbb{R}^d$