Oder-bounded sets in locally solid Riesz spaces


Let E be Dedekind complete, Hausdorff, locally solid Riesz space and P an order bounded interval. We give a new proofs of Nakano’s theorem, that if E has Fatou property, P is complete, that the restrictions on P, of all topologies on E having Lebesgue property, are identical; we also give a measure-theoretic proof of the result that if (E,T) is a Dedekind complete, Hausdorff, locally convex-solid Riesz space with Lebesque property, then P is weakly compact and E is a regular Riesz subspace of E".

DOI Code: 10.1285/i15900932v28n1p119

Keywords: locally solid; band; Lebesgue property; Fatou property; order intervals; order direct sum

Classification: 46A40; 46A50; 46B42

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