The Egoroff property and its relation to the order topology in the theory of Riesz spaces

Citation

Chow, Theresa Kee Yu (1969) The Egoroff property and its relation to the order topology in the theory of Riesz spaces. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/T2KV-BF37. https://resolver.caltech.edu/CaltechETD:etd-10072002-143502

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. A sequence(f[subscript]n : n = 1, 2, ...) in a Riesz space L is order convergent to an element [...] whenever there exists a sequence [...] in L such that [...] holds for all n. Sequential order convergence defines the order topology on L. The closure of a subset S in this topology is denoted by cl(S). The pseudo order closure S' of a subset S is the set of all [...] such that there exists a sequence in S which is order convergent to f. If S' = cl(S) for every convex subset S, then S' = cl(S) for every subset S. L has the Egoroff property if and only if S' = cl(S) for every order bounded subset S of L. A necessary and sufficient condition for L to have the property that S' = cl(S) for every subset S of L is that L has the strong Egoroff property. A sequence(f[subscript]n : n = 1, 2, ...) in a Riesz space L is ru-convergent to an element [...] whenever there exists a real sequence [...] and an element [...] such that [...] holds for all n. Sequential ru-convergence defines the ru-topology on L. The closure of a subset S in this topology is denoted by [...]. The pseudo ru-closure S'[subscript ru] of a subset S is the set of all [...] such that there exists a sequence in S which is ru-convergent to f. If L is Archimedean, then [...] for every convex subset S implies that [...] for every subset S. A characterization of those Archimedean Riesz spaces L with the property that [...] for every subset S of L is obtained. If [...] is a monotone seminorm on a Riesz space L, then a necessary and sufficient condition for [...] in L implies [...] is that the set [...] is order closed. For every monotone seminorm [...] on L, the largest [...]-Fatou monotone serninorm bounded by [...] is the Minkowski functional of the order closure of [...]. A monotone seminorm p on a Riesz space L is called strong Fatou whenever [...]. A characterization of those Riesz spaces L which have the following property is given: "For every monotone seminorm [...], the largest strong Fatou monotone seminorm bounded by [...] : [...]." A similar characterization for Boolean algebras is also obtained.

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