Brouwer fixed point theorem in (L0)d

The classical Brouwer fixed point theorem states that in Rd every continuous function from a convex, compact set on itself has a fixed point. For an arbitrary probability space, let L0=L0(Ω,A,P) be the set of random variables. We consider (L0)d as an L0-module and show that local, sequentially continuous functions on L0-convex, closed and bounded subsets have a fixed point which is measurable by construction. MSC:47H10, 13C13, 46A19, 60H25.