Accessible cylindrical measure and Bochner's theorem (Q1074065)

The Bochner problem on a locally convex space admitting an accessible measure is investigated. Let \(\mu\) be a probability measure on a localy convex Hausdorff space E. Denote by \(\tau_{\mu}\) the usual \(L^ 0(\mu)\)-topology on E' (the topological dual). The space \(K_{\mu}=(E',\tau_{\mu})'\) is called the kernel of \(\mu\). For a subset \(D\subset E\), \(\mu\) is called D-accessible if \(D\subset K_{\mu}\), i.e., for every \(x\in D\), \(x'\mapsto <x,x'>\) is \(\tau_{\mu}\)-continuous. The following Bochner problem is considered. Let F,E be locally convex Hausdorff spaces and \(u: F\to E\) be a continuous linear mapping. Suppose that E admits a u(F)-accessible probability measure. Then for every finitely additive cylindrical meaure \(\nu\) with the continuous characteristic functional \(\nu\hat.(x)\) \((x\in E)\), is the image \(u'(\nu)\) \(\sigma\)-additive on \(F'\) \((u': E'\to F'\) is the transpose of u)? That is to say, for every continuous positive-definite function \(\phi\) on E with \(\phi(0)=1\), does there exist a \(\sigma\)-additive probability measure \(\lambda\) on \(F'\) with the characteristic function \(\phi\circ u?\) The main result is Theorem. If F is barrelled and E has the metric approximation property, then the Bochner problem is affirmative. The proof is based on the Xia's inequality which implies that \(u':E'\to F'\) is completely summing.