Definition 0.1. Let G_lc be a locally compact topological groupoid endowed with a Haar system

A representation of G_lc together with its associated Haar system ν is defined as a triple

where:

• μ is a quasi-invariant measure defined on U_Glc;
• U_Glc ∗ℋ is an analytical fibered Hilbert space, or Hilbert bundle, over U_Glc; and
• 

  is a Borel groupoid morphism whose restriction to U_Glc is the identification map. Thus the unit space

  

  is identified via L with U_Glc.

For x ∈ G_lc, write

where

is a Hilbert-space isomorphism.