A DANIELL-KOLMOGOROV THEOREM FOR SUPREMUM PRESERVING UPPER PROBABILITIES HUGO JANSSEN, GERT DE COOMAN, AND ETIENNE E. KERRE Abstract. Possibility measures are interpreted as upper probabilities that are in particular supremum preserving. We define a possibilistic process as a special family of possibilistic vari- ables, 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 prob- abilistic Daniell-Kolmogorov theorem, in the important special case that the variables assume values in compact spaces, and that the possibility measures involved are regular. 1. Introduction A process p can be informally defined as a variable which changes in time. If we consider a time set T , and denote by X the set in which the variable p may take its values, then formally, a process p in X is a T − X -mapping, and p(t) is the value of the process at time t, t ∈ T . In many practical applications, processes are real , that is, they assume real values: X = R. A process p is called uncertain if there is not enough information available in order to specify its value p(t) in X unequivocally for all times t ∈ T . Such processes can be modelled by considering a collection P of processes in X , one of which, but unknown to us, is the actual process. Information about the process essentially leads to a restriction on the candidate processes in P , which can take a number of forms. It could take the form of a subset of P , or of an additive probability on P . In the latter case, we may speak of a stochastic process. Arguably the most general case, which encompasses the cases just mentioned, is when the available information is represented by an imprecise probability model [17] on P . We are interested in the values that an uncertain process may assume at given times. This leads to the consideration of a family of mappings f t : P → X , indexed by the time set T . Each f t maps a process p in P to its corresponding value at time t: f t (p)= p(t). Of course, these mappings (and their products) may be used to transport the given imprecise probability model on P to imprecise probability models on X , that is, to information about the values which the process may assume at the various corresponding times. A very general formal model for an uncertain process therefore consists of a nonempty set Ω, called basic space , a nonempty set X , called sample space , and a family of Ω − X -mappings (f t | t ∈ T ), indexed by a nonempty time set T . Information about the process is represented by an imprecise probability model on the basic space. The paper provides a closer look at uncertain processes of a special type, namely for which the given imprecise probability model is a possibility measure, that is, a supremum preserving upper probability (see the next section for a more precise definition). For such possibilistic processes , we intend to show that it is possible to prove a counterpart to a fundamental result in the theory of stochastic processes, namely the Daniell-Kolmogorov theorem. Key words and phrases. Possibilistic process, Daniell-Kolmogorov theorem, regularity, upper semicontinuity, possibility measure, upper probability. 1