Haar measure for compact right topological groups

Abstract:

Compact right topological groups arise in topological dynamics and in other settings. Following H. Furstenberg’s seminal work on distal flows, R. Ellis and I. Namioka have shown that the compact right topological groups of dynamical type always admit a probability measure invariant under the continuous left translations; however, this invariance property is insufficient to identify a unique probability measure (in contrast to the case of compact topological groups). In the present paper, we amplify on the proofs of Ellis and Namioka to show that a right invariant probability measure on the compact right topological group $G$ exists provided $G$ admits an appropriate system of normal subgroups, that it is uniquely determined and that it is also invariant under the continuous left translations. Using Namioka’s work, we show that $G$ has such a system of subgroups if its topological centre contains a countable dense subset, or if it is a closed subgroup of such a group.