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On the generalised continuity of the semivariation in locally convex spaces. (English) Zbl 0788.46052
Summary: If Condition (GB) introduced by the author in [Measure theory, Proc. 2nd Winter Sch., Liptovský Jan/Czech. 1990, 28-31 (1990)] by the author is fulfilled, then the everywhere convergence of the net of measurable functions implies the convergence of these functions with respect to the semivariation on a set of the finite variation of the measure \(m: \Sigma\to L(X,Y)\), where \(\Sigma\) is a \(\sigma\)-algebra of subsets of the set \(T\neq 0\), \(X\), \(Y\) are both locally convex spaces. The generalized strong continuity of the semivariation of the measure, introduced in this paper, implies Condition (GB).

46G10 Vector-valued measures and integration
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