Haar systems on locally compact groupoids

Abstract

Haar systems are generalizations of Haar measures on groups to groupoids. Naturally, important research directions in the field try to generalize the well known existence of a Haar measure on a locally compact group to the existence of Haar systems in different groupoid settings. The groupoid case differs significantly from the group case, evidenced by a result of Deitmar, showing that non-existence is possible even for compact groupoids. We first present the classical theory of locally compact groups and Haar Measures on them. We motivate our investigation by constructing full C∗-algebras on locally compact groups, which uses the existence of Haar measures. Then, we cover the theory of locally compact groupoids and present Renault's result that provides a complete characterization of the existence of Haar systems for the r-discrete locally compact groupoid setting, which are precisely the ones where the range map is a local homeomorphism. We present a question from Williams that investigates if the open range map assumption is redundant for second countable, locally compact and transitive groupoids. Finally, we present Buneci's counter-example that answers this question in the negative.