Continuity of Quasi-Invariant Measures and Zero-One Laws on Groups

Let G be a locally compact metrizable group acting on a probability space (X, B, μ). First, we give several equivalent conditions for the continuity of δg ∗ μ in g ∈ G. Then for a quasi-invariant and ergodic probability measure on a measurable group, we prove a zero-one law for translated subgroups, which is a natural extension of Kallianpur′s zero-one law for Gaussian measures, and apply it to the Wiener measure on a loop group. We also give an answer to a problem posed by J. Zinn (Proc. Amer. Math. Soc.44, No. 1 (1974), 179-185).