Mildly Mixing Actions of Locally Compact Groups

Let G be a locally compact second countable group, and let (g, x) → gx be a properly ergodic nonsingular action of G on a probability space (X, I, μ). This action is called mildly mixing if, for every properly ergodic non-singular action of G on a probability space (Y, I, v), the product action of G on (X × Y,I×v, μ×ν) is again ergodic. In this paper the notion of mild mixing is linked with certain ergodic proterties of a natural G-action on a space of functions on X. If G is abelian, these methods yield a new cirterion for the existence of a finite G-invariant measure equivalent to μ. We porve furthermore that a given G-action can only be mildly mixing if it is finite measure preserving. The last section contains some examples and results concerning completely rigid ergodic transformation.