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

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≥ 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→ (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→ (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 ℝ^n to (L^0)^n.