On the Expected Value of Fuzzy Events

Generalizing a first approach by L. A. ZADEH (J. Math. Anal. Appl. 23, 1968), expected values of fuzzy events are studied which are (up to standard boundary conditions) only required to be monotone. They can be seen as an extension of capacities, i.e., monotone set functions satisfying standard boundary conditions. Some of these expected values can be characterized axiomatically, others are based on some distinguished integrals (Choquet, Sugeno, Shilkret, universal, and decomposition integral).

[1]  Michel Grabisch,et al.  K-order Additive Discrete Fuzzy Measures and Their Representation , 1997, Fuzzy Sets Syst..

[2]  R. Nelsen.  An Introduction to Copulas , 1998 .

[3]  D. Butnariu,et al.  Triangular Norm-Based Measures and Games with Fuzzy Coalitions , 1993 .

[4]  Ehud Lehrer,et al.  Decomposition-integral: unifying Choquet and the concave integrals , 2014 .

[5]  M. J. Frank.  On the simultaneous associativity ofF(x, y) andx+y−F(x, y) , 1978 .

[6]  Francesc Esteva,et al.  Review of Triangular norms by E. P. Klement, R. Mesiar and E. Pap. Kluwer Academic Publishers , 2003 .

[7]  G. Klir,et al.  Generalized Measure Theory , 2008 .

[8]  Ehud Lehrer,et al.  A New Integral for Capacities , 2005 .

[9]  E. Klement.  Construction of Fuzzy σ-algebras using triangular norms , 1982 .

[10]  Jean-Luc Marichal,et al.  An axiomatic approach of the discrete Sugeno integral as a tool to aggregate interacting criteria in a qualitative framework , 2001, IEEE Trans. Fuzzy Syst..

[11]  O. Stolz.  Ueber einen zu einer unendlichen Punktmenge gehörigen Grenzwerth , 1884 .

[12]  E. Klement.  Characterization of fuzzy measures constructed by means of triangular norms , 1982 .

[13]  Erich Peter Klement,et al.  Fuzzy probability measures , 1981 .

[14]  Radko Mesiar,et al.  K-Order Additive Fuzzy Measures , 1999, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

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

[16]  Radko Mesiar,et al.  A Universal Integral as Common Frame for Choquet and Sugeno Integral , 2010, IEEE Transactions on Fuzzy Systems.

[17]  G. Choquet.  Theory of capacities , 1954 .

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

[19]  Dan Butnariu,et al.  Triangular norm-based measures and their Markov kernel representation , 1991 .

[20]  N. Shilkret.  Maxitive measure and integration , 1971 .

[21]  Radko Mesiar.  Fuzzy measures and integrals , 2005, Fuzzy Sets Syst..

[22]  Radko Mesiar,et al.  Universal integrals based on copulas , 2014, Fuzzy Optimization and Decision Making.

[23]  Ky Fan.  Entfernung zweier zufälligen Größen und die Konvergenz nach Wahrscheinlichkeit , 1943 .

[24]  M. Bhaskara Rao,et al.  Theory of Charges: A Study of Finitely Additive Measures , 2012 .

[25]  R. Mesiar,et al.  ”Aggregation Functions”, Cambridge University Press , 2008, 2008 6th International Symposium on Intelligent Systems and Informatics.

[26]  D. Schmeidler.  Integral representation without additivity , 1986 .

[27]  M. Sugeno,et al.  Pseudo-additive measures and integrals , 1987 .

[28]  Jun Li,et al.  Atoms of weakly null-additive monotone measures and integrals , 2014, Inf. Sci..

[29]  E. Pap.  Null-Additive Set Functions , 1995 .

[30]  Constantin Carathéodory,et al.  Vorlesungen über reelle Funktionen , 1968 .

[31]  Endre Pap,et al.  Handbook of measure theory , 2002 .

[32]  S. Weber,et al.  Fundamentals of a Generalized Measure Theory , 1999 .

[33]  Endre Pap,et al.  Generalized real analysis and its applications , 2008, Int. J. Approx. Reason..

[34]  D. Schmeidler.  Subjective Probability and Expected Utility without Additivity , 1989 .

[35]  Glenn Shafer,et al.  A Mathematical Theory of Evidence , 2020, A Mathematical Theory of Evidence.

[36]  Rudolf Kruse,et al.  Fuzzy integrals and conditional fuzzy measures , 1983 .

[37]  菅野 道夫,et al.  Theory of fuzzy integrals and its applications , 1975 .

[38]  Hung T. Nguyen,et al.  A First Course in Fuzzy Logic , 1996 .

[39]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[40]  Dan Butnariu,et al.  Triangular Norm-Based Measures , 1993 .

[41]  Jun Li,et al.  Superdecomposition integrals , 2015, Fuzzy Sets Syst..

[42]  Gregory T. Adams,et al.  The fuzzy integral , 1980 .

[43]  Erich Peter Klement,et al.  Characterization of finite fuzzy measures using Markoff-kernels , 1980 .

[44]  Dan Butnariu,et al.  Fuzzy measurability and integrability , 1986 .

[45]  M. J. Frank.  On the simultaneous associativity ofF(x,y) andx +y -F(x,y) , 1979 .

[46]  Jun Li,et al.  Integrals based on monotone set functions , 2015, Fuzzy Sets Syst..

[47]  Jun Li,et al.  Lusin's theorem on monotone measure spaces , 2011, Fuzzy Sets Syst..

[48]  S. Weber.  ⊥-Decomposable measures and integrals for Archimedean t-conorms ⊥ , 1984 .

[49]  Arthur P. Dempster,et al.  Upper and Lower Probabilities Induced by a Multivalued Mapping , 1967, Classic Works of the Dempster-Shafer Theory of Belief Functions.

[50]  G. Vitali,et al.  Sulla definizione di integrale delle funzioni di una variabile , 1925 .

[51]  Radko Mesiar,et al.  Decomposition integrals , 2013, Int. J. Approx. Reason..

[52]  H. Bauer,et al.  Probability Theory and Elements of Measure Theory , 1982 .

[53]  D. Denneberg.  Non-additive measure and integral , 1994 .

[54]  Siegfried Weber,et al.  Generalized measures , 1991 .