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Let $(\xi,\eta)$ be a pair of jointly stationary, ergodic random measures ofequal finite intensity. A balancing allocation is a translation-invariant(equivariant) map $T:\mathbb{R}^d\to\mathbb{R}^d$ such that the image measureof $\xi$ under $T$ is $\eta$. We show that as soon as $\xi$ does not chargesmall sets, i.e.\ does not give mass to $(d-1)$-rectifiable sets, there isalways a balancing allocation $T$ which is measurably depending only on$(\xi,\eta)$, i.e. $T$ is a factor.

Transportation of random measures not charging small sets