Regularity of the Semigroup of Regular Probability Measures on Locally Compact Hausdorff Topological Groups

Let G be a locally compact Hausdorff group, and let P(G) denote the class of all regular probability measures on G. It is well known that P(G) forms a semigroup under the convolution of measures. In this paper, we prove that P(G) is not algebraically regular in the sense that not every element has a generalized inverse. Additionally, we attempt to identify algebraically regular elements in some exceptional cases. Several supporting examples are provided to justify these assumptions.