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

Abstract (Summary)

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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.
Bibliographical Information:

Advisor:W.A.J. Luxemburg

School:California Institute of Technology

School Location:USA - California

Source Type:Master's Thesis

Keywords:mathematics

ISBN:

Date of Publication:04/07/1969