On the Existence of Balancing Allocations and Factor Point Processes

In this article, we show that every stationary random measure on $\mathbb R^d$ that is essentially free (i.e., has no symmetries a.s.) admits a point process as a factor (i.e., as a measurable and translation-equivariant function of the measure). As a result, we improve the results of Last and Thorisson (2022) on the existence of a factor balancing allocation between ergodic pairs of stationary random measures $\Phi$ and $\Psi$ with equal intensities. In particular, we prove that such an allocation exists if $\Phi$ is diffuse and either $(\Phi,\Psi)$ is essentially free or $\Phi$ assigns zero measure to every $(d-1)$-dimensional affine hyperplane. The main result is deduced from an existing result in descriptive set theory, that is, the existence of lacunary sections. We also weaken the assumption of being essentially free to the case where a discrete group of symmetries is allowed.