Rough set approximations VS. measurable spaces

In this paper relationships between rough set ap- proximations and measurable spaces are examined. It is proved that the family of all definable sets in a serial crisp (fuzzy, respectively) rough set algebra forms a crisp (fuzzy respectively) algebra. For any crisp measurable space there must exist a crisp rough set algebra such that the family of all definable sets is the given crisp algebra. Also, for a fuzzy algebra generated by a crisp algebra there must exist a fuzzy rough set algebra such that the family of all definable sets is just the given fuzzy algebra.

[1]  Yiyu Yao,et al.  Generalization of Rough Sets using Modal Logics , 1996, Intell. Autom. Soft Comput..

[2]  Yee Leung,et al.  On characterizations of (I, J)-fuzzy rough approximation operators , 2005, Fuzzy Sets Syst..

[3]  Yiyu Yao,et al.  Relational Interpretations of Neigborhood Operators and Rough Set Approximation Operators , 1998, Inf. Sci..

[4]  Y. Yao,et al.  Generalized Rough Set Models , 1998 .

[5]  Wei-Zhi Wu,et al.  Neighborhood operator systems and approximations , 2002, Inf. Sci..

[6]  Yiyu Yao,et al.  Constructive and Algebraic Methods of the Theory of Rough Sets , 1998, Inf. Sci..

[7]  Wei-Zhi Wu,et al.  Generalized fuzzy rough sets , 2003, Inf. Sci..

[8]  Tsau Young Lin,et al.  Rough Approximate Operators: Axiomatic Rough Set Theory , 1993, RSKD.

[9]  Nehad N. Morsi,et al.  Axiomatics for fuzzy rough sets , 1998, Fuzzy Sets Syst..

[10]  Wen-Xiu Zhang,et al.  An axiomatic characterization of a fuzzy generalization of rough sets , 2004, Inf. Sci..

[11]  Janusz Zalewski,et al.  Rough sets: Theoretical aspects of reasoning about data , 1996 .

[12]  Yiyu Yao,et al.  Generalized probabilistic rough set models , 1996, Soft Computing in Intelligent Systems and Information Processing. Proceedings of the 1996 Asian Fuzzy Systems Symposium.

[13]  Erich Peter Klement,et al.  Fuzzy σ-algebras and fuzzy measurable functions , 1980 .

[14]  Anna Maria Radzikowska,et al.  A comparative study of fuzzy rough sets , 2002, Fuzzy Sets Syst..

[15]  J. K. Hunter,et al.  Measure Theory , 2007 .

[16]  L. Zadeh.  Probability measures of Fuzzy events , 1968 .

[17]  Wei-Zhi Wu,et al.  Constructive and axiomatic approaches of fuzzy approximation operators , 2004, Inf. Sci..