A note on conservative measures on semigroups (Q1186349)

Let \(S\) be a topological metric semigroup and let \(\mu\) be a nonnegative Borel measure on \(S\). Assume further that right translations on \(S\), the maps \(t_x: s\mapsto sx\), are closed. The paper addresses the question to what extent some ergodic theoretical or probabilistic invariance conditions on \(\mu\) impose algebraic properties on the support \(F\) of \(\mu\). Let \(\mu\) be a conservative measure, i.e., for all Borel sets \(B\), \(Bx^{-1}\supset B\) implies \(\mu(Bx^{-1}\setminus B)=0\). (Here \(Bx^{-1}\) is a notation for \(\{s\mid sx\in B\}\).) Assume further that \(\mu\) is bounded. Then the author shows that, necessarily, \(F\) generates a left group (a semigroup which is left simple and right cancellative). If \(\mu\) is only \(\sigma\)-finite the same result may be obtained for separable \(S\). For a general discussion of the interplay between invariance and algebraic properties, cf. the books by \textit{A. Mukherjea} and the author [Measures on topological semigroups: Convolution products and random walks. Lect. Notes Math. 547. Berlin etc.: Springer Verlag (1976; Zbl 0342.43001)] and \textit{J. F. Berglund} and \textit{K. H. Hofmann} [Compact semitopological semigroups and weakly almost periodic functions. Lect. Notes Math. 42. Berlin etc.: Springer Verlag (1967; Zbl 0155.18702)].