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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 $L^0({\mathcalF})$(briefly, an $L^0({\mathcal F})$-module) is said to be regular if $x=y$ forany given two elements $x$ and $y$ in $M$ such that there exists a countablepartition $\{A_n,n\in \mathbb N\}$ 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 thefundamental 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 $L^0$-linesegment, then $T$ must be $L^0$-affine.

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