Crossed Products on the Flow of Groupoid with Quasi-Invariant Measures

Let G be a second countable groupoid with Haar system {λ~u}, A be an abelian groupwhich left invariant acts on G. Then we have a C*-dynamic system (C*(G), A, β). In this paper wehave studied the existence of quasi-invariant measure with certain properties; using these measuressome important results about crossed products and groupoid C*-algebras have been obtained.