System of imprimitivity

Summary

A system of imprimitivity is a unitary representation of a group together with a projection-valued measure on a space on which the group acts. The group must move the projections in the same way as it moves the sets. Induced representations come with a canonical system of imprimitivity. The Mackey imprimitivity theorem says that, for transitive actions, every system of imprimitivity is of this form. This is the basis of the Mackey machine for classifying representations of group extensions.