Brouwer Fixed Point Theorem in (L^0)^d
Functional Analysis
2013-09-13 v3
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.
Cite
@article{arxiv.1305.2890,
title = {Brouwer Fixed Point Theorem in (L^0)^d},
author = {Samuel Drapeau and Martin Karliczek and Michael Kupper and Martin Streckfuß},
journal= {arXiv preprint arXiv:1305.2890},
year = {2013}
}