Mackey topologies and compactness in spaces of vector measures (Q2450377)

In this paper, \(\Sigma\) is a \(\sigma\)-algebra of subsets of a set \(\Omega\), \(B(\Sigma)\) the vector space of all bounded, \(\Sigma\)-measurable, scalar-valued functions on \(\Omega\) and \((E, \xi)\) a quasi-complete locally convex space; \(ca(\Sigma, E)\) denotes the space of all countably additive, \(E\)-valued measures on \(\Sigma\) with the topology of pointwise convergence on \(\Sigma\) and \(\mathcal{L}_{\tau, \xi}( B(\Sigma), E)\) the space of all linear continuous operators \( (B(\Sigma), \tau(B(\Sigma), ca(\Sigma))) \to E\) with the topology of pointwise convergence on \(B(\Sigma)\). There is a one-to-one correspondence between \(ca(\Sigma, E)\) and \(\mathcal{L}_{\tau, \xi} (B(\Sigma), E)\). The main result is: For an \(\mathcal{M} \subset ca(\Sigma, E)\), the following statements are equivalent: {\parindent=8mm \begin{itemize}\item[(i)] \(\mathcal{M}\) is relatively compact in \(ca(\Sigma, E)\). \item[(ii)] \(\mathcal{M}\) is relatively compact in \(\mathcal{L}_{\tau, \xi}( B(\Sigma), E)\). \item[(iii)] \(\mathcal{M}\) is \((\tau(B(\Sigma), ca(\Sigma)), \xi)\)-equicontinuous and for every \(f \in B(\Sigma)\), the set \(\{ \int f dm: m \in \mathcal{M} \} \) is relatively \(\xi\)-compact in \(E\). \item[(iv)] \(\mathcal{M}\) is uniformly \(\xi\)-countably additive and for every \(A \in \Sigma\), \(\{ m(A); m \in \mathcal{M} \}\) is relatively \(\xi\)-compact in \(E\). \item[(v)] \(\int f_{n} dm \to 0 \; (\xi)\) uniformly on \( m \in \mathcal{M}\) whenever \(\{ f_{n} \} \subset B(\Sigma) \) is a uniformly bounded sequence such that \( f_{n} \to 0\) pointwise on \(\Omega\); and for each \( f \in B(\Sigma)\), the set \( \{ \int f dm : m \in \mathcal{M} \}\) is relatively \(\xi\)-compact in \(E\). \end{itemize}} A generalized Nikodým-type convergence theorem is derived from this result.