A local Hahn–Banach theorem and its applications

Gao, Niushan; Leung, Denny H.; Xanthos, Foivos

doi:10.1007/s00013-018-1280-z

Cited by 2 publications

(1 citation statement)

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“…We begin our exploration with a result of the Hahn-Banach theorem spirit. Similar results of this type have been achieved in a recent paper [5], where the authors established a "localized" Hahn-Banach theorem on a vector space and applied it to study the uodual of a Banach lattice, resulting in a very transparent proof of Theorem 1.2. The following result is an extension of their approach, embracing solidity.…”

Section: Uniformly Locally Convex-solid Topologiessupporting

confidence: 71%

On local convexity in $\mathbb{L}^0$ and switching probability measures

Gao¹,

Leung²,

Xanthos³

2019

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In the paper, we investigate the following fundamental question. For a set K in L 0 (P), when does there exist an equivalent probability measure Q such that K is uniformly integrable in L 1 (Q). Specifically, let K be a convex bounded positive set in L 1 (P). Kardaras [6] asked the following two questions: (1) If the relative L 0 (P)topology is locally convex on K, does there exist Q ∼ P such that the L 0 (Q)-and L 1 (Q)-topologies agree on K? (2) If K is closed in the L 0 (P)-topology and there exists Q ∼ P such that the L 0 (Q)-and L 1 (Q)-topologies agree on K, does there exist Q ′ ∼ P such that K is Q ′ -uniformly integrable? In the paper, we show that, no matter K is positive or not, the first question has a negative answer in general and the second one has a positive answer. In addition to answering these questions, we establish probabilistic and topological characterizations of existence of Q ∼ P satisfying these desired properties. We also investigate the peculiar effects of K being positive.

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Section: Uniformly Locally Convex-solid Topologiessupporting

confidence: 71%