A paradoxical Pareto frontier in the cake-cutting context

We construct two finitely-additive measures that have all the properties required by discrete cake-cutting procedures, but for which the only Pareto optimal allocations can be cryptically described as the ones in which the whole cake is given to one of the players. This allows us to conclude that there is no discrete cake-cutting scheme for two players that will ensure a Pareto optimal, envy-free allocation.