Transportation of random measures not charging small sets

2023/03/01 by Martin Huesmann, Huesmann, Martin, Bastian Müller +1
Mathematics · #FOS: Mathematics #Mathematical Dynamics and Fractals #Point processes and geometric inequalities #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2303.00504

openalex publication_date 2023/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

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