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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).

##### MSC:
 46G10 Vector-valued measures and integration
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