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On the continuity of the semivariation in locally convex spaces. (English) Zbl 0796.46030
Summary: If the introduced Condition (GB) is fulfilled, then everywhere convergence of nets of measurable functions implies convergence in semivariation on a set of finite variation of a measure \({\mathbf m}: \Sigma\to L({\mathbf X},{\mathbf Y})\) which is \(\sigma\)-additive in the strong operator topology (\(\Sigma\) is a \(\sigma\)-algebra of subsets, and \({\mathbf X}\), \({\mathbf Y}\) are both locally convex spaces). In the case of the purely atomic measure Condition (GB) is fulfilled.

MSC:
46G10 Vector-valued measures and integration
28B05 Vector-valued set functions, measures and integrals
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References:
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