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Strict and Mackey topologies and tight vector measures. (English) Zbl 1414.46029

Let \(E=(E,\xi)\) be a Hausdorff sequentially complete locally convex space over \(\mathbb{C}\), \(\mathcal{B}_0\) the \(\sigma\)-algebra of Borel sets in a Hausdorff topological space and \(B(\mathcal{B}_0)\) the Banach space of all bounded \(\mathbb{C}\)-valued \(\mathcal{B}_0\)-measurable functions on \(X\). The author considers on \(B(\mathcal{B}_0)\) the Mackey topology \(\tau(B(\mathcal{B}_0),ca(\mathcal{B}_0))\), the topology \(\tau_u\) of uniform convergence, the compact-open topology \(\tau_c\), the finest linear topology \(\beta\) agreeing with \(\tau_c\) on \(\tau_u\)-bounded sets, i.e., the mixed topology \(\gamma[\tau_u,\tau_c]\) in the sense of A. Wiweger [Stud. Math. 20, 47–68 (1961; Zbl 0097.31301)] and the infimum \(\eta:=\beta\wedge \tau(B(\mathcal{B}_0),ca(\mathcal{B}_0))\). Note that \(\beta\) is a strict topology in the sense of J. Hoffmann-Jørgensen [Math. Scand. 30, 313–323 (1972; Zbl 0256.46036)]. It is shown that a measure \(m:\mathcal{B}_0\rightarrow E\) is \(\sigma\)-additive (tight, \(\sigma\)-additive and tight) iff the integration map \(T_m:B(\mathcal{B}_0)\rightarrow E\) is \(\tau(B(\mathcal{B}_0),ca(\mathcal{B}_0))\)-continuous (respectively, \(\beta\)-continuous, \(\eta\)-continuous). More generally, uniformly bounded subsets \(\mathcal{M}\) of \(ba(\mathcal{B}_0,E)\) are considered; e.g., it is shown that \(\mathcal{M}\) is uniformly tight iff \(\{T_m:m\in\mathcal{M}\}\) is \(\beta\)-equicontinuous. Morever, a version of the Nikodým convergence theorem is deduced.
Reviewer: Hans Weber (Udine)

MSC:

46G10 Vector-valued measures and integration
28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
28A25 Integration with respect to measures and other set functions
46A70 Saks spaces and their duals (strict topologies, mixed topologies, two-norm spaces, co-Saks spaces, etc.)
46E27 Spaces of measures
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