  We extend a classical result of de la Harpe and Karoubi, concerning almost
representations of compact groups, to proper groupoids admitting continuous
Haar measure systems. As an application, we establish the existence of
sufficiently many continuous representations of such groupoids on finite-rank
Hilbert bundles locally, and use this fact to prove a new generalization of the
classical Tannaka duality theorem to groupoids in a purely topological setting.
