The fundamental theorem of affine geometry in $(L^0{)}^n$

Abstract

Let $L^0$ be the algebra of equivalence classes of real valued random variables on a given probability space, and $(L^0{)}^n$ the $n$-ary Cartesian power of $L^0$ for each integer $n\ge 2$. We consider $(L^0{)}^n$ as a free module over $L^0$ and study affine geometry in $(L^0{)}^n$. One of our main results states that: an injective mapping $T:(L^0{)}^n\to (L^0{)}^n$ which is local and maps each $L^0$-line onto an $L^0$-line must be an $L^0$-affine linear mapping. The other main result states that: a bijective mapping $T:(L^0{)}^n\to (L^0{)}^n$ which is local and maps each $L^0$-line segment onto an $L^0$-line segment must be an $L^0$-affine linear mapping. These results extend the fundamental theorem of affine geometry from ${\mathbb{R}}^n$ to $(L^0{)}^n$.

10 pages