We prove that the concepts of completely mixing, mixing, and weakly mixing probability measures on a locally compact group in the class [SIN] are mutually equivalent. A probability measure on such a group is completely mixing if and only if it is ergodic and aperiodic. For measures that are not necessarily aperiodic ergodicity is shown to be equivalent to a mixing-like property which we call almost mixing. Some results on when a probability measure on a [SIN] group is ergodic are also developed. Our methods are based on the theory of boundaries of random walks.

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Keywords ergodicity, locally compact groups, mixing, Probability measures, random walks
Persistent URL dx.doi.org/10.1023/B:JOTP.0000040297.84097.57
Journal Journal of Theoretical Probability
Jaworski, W. (2004). Ergodic and mixing probability measures on [SIN] groups. Journal of Theoretical Probability, 17(3), 741–759. doi:10.1023/B:JOTP.0000040297.84097.57