2021/07/19 by Mingzhi Wu, Wu, Mingzhi, Tiexin Guo +3
Economics, Econometrics and Finance · Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2107.08625
openalex publication_date 2021/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (Ω,\mathcal F,P) be a probability space and L0(\mathcal F) the algebra of equivalence classes of real-valued random variables defined on (Ω,\mathcal F,P). A left module M over the algebra L0(\mathcal F)(briefly, an L0(\mathcal F)-module) is said to be regular if x=y for any given two elements x and y in M such that there exists a countable partition \An,n∈ \mathbb N\ of Ω to \mathcal F such that IAn⋅ x= IAn⋅ y for each n∈ \mathbb N, where IAn is the characteristic function of An and IAn its equivalence class. The purpose of this paper is to establish the fundamental theorem of affine geometry in regular L0(\mathcal F)-modules: let V and V^′ be two regular L0(\mathcal F)-modules such that V contains a free L0(\mathcal F)-submodule of rank 2, if T:V→ V^′ is stable and invertible and maps each L0-line segment to an L0-line segment, then T must be L0-affine.