Weakly admissible translates of probability measures on locally convex space (Q1079862)

Let \(\mu\), \(\nu\) be two probability measures on a locally convex space E. \(\mu\) is said to be absolutely continuous in the weak sense with respect to \(\nu\), if for every sequence \((y_ n)\) in E' such that \(<\cdot,y_ n>\to 0\) in \(\nu\)-measure, one has \(<\cdot,y_ n>\to 0\) in \(\mu\)- measure. Absolute continuity obviously implies absolute continuity in the weak sense. This notion is used to define the set \(H_{\mu}\) of all so-called weakly admissible translates of a given \(\mu\). Compared with the set \(A_{\mu}\) of admissible translates of \(\mu\), the set \(H_{\mu}\) has more interesting properties. \(H_{\mu}\) is always a measurable linear subspace of E and in the case of a centered Gaussian measure equal to \(A_{\mu}\). If \(\mu\) is a symmetric stable measure of characteristic exponent greater than 1, \(H_{\mu}\) is a Banach space, and if \(\mu\) has a discrete spectrum, \(H_{\mu}\) can be explicitly described. The last part of the paper is concerned with the dichotomy problem for stable measures. It is proved that two stable measures are either weakly equivalent or orthogonal.