On the continuity of Haar measure on topological groupoids

Abstract

It is shown that continuity of a family of invariant (Haar) measures on a topological groupoid G G is equivalent to the continuity of the implied convolution product f ∗ g f * g for all pairs of functions f f and g g . An example is given of a groupoid which admits no (continuous) Haar measure. It results, therefore, that the usual C ∗ {C^ * } -algebra associated with a Haar measure on G G cannot, in general, be constructed. Some remarks are included concerning the construction of Haar measures on the holonomy groupoid of a foliated manifold.