Measurable Structures of I -Fuzzy Rough Sets

In this paper, dual fuzzy rough approximation operators determined by a fuzzy implication operator \({\cal I}\) in infinite universes of discourse are first introduced. Measurable structures of \({\cal I}\)-fuzzy rough sets are then discussed. It is shown that the family of all definable sets in an \({\cal I}\)-fuzzy rough set algebra derived from a reflexive fuzzy space forms a σ-fuzzy algebra. In a finite universe of discourse, the family of all definable sets in a serial \({\cal I}\)-fuzzy rough set algebra is a fuzzy algebra, and conversely if a σ-fuzzy algebra is generated by a crisp algebra, then there must exist an \({\cal I}\)-fuzzy rough set algebra such that the family of all definable sets is exactly the given σ-fuzzy algebra.