On Characterization of Locally L0-Convex Topologies Induced by a Family of L0-Seminorms

Zapata-García, José Miguel

doi:10.2139/ssrn.2461722

Cited by 5 publications

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“…Remark 5.8. It is easy to see that the notion of a cyclic set includes as a special case the notion of a set having the relative countable concatenation property (also called a relatively σ-stable set, introduced independently in [44,46]). It is also obvious that the notion of a cyclic set is weaker than the notion of a universally complete set, and it is the latter that is frequently used in the study of related issues, so here we do not intend to make too much discussion about the notion of a cyclic set.…”

Section: On Universal Completeness and B-stabilitymentioning

confidence: 99%

“…In 2009, locally L 0 -convex modules were introduced in [5] in order to establish a generalized convex analysis suitable for the study of conditional convex risk measures. With the notion of a locally L 0 -convex module, the locally L 0 -convex topology for a random locally convex module was also presented in [5], it is later showed independently in [44] and in [46]that the truly useful part of the theory of a locally L 0 -convex module amounts to the theory of a random locally convex module endowed with the locally L 0 -convex topology. The (ε, λ)-topology is an abstract generalization of the usual topology of convergence in probability measure and has been widely used in functional analysis, probability theory and mathematical finance.…”

Section: Introductionmentioning

confidence: 99%

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Notions, Definitions, and Models

Guo

Susilo

2018

Introduction to Security Reduction

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First, we prove that a random metric space can be isometrically embedded into a complete random normed module, as an application it is easy to see that the notion of d-σ-stability in a random metric space can be regarded as a special case of the notion of σ-stability in a random normed module; as another application we give the final version of the characterization for a d-σ-stable random metric space to be stably compact. Second, we prove that an L p -normed L ∞ -module is exactly generated by a complete random normed module so that the gluing property of an L p -normed L ∞ -module can be derived from the σ-stability of the generating random normed module, as applications almost all the basic theory of module duals can be obtained from the theory of random conjugate spaces. Third, we prove that a random normed space is order complete iff it is (ε, λ)-complete, as an application it is proved that the d-decomposability of an order complete random normed space is exactly its d-σ-stability. Finally, we prove that an equivalence relation on the product space of a nonempty set X and a complete Boolean algebra B is regular iff it can be induced by a B-valued Boolean metric on X, as an application it is proved that a nonempty subset of a Boolean set (X, d) is universally complete iff it is a B-stable set defined by a regular equivalence relation.

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Section: On Universal Completeness and B-stabilitymentioning

confidence: 99%