Nonsingular transformations that are ergodic with isometric coefficients and not weakly doubly ergodic

We study two properties of nonsingular and infinite measure-preserving ergodic systems: weak double ergodicity, ergodicity with isometric coefficients. show that there exist transformations are coefficients but not weakly doubly ergodic; hence these notions equivalent for measure. also give type III? examples such systems, 0<??1. prove under certain hypotheses, systems mixing along the way we an example a uniformly rigid topological dynamical system sequence (ni) is measure theoretically any finite