The fundamental theorem of affine geometry in regular $L^0$-modules

TL;DR

This paper proves a fundamental theorem of affine geometry within regular modules over the algebra of random variables, showing that certain structure-preserving maps are necessarily affine transformations.

Contribution

It establishes that stable, invertible maps preserving $L^0$-line segments in regular $L^0$-modules are necessarily $L^0$-affine, extending affine geometry to a probabilistic module setting.

Findings

Maps preserving $L^0$-line segments are affine transformations.

The theorem applies to regular $L^0$-modules with a rank 2 free submodule.

The result generalizes classical affine geometry to a stochastic module context.

Abstract

Let $(\Omega,{\mathcal F},P)$ be a probability space and $L^0({\mathcal F})$ the algebra of equivalence classes of real-valued random variables defined on $(\Omega,{\mathcal F},P)$. A left module $M$ over the algebra $L^0({\mathcal F})$(briefly, an $L^0({\mathcal F})$-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 $\{A_n,n\in \mathbb N\}$ of $\Omega$ to $\mathcal F$ such that ${\tilde I}_{A_n}\cdot x={\tilde I}_{A_n}\cdot y$ for each $n\in \mathbb N$, where $I_{A_n}$ is the characteristic function of $A_n$ and ${\tilde I}_{A_n}$ its equivalence class. The purpose of this paper is to establish the fundamental theorem of affine geometry in regular $L^0({\mathcal F})$-modules: let $V$ and $V^\prime$ be two regular $L^0({\mathcal F})$-modules such that $V$ contains a free $L^0({\mathcal F})$-submodule of rank $2$, if $T:V\to V^\prime$ is stable and invertible and maps each $L^0$-line segment to an $L^0$-line segment, then $T$ must be $L^0$-affine.