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- W3183290475 abstract "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 ${A_n,nin mathbb N}$ of $Omega$ to $mathcal F$ such that ${tilde I}_{A_n}cdot x={tilde I}_{A_n}cdot y$ for each $nin mathbb N$, 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 $T:Vto V^prime$ is stable and invertible and maps each $L^0$-line segment onto an $L^0$-line segment, then $T$ must be $L^0$-affine." @default.
- W3183290475 created "2021-08-02" @default.
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- W3183290475 date "2021-07-19" @default.
- W3183290475 modified "2023-09-27" @default.
- W3183290475 title "The fundamental theorem of $L^0$-affine geometry in regular $L^0$-modules" @default.
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