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184 P. Carrillo Rouse• mW G .2/ ! G , called the product map (where G .2/ Df.; / 2 G G W s./ Dr./g),• uW G .0/ ! G the unit map, and• i W G ! G the inverse mapsuch that, if we write m.; / D , u.x/ D x and i./ D 1 ,wehave1. . ı/ D . / ı for all ; ; ı 2 G when this is possible,2. x D and x D for all ; 2 G with s./ D x and r./ D x,3. 1 D u.r.// and 1 D u.s.// for all 2 G ,4. r. / D r./ and s. / D s./.Generally, we denote a groupoid by G G .0/ where the parallel arrows are the sourceand target maps and the other maps are given.Now, a Lie groupoid is a groupoid in which every set and map appearing inthe last definition is C 1 (possibly with borders), and the source and target mapsare submersions. For A; B subsets of G .0/ we use the notation GA B for the subsetf 2 G W s./ 2 A; r./ 2 Bg.Throughout this paper, G G .0/ is a Lie groupoid. We recall how to define analgebra structure in Cc1 .G / using smooth Haar systems.Definition 2.2. A smooth Haar system over a Lie groupoid consists of a family ofmeasures x in G x for each x 2 G .0/ such that• for 2 Gx y we have the following compatibility condition:ZZf./d x ./ D f. ı / d y ./;G x G y• for each f 2 Cc 1.G / the map Zx 7! f./d x ./G xbelongs to C 1 c .G .0/ /.A Lie groupoid always possesses a smooth Haar system. In fact, if we fix a smooth(positive) section of the 1-density bundle associated to the Lie algebroid we obtain asmooth Haar system in a canonical way. The advantage of using 1-densities is that themeasures are locally equivalent to the Lebesgue measure. We suppose for the rest ofthe paper an underlying smooth Haar system given by 1-densities (for complete details