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By Jean Renault

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12. D e f i n i t i o n : An i n v a r i a n t measure is a q u a s i - i n v a r i a n t measure whose modular function is equal to 1. 13. D e f i n i t i o n : ([55] p. 448). Suppose G p r i n c i p a l . 14, D e f i n i t i o n : i f i t is properly ergodic and contains no i n v a r i a n t measure. A principal groupoid is of type I i f i t has type I q u a s i - o r b i t s only. The notion of i n v a r i a n t measure can be extended as f o l l o w s . Before g i v i n g the d e f i n i t i o n , recall that ZI(G, IR) is the group of continuous homomorphisms of G into IR.

In f a c t , our construction closely fellows [23]: the space Cc(G) of continuous functions with compact support is made into a * -algebra and endowed with the smallest C*-norm making i t s representations continuous ; C*(G) is i t s completion. The d e t a i l s are in Section 1. We r e f r a i n from putting any modular function in the d e f i n i t i o n of the i n v o l u t i o n , since none is a v a i l a b l e . However, this is a minor change and the C * - a l g e b r a so obtained is isomorphic to the usual one in the case of a transformation group.

We have shown R (c) c R ( c ' ) , R (c'). The e q u a l i t y T(c) = T(c') (iii) hence R (c) = r e s u l t s from the d e f i n i t i o n o f a T-set. Clear. D. S i m i l a r proofs y i e l d s i m i l a r r e s u l t s about the asymptotic range of a cocycle at a u n i t u. 6. P r o p o s i t i o n : Let G, A, c be as before and u ~ Go . Then (i) (ii) (iii) (iv) (v) RU(c) R~(c) : RU(c). R~(c) is a closed subsemi-group o f A R~(c) depends only on the class R~(e) = of c. {e} I f u ~ v, RU(c) = RV(c).

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