M. R. Buneci
Necessary and sufficient conditions for the continuity of a pre-Haar system at a unit with singleton orbit

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 invariance and smoothness conditions called respectively "left invariance" and "continuity". Unlike the case of locally compact group, Haar system on groupoid need not exists.

In this paper we shall consider a locally compact groupoid G, and we shall denote by G_s⁽⁰⁾ the set of units with singleton orbit and by G_s the reduction G|_Gs⁽⁰⁾ of G to G_s⁽⁰⁾. We shall prove that if G admits Haar systems, then the restriction of the range map at G_s is an open map from G_s to G_s⁽⁰⁾. Conversely, we shall prove that if this map is open at every x in G_s, then the continuity condition of a Haar system holds at every unit with singleton orbit.