Vector-valued weakly analytic measures (Q1265211)

The classical F. and M. Riesz theorem states that an analytic measure \(\mu\) on \(\mathbb T\), i.e. a measure with \(\widehat\mu(n)=0\), \(n<0\), is absolutely continuous, which again is known to be equivalent to the continuity of the translation \(\mathbb R\to M(\mathbb T)\), \(t\to \delta_t*\mu\). In this paper representations \(T_t\) in spaces of vector valued measures (measures taking values in some dual space \(Y^*\)) are considered. The authors prove Bochner measurability of a large class of weakly analytic transformations \(T_t\). Since it is known that analytic vector valued measures need not translate continuously, the following restrictions on the Banach spaces \(Y^*\) are needed to extend the F. and M. Riesz theorem: \(Y^*\) has the analytic Radon-Nikodym property, i.e. any measure \(\mu\) with values in \(Y^*\) satisfying \(\int_{\mathbb T}e^{-int} d\mu(t)=0\), \(n<0\), has a Radon-Nikodym derivative in \(L^1(\mathbb T,\lambda,Y^*)\).