On local convexity in \mathbbL⁰ and switching probability measures

2019/02/03 by Gao, Niushan, Leung, Denny H., Xanthos, Foivos
#FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR)

paper · doi:10.48550/arxiv.1902.00992

Abstract

In the paper, we investigate the following fundamental question. For a set K in \mathbbL⁰(ℙ), when does there exist an equivalent probability measure ℚ such that K is uniformly integrable in \mathbbL¹(ℚ). Specifically, let K be a convex bounded positive set in \mathbbL¹(ℙ). Kardaras [6] asked the following two questions: (1) If the relative \mathbbL⁰(ℙ)-topology is locally convex on K, does there exist ℚ∼ ℙ such that the \mathbbL⁰(ℚ)- and \mathbbL¹(ℚ)-topologies agree on K? (2) If K is closed in the \mathbbL⁰(ℙ)-topology and there exists ℚ∼ ℙ such that the \mathbbL⁰(ℚ)- and \mathbbL¹(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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