Semigroups with invariant Radon measures

Gowrisankaran, Chandra

doi:10.1090/S0002-9939-1973-0313480-0

Semigroups with invariant Radon measures
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Proc. Amer. Math. Soc. 38 (1973), 400-404 Request permission

Abstract:

Let $S$ be a commutative semigroup which is a topological space such that the translations are both continuous and open maps. The main result states that if (1) either $S$ is Suslin such that there is at least one point of continuity for the semigroup mapping $S \times S \to S$ or $S$ is polish and (2) $\exists$ a nontrivial Radon measure on $S$ such that $\mu (V) = \mu (x + V)$ for $V$ open $\subset S$ and $x \in S$, then $S$ can be embedded as an open subsemigroup of a locally compact group. It is also shown that if $S$ is polish and a cancellation semigroup then $S$ can be embedded as an open subsemigroup of a group.

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