Brouwer Fixed Point Theorem in (L^0)^d

The classical Brouwer fixed point theorem states that in R^d every continuous function from a convex, compact set on itself has a fixed point. For an arbitrary probability space, let L^0 = L^0 (Ω, A,P) be the set of random variables. We consider (L^0)^d as an L^0-module and show that local, sequentially continuous functions on closed and bounded subsets have a fixed point which is measurable by construction.

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