2018/12/20 by Wu, Mingzhi, Long, Long
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1812.08397
Let L0 be the algebra of equivalence classes of real valued random variables on a given probability space, and (L0)n the n-ary Cartesian power of L0 for each integer n≥ 2. We consider (L0)n as a free module over L0 and study affine geometry in (L0)n. One of our main results states that: an injective mapping T: (L0)n→ (L0)n which is local and maps each L0-line onto an L0-line must be an L0-affine linear mapping. The other main result states that: a bijective mapping T: (L0)n→ (L0)n which is local and maps each L0-line segment onto an L0-line segment must be an L0-affine linear mapping. These results extend the fundamental theorem of affine geometry from \mathbb Rn to (L0)n.