×

Sequential compactness for the weak topology of vector measures in certain nuclear spaces. (English) Zbl 1023.28008

Let \(S\) be a completely regular Hausdorff space, \(X\) a semi-Montel space and \(M_t(S,X)\) the space of all \(X\)-valued Radon measures on \(S\) endowed with the weakest topology which makes \(\mu\to \int f d\mu\) continuous for every continuous bounded real-valued function \(f\) on \(S\). The author proves that a subset \(V\) of \(M_t(S,X)\) is relatively compact if \(\{x^*\mu: \mu\in V\}\) is relatively compact in \(M_t(S,\mathbb{R})\) for every \(x^*\in X^*\). Moreover, it is given a criterion for metrizability of \(V\).
Reviewer: Hans Weber (Udine)

MSC:

28B05 Vector-valued set functions, measures and integrals
46G10 Vector-valued measures and integration
28C15 Set functions and measures on topological spaces (regularity of measures, etc.)