Uniform integrability and local convexity in L⁰

2012/11/02 by Constantinos Kardaras, Kardaras, Constantinos
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #Advanced Banach Space Theory #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR) #Risk and Portfolio Optimization #Stochastic processes and financial applications #math.FA #math.PR

paper · pdf · doi:10.48550/arxiv.1211.0475

14 pages

arxiv created 2012/11/02 · openalex publication_date 2012/11/02 · arxiv updated 2012/11/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let L⁰ be the vector space of all (equivalence classes of) real-valued random variables built over a probability space (Ω, F, P), equipped with a metric topology compatible with convergence in probability. In this work, we provide a necessary and sufficient structural condition that a set X ⊆ L⁰ should satisfy in order to infer the existence of a probability Q that is equivalent to P and such that X is uniformly Q-integrable. Furthermore, we connect the previous essentially measure-free version of uniform integrability with local convexity of the L⁰-topology when restricted on convex, solid and bounded subsets of L⁰.

Citations

• Foundations of Modern Probability

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