Fuzzy measure theory considers a number of special classes of measures, each of which is characterized by a special property. Some of the measures used in this theory are plausibility and belief measures, fuzzy set membership function and the classical probability measures. In the fuzzy measure theory, the conditions are precise, but the information about an element alone is insufficient to determine which special classes of measure should be used. The central concept of fuzzy measure theory is the fuzzy measure (also capacity, see ^{[1]}) which was introduced by Choquet in 1953 and independently defined by Sugeno in 1974 in the context of fuzzy integrals.
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Fuzzy measure can be considered as generalization of the classical probability measure. A fuzzy measure g over a set X (the universe of discourse with the subsets E, F, ...) satisfies the following conditions:
A fuzzy measure g is called normalized if g(X) = 1.
For any , a fuzzy measure is:
Understanding the properties of fuzzy measures is useful in application. When a fuzzy measure is used to define a function such as the Sugeno integral or Choquet integral, these properties will be crucial in understanding the function's behavior. For instance, the Choquet integral with respect to an additive fuzzy measure reduces to the Lebesgue integral. In discrete cases, a symmetric fuzzy measure will result in the ordered weighted averaging (OWA) operator. Submodular fuzzy measures result in convex functions, while supermodular fuzzy measures result in concave functions when used to define a Choquet integral.
Let g be a fuzzy measure, the Möbius representation of g is given by the set function M, where for every ,
The equivalent axioms in Möbius representation are:
A fuzzy measure in Möbius representation M is called normalized if
Möbius representation can be used to give an indication of which
subsets of X interact with one another. For instance, an
additive fuzzy measure has Möbius values all equal to zero except
for singletons. The fuzzy measure g in standard
representation can be recovered from the Möbius form using the Zeta
transform:
Since fuzzy measures are defined on the power set, even in discrete cases the number of variables can be quite high (2^{X}). For this reason, in the context of Multicriteria decision analysis and other disciplines, simplification assumptions on the fuzzy measure have been introduced so that it is less computationally expensive to determine and use. For instance, when it is assumed the fuzzy measure is additive, it will hold that and the values of the fuzzy measure can be evaluated from the values on X. Similarly, a symmetric fuzzy measure is defined uniquely by X values. Two important fuzzy measures that can be used are the Sugeno or λfuzzy measure and kadditive measures, introduced by Sugeno^{[2]} and Grabisch^{[3]} respectively.
The Sugeno λmeasure is a special case of fuzzy measures defined iteratively. It has the following definition:
Let be a finite set and let . A Sugeno λmeasure is a function g from 2^{X} to [0, 1] with properties:
As a convention, the value of g at a singleton set is called a density and is denoted by . In addition, we have that λ satisﬁes the property
Tahani and Keller ^{[4]}as well as Wang and Klir have showed that that once the densities are known, it is possible to use the previous polynomial to obtain the values of λ uniquely.
The kadditive fuzzy measure limits the interaction between the subsets to size  E  = k. This drastically reduces the number of variables needed to define the fuzzy measure, and as k can be anything from 1 (in which case the fuzzy measure is additive) to  X  , it allows for a compromise between modeling ability and simplicity.
A discrete fuzzy measure g on a set X is called kadditive () if its Möbius representation verifies M(E) = 0, whenever  E  > k for any , and there exists a subset F with k elements such that .
In game theory, the Shapley value or Shapley index is used to indicate the weight of a game. Shapley values can calculated for fuzzy measures in order to give some indication of the importance of each singleton. In the case of additive fuzzy measures, the Shapley value will be the same as each singleton.
For a given fuzzy measure g, and  X  = n, the Shapley index for every is:
The Shapley value is the vector
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Fuzzy measure theory considers a number of special classes of measures, each of which is characterized by a special property. Some of the measures used in this theory are plausibility and belief measures, fuzzy set membership function and the classical probability measures. In the fuzzy measure theory, the conditions are precise, but the information about an element alone is insufficient to determine which special classes of measure should be used. The central concept of fuzzy measure theory is the fuzzy measure (also capacity, see ^{[1]}) which was introduced by Choquet in 1953 and independently defined by Sugeno in 1974 in the context of fuzzy integrals.
Contents 
Fuzzy measure can be considered as generalization of the classical probability measure. A fuzzy measure g over a set X (the universe of discourse with the subsets E, F, ...) satisfies the following conditions:
A fuzzy measure g is called normalized if $g(X)=1$.
For any $E,F\; \backslash subseteq\; X$, a fuzzy measure is:
Understanding the properties of fuzzy measures is useful in application. When a fuzzy measure is used to define a function such as the Sugeno integral or Choquet integral, these properties will be crucial in understanding the function's behavior. For instance, the Choquet integral with respect to an additive fuzzy measure reduces to the Lebesgue integral. In discrete cases, a symmetric fuzzy measure will result in the ordered weighted averaging (OWA) operator. Submodular fuzzy measures result in convex functions, while supermodular fuzzy measures result in concave functions when used to define a Choquet integral.
Let g be a fuzzy measure, the Möbius representation of g is given by the set function M, where for every $E,F\; \backslash subseteq\; X$,
The equivalent axioms in Möbius representation are:
A fuzzy measure in Möbius representation M is called normalized if $\backslash sum\_\{E\; \backslash subseteq\; X\}M(E)=1.$
Möbius representation can be used to give an indication of which subsets of X interact with one another. For instance, an additive fuzzy measure has Möbius values all equal to zero except for singletons. The fuzzy measure g in standard representation can be recovered from the Möbius form using the Zeta transform:
Since fuzzy measures are defined on the power set, even in discrete cases the number of variables can be quite high ($2^X$). For this reason, in the context of Multicriteria decision analysis and other disciplines, simplification assumptions on the fuzzy measure have been introduced so that it is less computationally expensive to determine and use. For instance, when it is assumed the fuzzy measure is additive, it will hold that $g(E)\; =\; \backslash sum\_\{i\; \backslash in\; E\}\; g(\backslash \{i\backslash \})$ and the values of the fuzzy measure can be evaluated from the values on X. Similarly, a symmetric fuzzy measure is defined uniquely by X values. Two important fuzzy measures that can be used are the Sugeno or $\backslash lambda$fuzzy measure and kadditive measures, introduced by Sugeno^{[2]} and Grabisch^{[3]} respectively.
The Sugeno $\backslash lambda$measure is a special case of fuzzy measures defined iteratively. It has the following definition:
Let $X\; =\; \backslash left\backslash lbrace\; x\_1,\backslash dots,x\_n\; \backslash right\backslash rbrace$ be a finite set and let $\backslash lambda\; \backslash in\; (1,+\backslash infty)$. A Sugeno $\backslash lambda$measure is a function g from $2^X$ to [0, 1] with properties:
As a convention, the value of g at a singleton set $\backslash left\backslash lbrace\; x\_i\; \backslash right\backslash rbrace$ is called a density and is denoted by $g\_i\; =\; g(\backslash left\backslash lbrace\; x\_i\; \backslash right\backslash rbrace)$. In addition, we have that $\backslash lambda$ satisﬁes the property
Tahani and Keller ^{[4]}as well as Wang and Klir have showed that once the densities are known, it is possible to use the previous polynomial to obtain the values of $\backslash lambda$ uniquely.
The kadditive fuzzy measure limits the interaction between the subsets $E\; \backslash subseteq\; X$ to size $E=k$. This drastically reduces the number of variables needed to define the fuzzy measure, and as k can be anything from 1 (in which case the fuzzy measure is additive) to $X$, it allows for a compromise between modelling ability and simplicity.
A discrete fuzzy measure g on a set X is called kadditive ($1\; \backslash leq\; k\; \backslash leq\; X$) if its Möbius representation verifies $M(E)\; =\; 0$, whenever $E\; >\; k$ for any $E\; \backslash subseteq\; X$, and there exists a subset F with k elements such that $M(F)\; \backslash neq\; 0$.
In game theory, the Shapley value or Shapley index is used to indicate the weight of a game. Shapley values can calculated for fuzzy measures in order to give some indication of the importance of each singleton. In the case of additive fuzzy measures, the Shapley value will be the same as each singleton.
For a given fuzzy measure g, and $X=n$, the Shapley index for every $i,\backslash dots,n\; \backslash in\; X$ is:
The Shapley value is the vector $\backslash mathbf\{\backslash phi\}(g)\; =\; (\backslash psi(1),\backslash dots,\backslash psi(n)).$
