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

TL;DR

This paper extends the Brouwer fixed point theorem to the space of random variables, showing that certain continuous functions on bounded subsets have fixed points that are measurable, within the framework of L^0-modules.

Contribution

It introduces a fixed point theorem for local, sequentially continuous functions in (L^0)^d, generalizing classical results to a probabilistic setting.

Findings

Fixed points exist for local, sequentially continuous functions on closed, bounded subsets of (L^0)^d.

Fixed points are constructed to be measurable.

The theorem applies within the framework of L^0-modules.

Abstract

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 (\Omega, 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.