Measure theory on granular fuzzy sets

A granular fuzzy set theory is modeled on fuzzy sets whose membership functions are defined on sets of sets (granules). The grade is interpreted literally; for example, that the grade of x is 1/2 means one half of the granule x belongs to the fuzzy set. By taking the union of these subgranules, one get a crisp set representation of a fuzzy set. In other words, a granular fuzzy set is a fuzzy set that has a crisp set representation. A measure theory based on such granular fuzzy sets is developed. The measure theory of crisp sets is imported to fuzzy sets via their crisp representations. Using such a notion, a belief function can be shown to be the inner probability of a probability theory of fuzzy sets