Regularity Of The Semi-group Of Regular Probability Measures On Compact Hausdorff Topological Groups

TL;DR

This paper investigates the algebraic regularity of the semi-group of probability measures on compact Hausdorff topological groups, showing it is not regular but can embed into regular semi-groups.

Contribution

It proves that the semi-group of probability measures on compact groups is not algebraically regular, and identifies cases where it can embed into regular semi-groups.

Findings

P(G) is not algebraically regular for compact topological groups

P(G) cannot be a group under convolution

Certain regular semi-groups can contain P(G) as an embedded subset

Abstract

There are many deep results on the structure of REGULAR probability measures $P(G)$ on compact/locally compact, Hausdorff topological groups G. See, for instance, the classic monographs by KR Parthasarathy, Ulf Grenander, A.Mukherjea and Nicolas A.Tserpes. It is known that the set $P(G)$ forms a semi-group under convolution. Wendel in his remarkable paper, proved a basic result regarding support of convolution of two probability measures. Consequently, he established that the semi-group $P(G)$ is not a group. In this short paper, it is proved that for a compact topological group G, the semi-group P(G) of probability measures is not algebraically regular. However, there are concrete regular semi-groups in which P(G) can be embedded.