Transportation of random measures not charging small sets

TL;DR

This paper proves that for certain stationary random measures that do not charge small sets, there exists a translation-invariant allocation map that balances them and depends measurably on the measures.

Contribution

It establishes the existence of a measurable, translation-invariant balancing allocation for random measures not charging small sets.

Findings

Existence of a factor balancing allocation when measures do not charge small sets

Allocation depends measurably only on the measures

Applicable to jointly stationary, ergodic random measures of equal finite intensity

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.