  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.
