Regularity of the Semigroup of Regular Probability Measures on Locally Compact Hausdorff Topological Groups in which every element is of finite order

TL;DR

This paper investigates the algebraic regularity of probability measures on locally compact groups where all elements have finite order, extending previous results and providing new characterizations and examples.

Contribution

It extends the characterization of regular elements in the semigroup of probability measures from compact to locally compact groups and describes regular elements in specific countable and uncountable groups.

Findings

Characterization of algebraically regular elements in P(G) for locally compact G

Complete description of regular elements in countable and uncountable groups with finite order elements

Use of Fourier transform techniques for compact Lie groups

Abstract

Let G be a locally compact Hausdorff group in which every element is of finite order, and let P(G) denote the class of all regular probability measures on G. In this note, it is observed that a characterization of algebraically regular elements in certain subsemigroups of P(G) (Theorem 4.1 [11]) for compact G remains true for locally compact G. In addition, a complete description of algebraically regular elements in P(G) has been established when G is countable or uncountable where every proper subgroup is countable. In this case the standing assumption that every element is of finite order is not required. For compact Lie groups, Fourier transform techniques are also used to get more information on P(G). Several concrete examples are provided to illustrate the observations.