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Locally Compact Groupoids and Haar Systems
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abelian group amenable ample semi-group approximate identity asymptotic range automorphism group c e Z G,A C*-algebra Cartan subalgebra cocycle compact open G-sets continuous 2-cocycle continuous function continuous G-set defined definition denoted dimension group dimension range dvQ(x dxu(x dxu(y dy(u elementary groupoid equivalence relation ergodic exists f e C G f e CC(G finite function on G G x G given groupoid G groupoid with Haar Haar measure Hence Hilbert space homomorphism induced measure inductive limit topology inverse semi-group isometry isomorphism left Haar system Lemma let c e Let G locally compact groupoid locally compact space Math measure on G Min(c modular function multiplier algebra non-negative non-singular Borel G-sets norm open set open subset probability measure Proof Proposition quasi-invariant measure r-discrete groupoid Radon-Nikodym derivative regular representation second countable sequence subset of G theorem topological groupoid transformation group u e G unit space x e G