A uniformly spread measure criterion

TL;DR

This paper proves that measures in Euclidean space that are uniformly close under all shifts are also close to the Lebesgue measure, establishing a stability property of measures.

Contribution

It introduces a criterion based on uniform spread and shift closeness to characterize measures near Lebesgue measure.

Findings

Measures with uniformly close shifts are close to Lebesgue measure.

The criterion provides a new way to identify Lebesgue-like measures.

The result has implications for measure stability and approximation.

Abstract

We prove that if all shifts of a measure in the Euclidean space are close in a sense to each other, then this measure is close to the Lebesgue one.