A Daniell-Kolmogorov theorem for supremum preserving upper probabilities

Possibility measures are interpreted as upper probabilities that are in particular supremum preserving. We define a possibilistic process as a special family of possibilistic variables, and show how its possibility distribution functions can be constructed. We introduce and study the notions of inner and outer regularity for possibility measures. Using these notions, we prove an analogon for possibilistic processes (and possibility measures) of the well-known probabilistic Daniell-Kolmogorov theorem, in the important special case that the variables assume values in compact spaces, and that the possibility measures involved are regular.