Abstract: 
We study weak mixing and double ergodicity for nonsingular
actions of locally compact Polish abelian groups. We show that if $T$ is a
nonsingular action of $G$, then $T$ is weakly mixing if and only if for all
cocompact subgroups $A$ of $G$
the action of $T$ restricted to $A$ is weakly mixing. We show that a doubly
ergodic nonsingular action is weakly mixing and
construct an infinite measure-preserving flow that is weakly mixing but not
doubly ergodic. We also construct an infinite measure-preserving
flow whose cartesian square is ergodic.

1. Williams College
   Williamstown, MA 01267, U.S.A.
   and
   Emmanuel College
   Cambridge, CB2 3AP, UK
   
2. Williams College
   Williamstown, MA 01267, U.S.A.
   and
   Department of Mathematics
   University of Texas Austin, TX 78712, U.S.A.
   
3. Department of Mathematics
   Williams College
   Williamstown, MA 01267, U.S.A.
   
4. University of Virginia
   Charlottesville, VA 22903, U.S.A.
   and
   Department of Mathematics
   Princeton University
   Princeton, NJ 08544, U.S.A.
   
5. Harvard University
   Cambridge, MA 02138, U.S.A.
   and
   Department of Mathematics
   Stanford University
   Palo Alto, CA 94305, U.S.A.