On conditions for a cylindrical measure to be countably additive in a dual locally convex space
Let \(\lambda\) be a cylindrical measure on the topological dual space \(E'\) of a locally convex space \(E\). Suppose that \(\lambda\) is induced by a linear operator from \(E\) to the space \(L_0\) of random variables. In the paper the author's study of the relationship of the following properties is continued: (A) \(\lambda\) is \(e\)-tight, i.e., it is tight with respect to equicontinuous subsets of \(E'\). (B) \(\lambda\) is \(\sigma\)-additive. (C) \(Fi\) images of the bounded subsets of \(E\) are lattice bounded in \(L_0\). The implications \(\text{(A)}\Rightarrow \text{(B)}\Rightarrow \text{(C)}\) are obvious. Previously the author has shown that for metrizable spaces and for LF-spaces (C) implies (A), but for an arbitrary DF-space \(E\) the condition (C) may not imply (A). In the given paper it is shown that, nevertheless, if either \(E\) is the strong dual of a metrizable space, or \(E\) is any DF-space and in (C) the operator \(Fi\) is Mackey continuous, then (C) implies (B). The paper contains also some other results in this direction.
scientific article; zbMATH DE number 830721