Transportation of random measures not charging small sets

Abstract

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.