An embedding theorem for uniform measures on topological groups

TL;DR

This paper generalizes the embedding theorem for uniform measures on topological groups, showing that natural mappings between spaces of uniform measures are injective and topologically embedding, extending previous results.

Contribution

It extends the embedding theorem to spaces of uniform measures on topological groups and their quotients, providing new insights into their structure and relationships.

Findings

Natural mappings between uniform measure spaces are injective.

Restrictions to positive cones are topological embeddings.

Results apply to quotients of topological groups by neutral subgroups.

Abstract

The embedding theorem of Roelcke and Dierolf for the completions of four standard uniform structures on topological groups and their quotients holds more generally for spaces of uniform measures. The natural mappings between the four spaces of uniform measures on a topological group are injective and their restrictions to positive cones are topological embeddings. The same holds for spaces of uniform measures on the quotient of a topological group by a neutral subgroup.