  This paper proves various results concerning non-ergodic actions of locally
compact groups and particularly Borel cocycles defined over such actions. The
general philosophy is to reduce the study of the cocycle to the study of its
restriction to each ergodic component of the action, while being careful to
show that all objects arising in the analysis depend measurably on the ergodic
component. This allows us to prove a version of the superrigidity theorems for
cocycles defined over non-ergodic actions.
