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

- Authors
- Samuel Drapeau, Michael Kupper, Martin Karliczek and Martin Streckfuß
- Date
- 2013
- Journal
- Fixed Point Theory and Applications:2013:301
- Abstract
- The classical Brouwer fixed point theorem states that in \(\mathbb{R}^d\) every continuous function from a convex, compact set on itself has a fixed point. For an arbitrary probability space, let \(L^0=L^0(\Omega,\mathcal{A},P)\) be the set of random variables. We consider \((L^0)^d\) as an \(L^0\)-module and show that local, sequentially continuous functions on \(L^0\)-convex, closed and bounded subsets have a fixed point which is measurable by construction.
- Keywords
- Conditional Simplex, Fixed Points in \((L^0)^d\), Brouwer
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