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The Orlicz-Pettis theorem for topological Riesz spaces. (English) Zbl 0885.40002

Summary: A finitely additive vector measure from a \(\sigma\)-ring to a Riesz space is countably additive (exhaustive) for all Hausdorff Lebesgue topologies on the range space, or for none of them. In particular, subseries convergent series are the same for all Hausdorff Lebesgue topologies on a Riesz space.

MSC:

40A99 Convergence and divergence of infinite limiting processes
46A40 Ordered topological linear spaces, vector lattices
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