The Mackey Machine for Crossed Products by Regular Groupoids. I

We first describe a Rieffel induction system for groupoid crossed products. We then use this induction system to show that, given a regular groupoid $G$ and a dynamical system $(A,G,\alpha)$, every irreducible representation of $A\rtimes G$ is induced from a representation of the group crossed product $A(u)\rtimes S_u$ where $u\in G\unit$, $A(u)$ is a fibre of $A$, and $S_u$ is a stabilizer subgroup of $G$.