Convolution algebras for topological groupoids with locally compact fibres

The aim of this paper is to introduce various convolution algebras associated with a topological groupoid with locally compact fibres. Instead of working with continuous functions on $G$, we consider functions having a uniformly continuity property on fibres. We assume that the groupoid is endowed with a system of measures (supported on its fibres) subject to the "left invariance" condition in the groupoid sense.