  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.
