A short proof of the converse to a theorem of Steinhaus

TL;DR

This paper provides a concise proof of the converse to Steinhaus's theorem, linking the difference set property of measurable sets to absolute continuity of measures.

Contribution

It offers a simplified proof of the characterization of absolutely continuous measures via difference sets, complementing Steinhaus's original result.

Findings

Difference set contains an open interval iff measure is absolutely continuous

Short proof simplifies understanding of measure properties

Clarifies the relationship between difference sets and measure types

Abstract

A result of H. Steinhaus states that any positive Lebesgue measurable set has a property that its difference set contains an open interval around the origin. Y. V. Mospan proved that this result is the characterization of absolutely continuous measure. In this note we give a short proof of it.