Oder-bounded sets in locally solid Riesz spaces

Abstract

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".