Uniformly spread measures and vector fields

TL;DR

This paper proves the equivalence of two different concepts of uniform spreading of measures in Euclidean space, one based on transportation and the other on vector fields, clarifying their relationship.

Contribution

It establishes the equivalence between two notions of uniform spreading of measures, linking transportation-based and vector field-based approaches.

Findings

Two notions of uniform spreading are shown to be equivalent.

Provides a unified understanding of measure spreading in Euclidean space.

Bridges concepts of transportation and vector fields in measure theory.

Abstract

We show that two different ideas of uniform spreading of locally finite measures in the d-dimensional Euclidean space are equivalent. The first idea is formulated in terms of finite distance transportations to the Lebesgue measure, while the second idea is formulated in terms of vector fields connecting a given measure with the Lebesgue measure.