The Egoroff Property and Related Properties in the Theory of Riesz Spaces

Citation

Holbrook, John Arthur Rankin (1965) The Egoroff Property and Related Properties in the Theory of Riesz Spaces. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/BN4B-9460. https://resolver.caltech.edu/CaltechETD:etd-01142003-101748

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. A Riesz space L is said to be Egoroff, if, whenever [...] and [...], there is a sequence [...] in L such that [...] and, for each n,m, there exists an index k(n,m) such that [...]. This notion was introduced, in rather a different form, by Nakano. Banach function spaces are Egoroff, and Lorentz showed that, for any function seminorm [...], the maximal seminorm [...] among those which are dominated by [...] and which are [...] (a monotone seminorm [...] is [...] if [...]) is precisely the "Lorentz seminorm" [...], where [...]. In this thesis the extent to which [...] holds in general Riesz spaces is determined. In fact, [...] for every monotone seminorm [...] on a Riesz space L if, and only if, L is "almost-Egoroff". The almost-Egoroff property is closely related to the Egoroff property and, indeed, coincides with it in the case of Archimedean spaces. Analogous theorems for Boolean algebras are discussed. The almost-Egoroff property is shown to yield a number of results which ensure that, under certain conditions, a monotone seminorm is [...] when restricted to an appropriate super order dense ideal. Riesz spaces L possessing an integral, Riesz norm [...](i.e., a Riesz norm such that [...] are considered also, since in many cases these are known to be Egoroff. In particular if [...] is normal on L (i.e., [...] a directed system, [...] ), then L is Egoroff. In this connection, a pathological space, possessing an integral Riesz norm which is nowhere normal, is constructed.

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