GROUP INVARIANT BOUNDED LINEAR FUNCTIONS ON DEDEKIND COMPLETE TOTALLY ORDERED RIESZ SPACES

In this paper we consider the set B of all bounded subsets of V, where V is a totally ordered Dedekind complete Riesz space equipped with the order topology. We show the existence of nontrivial bounded linear functions on B that are invariant under group actions of the symmetric group of B. To do this, we construct a set of “approximately” group invariant bounded linear functions and we show, using Tychonff’s Theorem (that is equivalent to the Axiom of Choice), that this set has a cluster point. This cluster point is the group invariant bounded linear function on B that we are looking for.