Inmathematics,fuzzy measure theory considers generalizedmeasures in which the additive property is replaced by the weaker property of monotonicity. The central concept of fuzzy measure theory is the fuzzy measure (alsocapacity, see^[1]), which was introduced byChoquet in 1953 and independently defined by Sugeno in 1974 in the context offuzzy integrals. There exists a number of different classes of fuzzy measures includingplausibility/belief measures,possibility/necessity measures, andprobability measures, which are a subset ofclassical measures.

Let${\displaystyle \mathbf{X}}$ be auniverse of discourse,${\displaystyle \mathcal{C}}$ be aclass ofsubsets of${\displaystyle \mathbf{X}}$, and${\displaystyle E,F\in \mathcal{C}}$. Afunction${\displaystyle g:\mathcal{C}\to \mathbb{R}}$ where

1. ${\displaystyle \mathrm{\varnothing }\in \mathcal{C}\Rightarrow g(\mathrm{\varnothing })=0}$
2. ${\displaystyle E\subseteq F\Rightarrow g(E)\le g(F)}$

is called afuzzy measure. A fuzzy measure is callednormalized orregular if${\displaystyle g(\mathbf{X})=1}$.

A fuzzy measure is:

• additive if for any${\displaystyle E,F\in \mathcal{C}}$ such that${\displaystyle E\cap F=\mathrm{\varnothing }}$, we have${\displaystyle g(E\cup F)=g(E)+g(F).}$;
• supermodular if for any${\displaystyle E,F\in \mathcal{C}}$, we have${\displaystyle g(E\cup F)+g(E\cap F)\ge g(E)+g(F)}$;
• submodular if for any${\displaystyle E,F\in \mathcal{C}}$, we have${\displaystyle g(E\cup F)+g(E\cap F)\le g(E)+g(F)}$;
• superadditive if for any${\displaystyle E,F\in \mathcal{C}}$ such that${\displaystyle E\cap F=\mathrm{\varnothing }}$, we have${\displaystyle g(E\cup F)\ge g(E)+g(F)}$;
• subadditive if for any${\displaystyle E,F\in \mathcal{C}}$ such that${\displaystyle E\cap F=\mathrm{\varnothing }}$, we have${\displaystyle g(E\cup F)\le g(E)+g(F)}$;
• symmetric if for any${\displaystyle E,F\in \mathcal{C}}$, we have${\displaystyle |E|=|F|}$ implies${\displaystyle g(E)=g(F)}$;
• Boolean if for any${\displaystyle E\in \mathcal{C}}$, we have${\displaystyle g(E)=0}$ or${\displaystyle g(E)=1}$.

Understanding the properties of fuzzy measures is useful in application. When a fuzzy measure is used to define a function such as theSugeno integral orChoquet 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 theLebesgue integral. In discrete cases, a symmetric fuzzy measure will result in theordered 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.

Letg be a fuzzy measure. The Möbius representation ofg is given by the set functionM, where for every${\displaystyle E,F\subseteq X}$,

${\displaystyle M(E)=\sum _{F\subseteq E}(-1{)}^{|E\mathrm{\setminus }F|}g(F).}$

The equivalent axioms in Möbius representation are:

1. ${\displaystyle M(\mathrm{\varnothing })=0}$.
2. ${\displaystyle \sum _{F\subseteq E|i\in F}M(F)\ge 0}$, for all${\displaystyle E\subseteq \mathbf{X}}$ and all${\displaystyle i\in E}$

A fuzzy measure in Möbius representationM is callednormalizedif${\displaystyle \sum _{E\subseteq \mathbf{X}}M(E)=1.}$

Möbius representation can be used to give an indication of which subsets ofX interact with one another. For instance, an additive fuzzy measure has Möbius values all equal to zero except for singletons. The fuzzy measureg in standard representation can be recovered from the Möbius form using the Zeta transform:

${\displaystyle g(E)=\sum _{F\subseteq E}M(F),\mathrm{\forall }E\subseteq \mathbf{X}.}$

Fuzzy measures are defined on asemiring of sets ormonotone class, which may be as granular as thepower set ofX, and even in discrete cases the number of variables can be as large as 2^|X|. For this reason, in the context ofmulti-criteria 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 isadditive, it will hold that${\displaystyle g(E)=\sum _{i\in E}g(\{i\})}$ and the values of the fuzzy measure can be evaluated from the values onX. Similarly, asymmetric fuzzy measure is defined uniquely by |X| values. Two important fuzzy measures that can be used are the Sugeno- or${\displaystyle \lambda }$-fuzzy measure andk-additive measures, introduced by Sugeno^[2] and Grabisch^[3] respectively.

The Sugeno${\displaystyle \lambda }$-measure is a special case of fuzzy measures defined iteratively. It has the following definition:

Let${\displaystyle \mathbf{X}=\left\{x_1,\dots ,x_n\right\}}$ be a finite set and let${\displaystyle \lambda \in (-1,+\mathrm{\infty })}$. ASugeno${\displaystyle \lambda }$-measure is a function${\displaystyle g:2^X\to [0,1]}$ such that

1. ${\displaystyle g(X)=1}$.
2. if${\displaystyle A,B\subseteq \mathbf{X}}$ (alternatively${\displaystyle A,B\in 2^{\mathbf{X}}}$) with${\displaystyle A\cap B=\mathrm{\varnothing }}$ then${\displaystyle g(A\cup B)=g(A)+g(B)+\lambda g(A)g(B)}$.

As a convention, the value of g at a singleton set${\displaystyle \left\{x_i\right\}}$is called a density and is denoted by${\displaystyle g_i=g(\left\{x_i\right\})}$. In addition, we have that${\displaystyle \lambda }$ satisfies the property

${\displaystyle \lambda +1=\prod _{i=1}^n(1+\lambda g_i)}$.

Tahani and Keller^[4] as well as Wang and Klir have shown that once the densities are known, it is possible to use the previouspolynomial to obtain the values of${\displaystyle \lambda }$ uniquely.

Thek-additive fuzzy measure limits the interaction between the subsets${\displaystyle E\subseteq X}$ to size${\displaystyle |E|=k}$. This drastically reduces the number of variables needed to define the fuzzy measure, and ask can be anything from 1 (in which case the fuzzy measure is additive) toX, it allows for a compromise between modelling ability and simplicity.

A discrete fuzzy measureg on a setX is calledk-additive (${\displaystyle 1\le k\le |\mathbf{X}|}$) if its Möbius representation verifies${\displaystyle M(E)=0}$, whenever${\displaystyle |E|>k}$ for any${\displaystyle E\subseteq \mathbf{X}}$, and there exists a subsetF withk elements such that${\displaystyle M(F)\ne 0}$.

Ingame theory, theShapley value or Shapley index is used to indicate the weight of a game. Shapley values can be 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 measureg, and${\displaystyle |\mathbf{X}|=n}$, the Shapley index for every${\displaystyle i,\dots ,n\in X}$ is:

${\displaystyle \varphi (i)=\sum _{E\subseteq \mathbf{X}\mathrm{\setminus }\{i\}}\frac{(n-|E|-1)!|E|!}{n!}[g(E\cup \{i\})-g(E)].}$

The Shapley value is the vector${\displaystyle \varphi (g)=(\psi (1),\dots ,\psi (n)).}$

• Probability theory
• Possibility theory

• Beliakov, Pradera and Calvo,Aggregation Functions: A Guide for Practitioners, Springer, New York 2007.
• Wang, Zhenyuan, and,George J. Klir,Fuzzy Measure Theory, Plenum Press, New York, 1991.

• Fuzzy Measure Theory at Fuzzy Image ProcessingArchived 2019-06-30 at theWayback Machine

• Exotic probabilities
• Measure theory
• Fuzzy logic

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