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

## TL;DR

This paper extends the fundamental theorem of affine geometry to the space of random variables, showing that certain structure-preserving maps must be affine linear in the context of $(L^0)^n$, a module over random variables.

## Contribution

It establishes that injective and bijective maps preserving lines and line segments in $(L^0)^n$ are necessarily affine linear, extending classical affine geometry results to a probabilistic setting.

## Key findings

- Injective maps preserving lines are affine linear.
- Bijective maps preserving line segments are affine linear.
- Results generalize classical affine geometry to random variable spaces.

## 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 $n\geq 2$. 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 $(L^0)^n$.