On the density of properly maximal claims in financial markets with transaction costs

We consider trading in a financial market with proportional transaction costs. In the frictionless case, claims are maximal if and only if they are priced by a consistent price process--the equivalent of an equivalent martingale measure. This result fails in the presence of transaction costs. A properly maximal claim is one which does have this property. We show that the properly maximal claims are dense in the set of maximal claims (with the topology of convergence in probability).