Fuzzy measures and generalized Mobius transform

Generalized Mobius transform is recalled and applied in some special cases. The relationship with the standard Mobius transform is shown. By means of the generalized Mobius transform, a general concept of k-order additivity independent of the cardinality of the underlying space is introduced. The relationship of the Choquet integral and the Lebesgue integral by means of the generalized Mobius transform is clarified. Also possibilistic Mobius transform and k-order possibility measures are introduced. Finally, some examples are given, including the characterization of de Finetti's discrete lower probabilities.