On the coherence of supremum preserving upper previsions

We study the relation between possibility measures and the theory of imprecise probabilities. It is shown that a possibility measure is a coherent upper probability iff it is normal. We also prove that a possibility measure is the restriction to events of the natural extension of a special kind of upper probability, defined on a class of nested sets. Next, we go from upper probabilities to upper previsions. We show that if a coherent upper prevision defined on the convex cone of all positive gambles is supremum preserving, then it must take the form of a Shilkret integral associated with a possibility measure. But at the same time, we show that a supremum preserving upper prevision is not necessarily coherent! This makes us look for alternative extensions of possibility measures that are not necessarily supremum preserving, through natural extension.