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Aggregation and Completion of Random Sets with Distributional fuzzy Measures

The two known information theories, probability and possibility theory, are based on t-conorm decomposable fuzzy measures, so that bijective mappings exist between their set-valued measures and their point-valued distributions. Further, their random set (Dempster-Shafer evidence theoretical) interpretations have simple topological structures, with bijective mappings between the subset focal elements and the point singletons. We introduce the concepts of distributional and aggregable random sets and random set completion, and first use them as a model in which to cast probability and possibility measures and distributions. Then, towards the goal of deriving new forms of information theory, general Sugeno conorm decomposable fuzzy measures and ring-like aggregable random sets with set-intersection structural aggregation are examined, but it is shown that in these two cases no new information theories are forthcoming.