By Jean Renault
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Therefore, i f ¢ is a p o s i t i v e measurable function such that ~(u) : f~(X) f@d~u= 1 f o r u in U, the function 6 defined in r ( s ) by ~(X) dZu(X ) has the required property. u. e. u and any positive measurable f, ~f(y) 6 od(y) d~U(y) = = ~f(y) ~(x) #Ix) d~d(y)(X) d~U(Y), #f(y) ~(xy) = l(~f(Y) ~(xy) O(xy) dZu(X) d~U(y), #(xy) dzU(y)) dZu(X), = #f(y) ~(Y) (#@(xy)dZu(X)) dxU(Y), = #f(y) #(y) dZU(y) ; therefore 31 If(y) 6od(y) dv(y) = ~f(y) ~(y) dv(y), = I f ( y s - I ) dv(y). D. Proposition : Let u be a q u a s i - i n v a r i a n t measure, v i t s induced measure and s a measurable G-set.
I t is known ( e . g . 2 derivative iff it theorem 4 . 3 , page 276) t h a t ~ is q u a s i - i n v a r i a n t in the is q u a s i - i n v a r i a n t under the group S. The h o r i z o n t a l Radon-Nikodym A(u,s) is the usual Radon-Nikodym cocycle o f the a c t i o n . 20 t h a t i t s modular f u n c t i o n i s ~(s)/A(u,s). Case o f an r - d i s c r e t e groupoid. Since the counting measure ~u is i n v a r i a n t under any G-set s, the v e r t i c a l Radon-Nikod3an 6 ( u , s ) is i d e n t i c a l l y equal to 1, independently of any measure u on GO.
10, e T(c). 5. ( i ) . xoC ~ B I ( G , T ) , that is, 42 Recall t h a t , given a groupoid G, a group A and c ~ Z 1 (G,A), one may d e f i n e the skew-product G(c), whose u n d e r l y i n g space is G x A and u n i t space is GO x A. I f G and A are t o p o l o g i c a l and c continuous, G(c) w i t h the t o p o l o g y o f G x A i s a t o p o l o g i c a l groupoid. Note t h a t i f G has an open range map [resp. a cover o f continuous G - s e t s ] , then so has G(c). The f o l l o w i n g c h a r a c t e r i z a t i o n o f the asymptotic range o f a cocycle i n terms o f the skew-product is taken from Pedersen  8 .