Equivalent-singular dichotomy for quasi-invariant ergodic measures (Q1068417)

The paper deals with the comparison of two probability measures \(\mu_ 1\) and \(\mu_ 2\) on locally convex vector spaces. The main result shows that \(\mu_ 1\) and \(\mu_ 2\) are either equivalent or mutually singular whenever \(\mu_ 1\) and \(\mu_ 2\) are quasi-invariant and ergodic with respect to some linear subspaces. The result applies to Hájek-Feldman's dichotomy for Gaussian measures and Fernique's dichotomy for a product measure and a Gaussian measure. On the other hand the paper yields a contribution to Chatterji's and Ramaswamy's conjecture, namely: If \(\mu_ 1\) and \(\mu_ 2\) are symmetric stable distributions with a discrete Lévy measure on a separable Banach space then the equivalent-singular dichotomy holds.