On norm almost periodic measures

TL;DR

This paper investigates the properties of norm almost periodic measures on locally compact Abelian groups, establishing equivalences with other forms of almost periodicity for different classes of measures.

Contribution

It introduces new equivalences between norm almost periodicity and other types of almost periodicity, expanding understanding of measure behavior on Abelian groups.

Findings

Norm almost periodicity is equivalent to equi-Bohr almost periodicity of convolutions.

For absolutely continuous measures, norm almost periodicity equals Stepanov almost periodicity of the density.

Provides characterizations linking measure properties to function almost periodicity.

Abstract

In this paper, we study norm almost periodic measures on locally compact Abelian groups. First, we show that the norm almost periodicity of $\mu$ is equivalent to the equi-Bohr almost periodicity of $\mu*g$ for all $g$ in a fixed family of functions. Then, we show that, for absolutely continuous measures, norm almost periodicity is equivalent to the Stepanov almost periodicity of the Radon--Nikodym density.