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We study weak mixing and double ergodicity for nonsingular ac- tions of locally compact Polish abelian groups. We show that if T is a nonsingular action of G, 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.

On weakly mixing and doubly ergodic nonsingular actions