The fundamental theorem of affine geometry in regular L0-modules

Abstract

Let (Ω,F,P) be a probability space and L0(F) the algebra of equivalence classes of real-valued random variables defined on (Ω,F,P). A left module M over the algebra L0(F) (briefly, an L0(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∈N} of Ω to F such that I˜An⋅x=I˜An⋅y for each n∈N, where IAn is the characteristic function of An and I˜An its equivalence class. The purpose of this paper is to establish the fundamental theorem of affine geometry in regular L0(F)-modules: let V and V′ be two regular L0(F)-modules such that V contains a free L0(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.