Generalized Possibility Measures

We define a generalized possibility measure (GPM) on a set X to be a function from the power set of X to the unit interval I = [0,1], whereby the measure of a finite union of subsets of X equals the maximum of their measures. We characterize GPMs as the infima of downward directed families of possibility measures. This helps to confer an intuitive meaning on GPMs. We discuss all directed families of possibility measures (or, equivalently, of possibility distributions) which generate a given GPM, and we identify the largest such family. We introduce extensions of GPMs to real functions on IX, obtained through a continuous triangular norm. We provide characterizations and alternative descriptions for those extensions.

[1]  M. H. Ghanim,et al.  Semi-induced fuzzy topologies , 1989 .

[2]  N. N. Morsi.  Nearness concepts in fuzzy neighbourhood spaces and in their fuzzy proximity spaces , 1989 .

[3]  M. H. Ghanim,et al.  On product fuzzy topological spaces , 1989 .

[4]  Ying-ming Liu,et al.  Intersection operation on union-preserving mappings in completely distributive lattices , 1981 .

[5]  Nehad N. Morsi,et al.  Fuzzy metric neighbourhood spaces , 1992 .

[6]  Nehad N. Morsi.  Dual fuzzy neighbourhood spaces I , 1991 .

[7]  Stephen E. Rodabaugh,et al.  Point-set lattice-theoretic topology , 1991 .

[8]  P. Kloeden,et al.  Metric spaces of fuzzy sets , 1990 .

[9]  L. Valverde,et al.  On Some Logical Connectives for Fuzzy Sets Theory , 1983 .

[10]  D. Dubois,et al.  Fuzzy sets in approximate reasoning. I, Inference with possibility distributions , 1991 .

[11]  R. Yager.  Connectives and quantifiers in fuzzy sets , 1991 .

[12]  B. Hutton.  Normality in fuzzy topological spaces , 1975 .

[13]  Robert Lowen,et al.  The categorical topology approach to fuzzy topology and fuzzy convergence , 1991 .

[14]  Nehad N. Morsi,et al.  Continuity of fuzzy multiplication in the N-Euclidean space , 1992 .

[15]  N. N. Morsi.  A short note on fuzzy neighbourhood spaces , 1987 .

[16]  M. Gupta,et al.  Theory of T -norms and fuzzy inference methods , 1991 .

[17]  R. Lowen,et al.  Fuzzy Uniform Spaces , 1981 .

[18]  Stephen E. Rodabaugh,et al.  A theory of fuzzy uniformities with applications to the fuzzy real lines , 1988 .

[19]  Liu Ying-Ming.  Structures of fuzzy order homomorphisms , 1987 .

[20]  Abdelwahab M. El-Abyad,et al.  Fuzzy inner product spaces , 1991 .

[21]  Robert Lowen,et al.  Fuzzy neighborhood spaces , 1982 .

[22]  S. Gottwald.  Set theory for fuzzy sets of higher level , 1979 .

[23]  Peter E. Kloeden,et al.  Characterization of compact subsets of fuzzy sets , 1989 .

[24]  R. Lowen,et al.  On the existence of natural non-topological, fuzzy topological spaces , 1985 .

[25]  M. Puri,et al.  Differentials of fuzzy functions , 1983 .

[26]  R. Lowen.  The order aspect of the fuzzy real line , 1985 .

[27]  U. Höhle.  Probabilistic uniformization of fuzzy topologies , 1978 .