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

Let $(\Omega,{\mathcal F},P)$ be a probability space and $L^0({\mathcal F})$ be 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 any given two elements $x$ and $y$ in $M$ must be equal whenever 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 $L^0$-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 onto an $L^0$-line segment, then $T$ must be $L^0$-affine.