This paper proves various results concerning nonergodic 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 nonergodic actions.

This research was partially supported by the National Science Foundation (grants DMS-0226121, DMS-0100438, and DMS-0204576). The first author was also partially supported by a PSC-CUNY grant.