On the convolution algebras ofH-invariant measures

Abstract

1. Introduction. We denote the Banach space of bounded regular Borel measures and the totality of probability measures on a locally compact (Hausdorff) space X by M(X) and P{X), respectively. Beside the norm topology, M(X) may be equipped with the weak, weak* and vague topologies, which are the topologies of pointwise convergence on C*(X), C0(X) and K(X), respectively, where Cb(X) denotes the totality of bounded continuous functions on X, C0(X) and K(X) the subspaces of functions vanishing at and functions with compact supports, respectively. In P(X), the weak, weak* and vague topologies coincide (p. 59, [21; [71). Let S be a locally compact semigroup, then M(S) is a Banach algebra and P(S) a topological (Hausdorff) semigroup under the convolution *. We refer to [7] for the continuity of * in the weak, weak* and vague topologies.