HAAR SYSTEMS AND TOPOLOGIES ON GROUPOIDS

For developing an algebraic theory of functions on a locally compact groupoid, one needs an analogue of Haar measure on locally compact groups. This analogue is a system of measures, called Haar system, subject to suitable and smoothness conditions called respectively left invariance and continuity. Unlike the case of locally compact group, Haar system on groupoid need not exists, and if it does, it will not usually be unique. However on locally compact second countable groupoids one can construct systems of measures satisfying left invariance condition. But the assumption has topological consequences for groupoid. It entails that the range map (and hence the domain map) is open (Proposition I. 4 (14)). A. K. Seda has proved that the continuity condition is crucial in construction of the groupoid C * -algebra ((13)). In this paper by a pre-Haar system we shall mean of system of Borel measures satisfying left invariance condition. The continuity condition is a consequence of left invariance condition for any locally compact second countable transitive groupoid. Let G be a second countable locally compact groupoid (not necessarily transitive) with a given a pre-Haar system. We shall prove that there is a reduction G|L such that we can replace the topology induced from G on G|L with a locally compact topology having the property that the pre-Haar system is Haar system on G|L. The reduction G|L is an inessential reduction with respect to each transitive measure induced by the pre-Haar system. If all orbits of G are locally closed, then the reduction G|L can be taken to be the entire G. Therefore, replacing the topology on G (and eventually passing to a reduction), any pre-Haar system can be viewed as a Haar system, and consequently, one can construct a C * -algebra associated with the pre-Haar system.