Transportation of random measures not charging small sets

Let $(\xi ,\eta )$ be a pair of jointly stationary, ergodic random measures of equal finite intensity. A balancing allocation is a translation-invariant (equivariant) map $T:{\mathbb{R}}^d\to {\mathbb{R}}^d$ such that the image measure of $\xi$ under $T$ is $\eta$. We show that as soon as $\xi$ 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 $(\xi ,\eta )$, i.e. $T$ is a factor.