Amenability and ergodic properties of topological groups: from Bogolyubov onwards

TL;DR

This paper revisits the concepts of amenability and ergodicity in topological groups, emphasizing their historical development beyond locally compact groups through classical works from the mid-20th century.

Contribution

It highlights the presence of amenability and ergodic properties in non-locally compact topological groups, tracing their origins in foundational mathematical literature.

Findings

Historical perspective on amenability and ergodicity

Extension of concepts to non-locally compact groups

Connections to classical mathematical works

Abstract

The purpose of this expository article is to revisit the notions of amenability and ergodicity, and to point out that they appear for topological groups that are not necessarily locally compact in articles by Bogolyubov (1939), Fomin (1950), Dixmier (1950), and Rickert (1967).