To represent a set whose members are known partially, the graded ill-known set is proposed. In this paper, we investigate calculations of function values of graded ill-known sets. Because a graded ill-known set is characterized by a possibility distribution in the power set, the calculations of function values of graded ill-known sets are based on the extension principle but generally complex. To reduce the complexity, lower and upper approximations of a given graded ill-known set are used at the expense of precision. We give a necessary and sufficient condition that lower and upper approximations of function values of graded ill-known sets are obtained as function values of lower and upper approximations of graded ill-known sets.
The standard techniques of lower and upper approximations, used in
order to define the inner and outer measures given a σ-additive measure, perhaps a probabilistic one, are applied to possibilistic measures. The conditions under which this approach can be reasonable and useful are investigated and the most elernentary properties of the resulting inner and outer possibilistic measures are briefly sketched.