An elementary proof of the decomposition of measures on the circle group

TL;DR

This paper provides an elementary proof for a theorem on measure decomposition on the circle group, showing that any measure can be expressed as a sum of measures with natural spectrum and a discrete measure.

Contribution

It offers a simplified, elementary proof of a measure decomposition theorem specific to the circle group, expanding understanding of measure structures on compact abelian groups.

Findings

Every measure on the circle group can be decomposed into measures with natural spectrum and a discrete measure.

The proof simplifies existing methods, making the theorem more accessible.

The result applies specifically to the circle group, a fundamental example of a compact abelian group.

Abstract

In this short note we give an elementary proof for the case of the circle group of the theorem of O. Hatori and E. Sato which states that every measure on the compact abelian group $G$ can be decomposed into a sum of two measures with natural spectrum and a discrete measure.