# The \(\lambda \)-additive measure in a new light: the \(Q_{\nu }\) measure and its connections with belief, probability, plausibility, rough sets, multi-attribute utility functions and fuzzy operators

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## Abstract

The aim of this paper is twofold. On the one hand, the \(\lambda \)-additive measure (Sugeno \(\lambda \)-measure) is revisited, and a state-of-the-art summary of its most important properties is provided. On the other hand, the so-called \(\nu \)-additive measure as an alternatively parameterized \(\lambda \)-additive measure is introduced. Here, the advantages of the \(\nu \)-additive measure are discussed, and it is demonstrated that these two measures are closely related to various areas of science. The motivation for introducing the \(\nu \)-additive measure lies in the fact that its parameter \(\nu \in (0,1)\) has an important semantic meaning as it is the fix point of the complement operation. Here, by utilizing the \(\nu \)-additive measure, some well-known results concerning the \(\lambda \)-additive measure are put into a new light and rephrased in more advantageous forms. It is discussed here how the \(\nu \)-additive measure is connected with the belief-, probability- and plausibility measures. Next, it is also shown that two \(\nu \)-additive measures, with the parameters \(\nu _1\) and \(\nu _2\), are a dual pair of belief- and plausibility measures if and only if \(\nu _1+\nu _2 = 1\). Furthermore, it is demonstrated how a \(\nu \)-additive measure (or a \(\lambda \)-additive measure) can be transformed to a probability measure and vice versa. Lastly, it is discussed here how the \(\nu \)-additive measures are connected with rough sets, multi-attribute utility functions and certain operators of fuzzy logic.

## Keywords

Belief Probability Plausibility \(\lambda \)-Additive measure Rough sets Multi-attribute utility functions## 1 Introduction

It is an acknowledged fact that the \(\lambda \)-additive measure (Sugeno \(\lambda \)-measure) (Sugeno 1974) is one of the most widely applied monotone measures (fuzzy measure). The usefulness, versatility and applicability of \(\lambda \)-additive measures have inspired numerous theoretical and practical researches since Sugeno’s original results were published in 1974 (see, e.g., Magadum and Bapat 2018; Mohamed and Xiao 2003; Chiţescu 2015; Chen et al. 2016; Singh 2018).

- (1)
belief measure if and only if \(0<\nu \le 1/2\)

- (2)
probability measure if and only if \(\nu =1/2\)

- (3)
plausibility measure if and only if \(1/2 \le \nu <1\).

The rest of this paper is structured as follows: In Sect. 2, we give an overview of the monotone (fuzzy) measures including the belief-, probability- and plausibility measures. In Sect. 3, the \(\nu \)-additive measure is introduced and its key properties are discussed. In Sect. 4, we demonstrate how the \(\nu \)-additive measure is related to the belief-, probability- and plausibility measures, and in Sect. 5, we show how a \(\nu \)-additive measure can be transformed to a probability measure and vice versa. Section 6 reveals some areas of science which the \(\nu \)-additive (\(\lambda \)-additive) measures are connected with. Lastly, in Sect. 7, we give a short summary of our findings and highlight our future research plans including the possible application of \(\nu \)-additive measure in network science.

In this study, we will use the common notations \(\cap \) and \(\cup \) for the intersection and union operations over sets, respectively. Also, will use the notation \(\overline{A}\) for the complement of set *A*.

## 2 Monotone measures

Now, we will introduce the monotone measures and give a short overview of them that covers the probability-, belief- and plausibility measures.

### Definition 1

*X*. Then, the function \(g: \varSigma \rightarrow [0,1]\) is a monotone measure on the measurable space \((X, \varSigma )\) iff

*g*satisfies the following requirements:

- (1)
\(g(\emptyset ) = 0\), \(g(X) = 1\)

- (2)
if \(B \subseteq A\), then \(g(B) \le g(A)\) for any \(A, B \in \varSigma \) (monotonicity)

- (3)if \(\forall i \in \mathbb {N}, A_i \in \varSigma \) and \((A_i) \) is monotonic \((A_1 \subseteq A_2 \subseteq \cdots \subseteq A_n \subseteq \cdots \) or \(A_1 \supseteq A_2 \supseteq \cdots \supseteq A_n \cdots )\), then$$\begin{aligned} \lim \limits _{i \rightarrow \infty } g(A_i) = g\left( \lim \limits _{i \rightarrow \infty } A_{i} \right) \text {(continuity).} \end{aligned}$$

If *X* is a finite set, then the continuity requirement in Definition 1 can be disregarded and the monotone measure is defined as follows.

### Definition 2

*X*iff

*g*satisfies the following requirements:

- (1)
\(g(\emptyset ) = 0\), \(g(X) = 1\)

- (2)
if \(B \subseteq A\), then \(g(B) \le g(A)\) for any \(A, B \in \mathscr {P}(X)\) (monotonicity).

Note that the monotone measures given by Definitions 1 and 2 are known as fuzzy measures, which were originally defined by Choquet (1954) and Sugeno (1974).

### 2.1 Some examples of monotone measures

#### 2.1.1 Dirac measure

### Definition 3

*X*, iff \(\forall A \in \mathscr {P}(X)\):

#### 2.1.2 Probability measure

### Definition 4

*X*. Then, the function \(Pr: \varSigma \rightarrow [0,1]\) is a probability measure on the space \((X, \varSigma )\) iff

*Pr*satisfies the following requirements:

- (1)
\(\forall A \in \varSigma : Pr(A) \ge 0\)

- (2)
\(Pr(X) = 1\)

- (3)\(\forall A_{1}, A_{2}, \ldots \in \varSigma \), if \(A_i \cap A_j = \emptyset , \forall i\ne j\), then$$\begin{aligned} Pr\left( \bigcup \limits _{i=1}^{\infty } A_i \right) =\sum \limits _{i=1}^{\infty }Pr(A_i). \end{aligned}$$

### Remark 1

If *X* is a finite set, then requirement (3) in Definition 4 can be reduced to the following requirement: for any disjoint \(A, B \in \mathscr {P}(X)\), \(Pr(A \cup B) = Pr(A) + Pr(B)\).

#### 2.1.3 Belief measure and plausibility measure

### Definition 5

*X*, iff

*Bl*satisfies the following requirements:

- (1)
\(Bl(\emptyset ) = 0, Bl(X) = 1\)

- (2)for any \(A_1, A_2, \ldots , A_n \in \mathscr {P}(X)\),$$\begin{aligned} \begin{aligned}&Bl( A_1 \cup A_2 \cup \cdots \cup A_n) \\&\quad \ge \sum \limits _{k=1}^n \sum \limits _{\begin{array}{c} 1 \le i_{1}< i_{2} \cdots \\ \cdots < i_{k} \le n \end{array}} (-1)^{k-1} Bl \left( A_{i_{1}} \cap A_{i_{1}} \cap \cdots \cap A_{i_{k}} \right) . \end{aligned} \end{aligned}$$(1)

Here, *Bl*(*A*) is interpreted as a grade of belief in that a given element of *X* belongs to *A*.

### Lemma 1

*Bl*is a belief measure on the finite set

*X*, then for any \(A \in \mathscr {P}(X)\),

### Proof

The inequality \(Bl(A) + Bl(\overline{A}) \le 1\) means that a lack of belief in \(x \in A\) does not imply a strong belief in \(x \in \overline{A}\). In particular, total ignorance is modeled by the belief function \(Bl_i\) such that \(Bl_i(A) = 0\) if \(A\ne X\) and \(Bl_i(A) = 1\) if \(A = X\).

The following proposition is about the monotonicity of belief measures.

### Proposition 1

If *X* is a finite set, *Bl* is a belief measure on *X*, \(A,B \in \mathscr {P}(X)\) and \(B \subseteq A\), then \(Bl(B) \le Bl(A)\).

### Proof

### Corollary 1

The belief measure given by Definition 5 is a monotone measure.

### Proof

Let *Bl* be a belief measure. It follows from Definition 5 that *Bl* satisfies criterion (1) for a monotone measure given in Definition 2. Moreover, the monotonicity of *Bl* is proven in Proposition 1; that is, *Bl* also satisfies criterion (2) in Definition 2. \(\square \)

### Definition 6

*X*, iff

*Pl*satisfies the following requirements:

- (1)
\(Pl(\emptyset ) = 0, Pl(X) = 1\)

- (2)for any \(A_1, A_2, \ldots , A_n \in \mathscr {P}(X)\),$$\begin{aligned} \begin{aligned}&Pl( A_1 \cap A_2 \cap \cdots \cap A_n) \\&\quad \le \sum \limits _{k=1}^n \sum \limits _{\begin{array}{c} 1 \le i_{1}< i_{2} \cdots \\ \cdots < i_{k} \le n \end{array}} (-1)^{k-1} Pl \left( A_{i_{1}} \cup A_{i_{2}} \cdots \cup A_{i_{k}} \right) . \end{aligned} \end{aligned}$$(2)

Here, *Pl*(*A*) is interpreted as the plausibility of *A*.

### Lemma 2

*Pl*is a plausibility measure on the finite set

*X*, then for any \(A \in \mathscr {P}(X)\),

### Proof

This result can be interpreted so that the plausibility of \(x \in A\) does not imply a strong plausibility of \(x \in \overline{A}\).

The following proposition is about the monotonicity of plausibility measures.

### Proposition 2

If *X* is a finite set, *Pl* is a plausibility measure on *X*, \(A,B \in \mathscr {P}(X)\) and \(B \subseteq A\), then \(Pl(B) \le Pl(A)\).

### Proof

### Corollary 2

The plausibility measure given by Definition 6 is a monotone measure.

### Proof

Let *Pl* be a plausibility measure. It follows from Definition 6 that *Pl* satisfies criterion (1) for a monotone measure given in Definition 2. Next, the monotonicity of *Pl* is proven in Proposition 2; that is, *Pl* also satisfies criterion (2) in Definition 2. \(\square \)

*A*of the finite set

*X*was defined by Shafer (1976) as

*Bl*is a belief function. The following proposition states an interesting connection between the belief measure and the plausibility measure.

### Proposition 3

*X*be a finite set and let \(\mu _1, \mu _2: \mathscr {P}(X) \rightarrow [0,1]\) be two monotone measures on

*X*such that

*X*if and only if \(\mu _2\) is a plausibility measure on

*X*, or (2) \(\mu _1\) is a plausibility measure on

*X*if and only if \(\mu _2\) is a belief measure on

*X*.

### Proof

*X*and \(\mu _2(A)\) is given as \(\mu _2(A) = 1-\mu _1(\overline{A})\) for any \(A \in \mathscr {P}(X)\), then \(\mu _2\) is a plausibility measure on

*X*. Let \(\mu _1\) be a belief measure on

*X*and \(\mu _2(A) = 1-\mu _1(\overline{A})\) for any \(A \in \mathscr {P}(X)\). Then, \(\mu _2(\emptyset )=0\) and \(\mu _2(X)=1\) trivially follow from the fact that \(\mu _1\) is a belief measure and \(\mu _2(A) = 1-\mu _1(\overline{A})\). That is, function \(\mu _2\) satisfies requirement (1) for a plausibility measure given in Definition 6. Furthermore, since function \(\mu _1\) is a belief measure, the inequality

*X*and \(\mu _2(A)\) is given as \(\mu _2(A) = 1-\mu _1(\overline{A})\) for any \(A \in \mathscr {P}(X)\), then \(\mu _1\) is a belief measure on

*X*. Let \(\mu _2\) be a plausibility measure on

*X*and \(\mu _2(A) = 1-\mu _1(\overline{A})\) for any \(A \in \mathscr {P}(X)\). These conditions trivially imply that \(\mu _1(\emptyset )=0\) and \(\mu _1(X)=1\); that is, function \(\mu _1\) satisfies requirement (1) for a belief measure given in Definition 5. Next, because function \(\mu _2\) is a plausibility measure, the inequality

Later, we will use the concept of dual pair of belief- and plausibility measures.

### Definition 7

*Bl*and

*Pl*be a belief measure and a plausibility measure, respectively, on set

*X*. Then

*Bl*and

*Pl*are said to be a dual pair of belief- and plausibility measures iff

In the Dempster–Shafer theory of evidence, a belief mass is assigned to each element of the power set \(\mathscr {P}(X)\), where *X* is a finite set. The belief mass is given by the so-called basic probability assignment *m* from \(\mathscr {P}(X)\) to [0, 1] that is defined as follows.

### Definition 8

*X*, iff

*m*satisfies the following requirements:

- (1)
\(m (\emptyset ) = 0\)

- (2)
\(\sum _{A\in \mathscr {P}(X)} m(A) = 1\).

*A*of

*X*for which \(m(A) > 0\) are called the focal elements of

*m*. Let \(x \in A\) and \(A \in \mathscr {P}(X)\). Then, the mass

*m*(

*A*) can be interpreted as the probability of knowing \(x \in A\) given the available evidence. Utilizing a given basic probability assignment

*m*, the belief

*Bl*(

*A*) for the set

*A*is

*Pl*(

*A*) is

*m*can be represented by its belief function

*Bl*as

*m*is the basic probability assignment of the belief measure

*Bl*. Note that plausibility measures and belief functions were introduced by Dempster (1967) under the names upper and lower probabilities, induced by a probability measure by a multi-valued mapping.

### Remark 2

*Pl*can also be demonstrated by utilizing the duality \(Pl(A) = 1-Bl(\overline{A})\) and the monotonicity of the belief measure

*Bl*. Namely, if \(B \subseteq A\), then \(\overline{A} \subseteq \overline{B}\) and so

## 3 Introduction to the \(Q_{\nu }\) measure

Relaxing the additivity property of the probability measure, the \(\lambda \)-additive measures were proposed by Sugeno (1974).

### Definition 9

*X*, iff \(Q_{\lambda }\) satisfies the following requirements:

- (1)
\(Q_{\lambda }(X) = 1\)

- (2)for any \(A, B \in \mathscr {P}(X)\) and \(A \cap B=\emptyset \),where \(\lambda \in (-1, \infty )\).$$\begin{aligned} Q_{\lambda }(A \cup B)=Q_{\lambda }(A)+Q_{\lambda }(B)+\lambda Q_{\lambda }(A)Q_{\lambda }(B), \end{aligned}$$(7)

Note that if *X* is an infinite set, then the continuity of function \(Q_{\lambda }\) is also required. Here, we will show that the \(\lambda \)-additive measures are monotone measures as well.

### Proposition 4

Every \(\lambda \)-additive measure is a monotone measure.

### Proof

Let \(Q_{\lambda }\) be a \(\lambda \)-additive measure on the set *X*. Then \(Q_{\lambda }(X)=1\) holds by definition. Next, by utilizing Eq. (7), we get \(Q_{\lambda }(X)=Q_{\lambda }(X \cup \emptyset ) =Q_{\lambda }(X)+Q_{\lambda }(\emptyset )(1+ \lambda Q_{\lambda }(X))\), which implies that \(Q_{\lambda }(\emptyset )=0\). Thus, \(Q_{\lambda }\) satisfies criterion (1) of a monotone measure given in Definition 2.

### Remark 3

### 3.1 The \(\lambda \)-additive complement and the Dombi form of negation

### Proposition 5

*X*is a finite set and \(Q_{\lambda }\) is a \(\lambda \)-additive measure on

*X*, then for any \(A \in \mathscr {P}(X)\) the \(Q_{\lambda }\) measure of the complement set \(\overline{A} = X \setminus A\) is

### Proof

### Remark 4

*X*is a finite set and \(Q_{\lambda }\) is a \(\lambda \)-additive measure on

*X*, then for any \(A \in \mathscr {P}(X)\) the \(Q_{\lambda }\) measure of the complement set \(\overline{A} = X \setminus A\) is

Following this line of thinking, here, we will introduce the \(\nu \)-additive measure and state some of its properties.

### Definition 10

*X*, iff \(Q_{\nu }\) satisfies the following requirements:

- (1)
\(Q_{\nu }(X) = 1\)

- (2)for any \(A, B \in \mathscr {P}(X)\) and \(A \cap B=\emptyset \),where \(\nu \in (0, 1)\).$$\begin{aligned} \begin{aligned}&Q_{\nu }(A \cup B)=Q_{\nu }(A)+Q_{\nu }(B) \\&\quad + \left( \left( \frac{1 - \nu }{\nu } \right) ^{2}-1 \right) Q_{\nu }(A)Q_{\nu }(B), \end{aligned} \end{aligned}$$(14)

Note that if *X* is an infinite set, then the continuity of function \(Q_{\nu }\) is also required. Here, we state a key proposition that we will frequently utilize later on.

### Proposition 6

*X*be a finite set, and let \(Q_{\lambda }\) and \(Q_{\nu }\) be a \(\lambda \)-additive and a \(\nu \)-additive measure on

*X*, respectively. Then,

### Proof

This proposition immediately follows from the definitions of the \(\lambda \)-additive measure and \(\nu \)-additive measure. \(\square \)

*X*, then by utilizing Eq. (13), the \(Q_{\nu }\) measure of the complement set \(\overline{A}\) is

### Proposition 7

*X*be a finite set, \(Q_{\nu }\) a \(\nu \)-additive measure on

*X*and let the set \(A_{\nu }\) be given as

### Proof

This result means that the \(\nu \)-additive complement operation may be viewed as a complement operation characterized by its fix point \(\nu \).

### 3.2 Main properties of the \(\nu \)-additive (\(\lambda \)-additive) measures

It is worth mentioning that the definition of the \(\nu \)-additive measure is the same as that of the \(\lambda \)-additive measure with an alternative parameterization. Thus, utilizing the fact that any \(\nu \)-additive measure is a \(\lambda \)-additive measure with \(\lambda = \left( \frac{1 - \nu }{\nu } \right) ^{2} -1\), some of the properties of \(\lambda \)-additive measures can be expressed in terms of \(\nu \)-additive measures and vice versa. In this section, we will discuss the main properties of these two measures. In many cases, to make the calculations simpler, we will use the \(\lambda \)-additive form to demonstrate some properties and then we will state them in terms of the \(\nu \)-additive measure as well. We will follow this approach from now on, and \(Q_{\lambda }\) will always denote a \(\lambda \)-additive measure with the parameter \(\lambda \in (-\,1, \infty )\) and \(Q_{\nu }\) will always denote a \(\nu \)-additive measure with the parameter \(\nu \in (0, 1)\).

#### 3.2.1 \(\nu \)-additive (\(\lambda \)-additive) measure of collection of disjoint sets

Here, we will outline the computation of the \(\nu \)-additive (\(\lambda \)-additive) measure of collection of pairwise disjoint sets.

### Proposition 8

*X*is a finite set, \(Q_{\lambda }\) is a \(\lambda \)-additive measure on

*X*and \(A_1, A_2, \ldots , A_n \in \mathscr {P}(X)\) are pairwise disjoint sets, then

### Proof

- (1)
In this case, the proposition trivially follows from the definition of the \(\lambda \)-additive measures.

- (2)Here, we will apply induction. By utilizing the definition of the \(\lambda \)-additive measures, the associativity of the union operation over sets and simple calculations, it can be shown thatholds for \(n=2\) and \(n=3\), where \(A_{1}, A_{2}, A_{3} \in \mathscr {P}(X)\), \(\lambda >-1\), \(\lambda \ne 0\). Now, let us assume that Eq. (19) holds for any \(A_{1}, A_{2}, \ldots , A_{n} \in \mathscr {P}(X)\), \(\lambda >-1\), \(\lambda \ne 0\). Let \(G_{n}\) be defined as follows:$$\begin{aligned} Q_{\lambda } \left( \bigcup \limits _{i=1}^n A_i \right) = \frac{1}{\lambda } \left( \prod \limits _{i=1}^{n} (1+\lambda Q_{\lambda }( A_i))-1 \right) \end{aligned}$$(19)With this notation, \(G_{n+1} = G_{n} \left( 1 + \lambda Q_{\lambda }(A_{n+1}) \right) \), and the equality that we seek to prove is$$\begin{aligned} G_{n} = \prod \limits _{i=1}^{n} (1+\lambda Q_{\lambda }( A_i)). \end{aligned}$$By utilizing the definition of the \(\lambda \)-additive measures and the associativity of the union operation over sets, we get$$\begin{aligned} Q_{\lambda } \left( \bigcup \limits _{i=1}^{n+1} A_i \right) = \frac{1}{\lambda } \left( G_{n+1}-1 \right) . \end{aligned}$$Now, utilizing the inductive condition, the last equation can be written as$$\begin{aligned} \begin{aligned} Q_{\lambda } \left( \bigcup \limits _{i=1}^{n+1} A_i \right)&= Q_{\lambda } \left( \bigcup \limits _{i=1}^{n} A_i \right) + Q_{\lambda }(A_{n+1}) \\&\quad + \lambda Q_{\lambda } \left( \bigcup \limits _{i=1}^{n} A_i \right) Q_{\lambda }(A_{n+1}). \end{aligned} \end{aligned}$$$$\begin{aligned} \begin{aligned}&Q_{\lambda } \left( \bigcup \limits _{i=1}^{n+1} A_i \right) = \frac{1}{\lambda } \left( G_{n}-1 \right) + Q_{\lambda }(A_{n+1}) \\&\qquad + \lambda \frac{1}{\lambda } \left( G_{n}-1 \right) Q_{\lambda }(A_{n+1}) \\&\quad = \frac{1}{\lambda } \left( G_{n}-1 \right) \left( 1 + \lambda Q_{\lambda }(A_{n+1}) \right) + Q_{\lambda }(A_{n+1}) \\&\quad = \frac{1}{\lambda } G_{n} \left( 1 + \lambda Q_{\lambda }(A_{n+1}) \right) - \frac{1}{\lambda } \\&\quad = \frac{1}{\lambda } \left( G_{n} \left( 1 + \lambda Q_{\lambda }(A_{n+1}) \right) - 1 \right) = \frac{1}{\lambda } \left( G_{n+1}-1 \right) . \end{aligned} \end{aligned}$$

### Remark 5

Proposition 8 can be stated in terms of the \(\nu \)-additive measure as follows.

### Proposition 9

*X*is a finite set, \(Q_{\nu }\) is a \(\nu \)-additive measure on

*X*and \(A_1, A_2, \ldots , A_n \in \mathscr {P}(X)\) are pairwise disjoint sets, then

#### 3.2.2 General forms for the \(\nu \)-additive (\(\lambda \)-additive) measure of union and intersection of two sets

The calculations of the \(\lambda \)-additive measure and \(\nu \)-additive measure of two disjoint sets are given in Definitions 9 and 10, respectively. Here, we will show how the \(\nu \)-additive (\(\lambda \)-additive) measure of two sets can be computed when these sets are not disjoint. We will also discuss how the \(\nu \)-additive (\(\lambda \)-additive) measure of intersection of two sets can be computed.

### Proposition 10

*X*is a finite set and \(Q_{\lambda }\) is a \(\lambda \)-additive measure on

*X*, then for any \(A, B \in \mathscr {P}(X)\),

### Proof

### Remark 6

Notice that if \(\lambda =0\), then Eq. (23) reduces to \(Q_{\lambda } ({A} \cup B) = Q_{\lambda } (A)+Q_{\lambda } (B) - Q_{\lambda } (A \cap B)\), which has the same form as the probability measure of union of two sets. Later, we will discuss how the \(\lambda \)-additive (\(\nu \)-additive) measure is related to the probability measure.

### Remark 7

### Corollary 3

*X*is a finite set and \(Q_{\lambda }\) is a \(\lambda \)-additive measure on

*X*, then for any \(A, B \in \mathscr {P}(X)\),

#### 3.2.3 Other properties of the \(\nu \)-additive (\(\lambda \)-additive) measure of union and the intersection of two sets

The following results are related to the \(\nu \)-additive measure of union and the intersection of two sets.

### Proposition 11

*X*be a finite set, \(Q_{\nu }\) be a \(\nu \)-additive measure on

*X*and let \(A, B \in \mathscr {P}(X)\). Then

- (1)if \(A \cup B=X\) (complementing case), then$$\begin{aligned} \begin{aligned}&Q_{\nu } ({A} \cap B)= Q_{\nu }(A) Q_{\nu }(B) \\&\quad -\left( \frac{\nu }{1-\nu } \right) ^{2} (1- Q_{\nu }(A))(1-Q_{\nu }(B)) \end{aligned} \end{aligned}$$(25)
- (2)if \(A \cap B=\emptyset \) (disjoint case), then$$\begin{aligned} \begin{aligned}&Q_{\nu } ({A} \cup B) \\&\quad =1- \Bigg ( (1-Q_{\nu }(A))(1-Q_{\nu }(B)) \\&\qquad - \left( \frac{1 - \nu }{\nu } \right) ^{2} Q_{\nu }(A) Q_{\nu }(B) \Bigg ). \end{aligned} \end{aligned}$$(26)

### Proof

- (1)Since the \(\nu \)-additive measure \(Q_{\nu }\) is identical to the \(\lambda \)-additive measure \(Q_{\lambda }\) with \(\lambda = \left( \frac{1 - \nu }{\nu } \right) ^{2} -1\), \(Q_{\nu }({A} \cap B) = Q_{\lambda } ({A} \cap B)\). Now, utilizing the fact that \(Q_{\nu }(A \cup B)\) = \(Q_{\lambda }(A \cup B)\) = 1 and Eq. (24), \(Q_{\lambda } ({A} \cap B)\) can be written asAnd by using the equation \(\lambda = \left( \frac{1 - \nu }{\nu } \right) ^{2} -1\), we get$$\begin{aligned} \begin{aligned} Q_{\lambda } ({A} \cap B)&= \frac{Q_{\lambda } (A)+Q_{\lambda } (B) + \lambda Q_{\lambda } (A)Q_{\lambda } (B) - 1}{1+ \lambda } \\&= \frac{(1+\lambda ) Q_{\lambda }(A) Q_{\lambda } (B)}{1+\lambda } \\&\quad - \frac{1-Q_{\lambda }(A) - Q_{\lambda }(B) + Q_{\lambda }(A)Q_{\lambda }(B)}{1+\lambda } \\&= Q_{\lambda }(A) Q_{\lambda }(B) - \frac{1}{1+\lambda } (1-Q_{\lambda }(A))(1-Q_{\lambda }(B)). \end{aligned} \end{aligned}$$$$\begin{aligned} \begin{aligned}&Q_{\nu } ({A} \cap B) \\&\quad = Q_{\nu }(A) Q_{\nu }(B) - \left( \frac{\nu }{1-\nu } \right) ^{2} (1- Q_{\nu }(A))(1-Q_{\nu }(B)). \end{aligned} \end{aligned}$$
- (2)Since \(A \cap B = \emptyset \), applying the definition of the \(\nu \)-additive measure gives$$\begin{aligned} \begin{aligned}&Q_{\nu }(A \cup B) \\&\quad = Q_{\nu }(A)+Q_{\nu }(B) + \left( \left( \frac{1 - \nu }{\nu } \right) ^{2}-1 \right) Q_{\nu }(A)Q_{\nu }(B) \\&\quad = 1-(1-Q_{\nu }(A) - Q_{\nu }(B) + Q_{\nu }(A)Q_{\nu }(B)) \\&\qquad + \left( \frac{1 - \nu }{\nu } \right) ^{2} Q_{\nu }(A)Q_{\nu }(B) \\&\quad = 1-\Bigg ( (1-Q_{\nu }(A))(1-Q_{\nu }(B)) \\&\qquad - \left( \frac{1 - \nu }{\nu } \right) ^{2} Q_{\nu }(A) Q_{\nu }(B) \Bigg ). \end{aligned} \end{aligned}$$

Note that the term \(\left( \frac{\nu }{1-\nu } \right) ^{2} (1- Q_{\nu }(A))(1-Q_{\nu }(B))\) in Eq. (25) may be regarded as the corrective term of the intersection; that is, if \(\nu \rightarrow 0\), then \(Q_{\nu } ({A} \cap B)= Q_{\nu }(A) Q_{\nu }(B)\). Similarly, the term \(\left( \frac{1 - \nu }{\nu } \right) ^{2} Q_{\nu }(A) Q_{\nu }(B)\) in Eq. (26) may be interpreted as the corrective term of the union; that is, if \(\nu \rightarrow 1\), then \(Q_{\nu }(A \cup B) = 1-(1-Q_{\nu }(A))(1-Q_{\nu }(B))\).

#### 3.2.4 Characterization by independent variables

*X*is a finite set, \(Q_{\lambda }\) is a \(\lambda \)-additive measure on

*X*, \(\lambda >-1\), \(\lambda \ne 0\), \(A_1, A_2, \ldots , A_n \in \mathscr {P}(X)\) are pairwise disjoint sets, and

### Proposition 12

*X*is a finite set, \(Q_{\lambda }\) is a \(\lambda \)-additive measure on

*X*, \(\lambda >-1\), \(\lambda \ne 0\), \(A_1, A_2, \ldots , A_n \in \mathscr {P}(X)\) are pairwise disjoint sets such that \(Q_{\lambda }(A_i)<1\), \(i \in \lbrace 1,2, \ldots , n \rbrace \), then the equation

### Proof

*f*(

*q*) be given as

*f*is

*q*) in the interval \((-\,1,\infty )\),

- (1)
Equation (31) implies that if \(S<1\), then \(f'(0)>0\). Since \(f'(0)>0\) and \(f'(q)\) is decreasing in the interval \((-\,1,\infty )\), \(f'(q)\) is positive in \((-\,1, 0)\). Therefore, \(f'(q) = 0\) has no root in \((-\,1,0)\). Based on Eq. (32), \(f'(\infty ) = -\infty \), and so \(f'(q) = 0\) has a unique root \(q^{*}\) in \((0,\infty )\). Since \(f(0) = 0\) and \(f'(q) > 0\) in \((0, q^{*})\), \(f(q) = 0\) has no root in \((0, q^{*})\). As \(f(q^{*})>0\) and \(f'(q)\) is negative and decreasing to \(-\infty \) in \((q^{*},\infty )\), \(f(q)=0\) has a unique root \(q_0\) in \((q^{*},\infty )\). Moreover, \(f(q)>0\) in \((0,q_0)\) and \(f(q)<0\) in \((q_0, \infty )\); that is, the unique root \(q_0\) is in \((0, \infty )\).

- (2)
It follows from Eq. (31) that if \(S=1\), then \(f'(0)=0\). Since \(f'(0)=0\) and \(f'(q)\) is decreasing in the interval \((-\,1,\infty )\), \(f'(q)\) is positive in the interval \((-\,1,0)\) and it is negative in the interval \((0, \infty )\). Thus, \(q=0\) is the only root of \(f'(q)=0\) in the interval \((-\,1,\infty )\). Moreover, since \(f(0)=0\), \(q=0\) is the only root of \(f(q)=0\). It means that if \(S=1\), then the only solution of Eq. (29) is \(\lambda =0\). Recall that \(\lambda \ne 0\); that is, in this case we do not get any solution to the equation in (28).

- (3)
Equation (31) implies that if \(S>1\), then \(f'(0)<0\). Since \(f'(0)<0\) and \(f'(q)\) is decreasing in the interval \((-\,1,\infty )\), \(f'(q)\) is negative in \((0,\infty )\). As \(f(0)=0\) and \(f'(q)\) is negative in \((0,\infty )\), \(f(q)=0\) has no root in \((0,\infty )\). On the one hand, as \(f(0)=0\) and \(f'(0)<0\), \(f(q)>0\) immediately to the left of zero. On the other hand, \(f(-\,1) < 0\). It means that there must be at least one root \(q_0\) of \(f(q)=0\) in \((-\,1,0)\). Since \(f'(q)\) is decreasing and \(f(0)=0\), \(q_0\) is the unique root of \(f(q)=0\) in (0, 1). \(\square \)

Proposition 12 tells us that Eq. (27) can be solved numerically for \(\lambda \) in the interval \((-\,1, 0)\) or in the interval \((0, \infty )\). Hence, the \(\lambda \)-additive measure \(Q_{\lambda }\) can be unambiguously characterized by *n* independent variables.

### 3.3 Dual \(\nu \)-additive (\(\lambda \)-additive) measures and their properties

Later, we will utilize the concept of the dual pair of \(\lambda \)-additive measures and the concept of the dual pair of \(\nu \)-additive measures.

### Definition 11

*X*. Then, \(Q_{\lambda _1}\) and \(Q_{\lambda _2}\) are said to be a dual pair of \(\lambda \)-additive measures iff

### Definition 12

*X*. Then, \(Q_{\nu _1}\) and \(Q_{\nu _2}\) are said to be a dual pair of \(\nu \)-additive measures iff

Later, we will utilize the following proposition.

### Proposition 13

*X*and let

### Proof

Here, we will demonstrate some key properties of the \(\nu \)-additive (\(\lambda \)-additive) measure related to a dual pair of \(\nu \)-additive (\(\lambda \)-additive) measures.

### Proposition 14

*X*. Then, \(Q_{\lambda _1}\) and \(Q_{\lambda _2}\) are a dual pair of \(\lambda \)-additive measures if and only if

### Proof

*X*, then \(\lambda _2 = - \frac{\lambda _1}{1+\lambda _1}\). Let \(Q_{\lambda _1}\) and \(Q_{\lambda _2}\) be a dual pair of \(\lambda \)-additive measures on

*X*. It means that \(Q_{\lambda _2} ({A}) = 1 - Q_{\lambda _1}(\overline{A})\) holds for any \(A \in \mathscr {P}(X)\). Next, let \(A,B \in \mathscr {P}(X)\) such that \(A \cap B=\emptyset \). Then, \(X=\overline{A \cap B} = \overline{A} \cup \overline{B}\). Now, noting that \(Q_{\lambda _2} ({A}) = 1 - Q_{\lambda _1}(\overline{A})\), the formula for the \(\lambda \)-additive measure of the intersection of two sets given by Eq. (24) and the fact that \(Q_{\lambda _1}(\overline{A} \cup \overline{B}) = Q_{\lambda _1}(X)=1\), we get

*X*. Let \(\lambda _2 = - \frac{\lambda _1}{1+\lambda _1}\). Here, we seek to show that \(Q_{\lambda _1}\) and \(Q_{\lambda _2}\) are a dual pair of \(\lambda \)-additive measures; that is, \(Q_{\lambda _2} ({A}) = 1 - Q_{\lambda _1}(\overline{A})\) holds for any \(A \in \mathscr {P}(X)\). Now, we will give an indirect proof of this. Let us assume that \(\lambda _2 = - \frac{\lambda _1}{1+\lambda _1}\), but either (1) \(Q_{\lambda _2} ({A}) > 1 - Q_{\lambda _1}(\overline{A})\), or (2) \(Q_{\lambda _2} ({A}) < 1 - Q_{\lambda _1}(\overline{A})\) holds for any \(A \in \mathscr {P}(X)\). We will show that this assumption leads to contradictions. Let \(A,B \in \mathscr {P}(X)\) such that \(A \cap B=\emptyset \). Then, \(X=\overline{A \cap B} = \overline{A} \cup \overline{B}\).

- (1)Here, as \(Q_{\lambda _2} ({A}) > 1 - Q_{\lambda _1}(\overline{A})\) holds for any \(A \in \mathscr {P}(X)\), \(Q_{\lambda _1}(\overline{A}) > 1- Q_{\lambda _2}({A})\) holds as well, and applying it to \(\overline{A} \cap \overline{B}\), we getUtilizing the formula for the \(\lambda \)-additive measure of the intersection of two sets given by Eq. (24) and the fact that \(Q_{\lambda _2}(\overline{A} \cup \overline{B}) = Q_{\lambda _2}(X)=1\), the right-hand side of Eq. (42) can be expressed as$$\begin{aligned} Q_{\lambda _1}(A \cup B) = Q_{\lambda _1}(\overline{\overline{A} \cap \overline{B}}) > 1- Q_{\lambda _2}(\overline{A} \cap \overline{B}). \end{aligned}$$(42)Now, applying the inequality \(Q_{\lambda _2} ({A}) > 1 - Q_{\lambda _1}(\overline{A})\) to set$$\begin{aligned} \begin{aligned}&1- Q_{\lambda _2}(\overline{A} \cap \overline{B}) \\&\quad = 1- \frac{Q_{\lambda _2}(\overline{A}) + Q_{\lambda _2}(\overline{B}) + \lambda _2 Q_{\lambda _2}(\overline{A}) Q_{\lambda _2}(\overline{B}) - 1}{1+ \lambda _2}. \end{aligned} \end{aligned}$$(43)
*B*, we have \(Q_{\lambda _2} ({B}) > 1 - Q_{\lambda _1}(\overline{B})\), and so utilizing the fact that \(\lambda _2 = - \frac{\lambda _1}{1+\lambda _1}\), \(Q_{\lambda _2} ({A}) > 1 - Q_{\lambda _1}(\overline{A})\) and \(Q_{\lambda _2} ({B}) > 1 - Q_{\lambda _1}(\overline{B})\), Proposition 13 yields the inequality relationsand$$\begin{aligned} Q_{\lambda _2} ({\overline{A}}) < 1- Q_{\lambda _1} (A) \end{aligned}$$(44)Next, noting Eqs. (44) and (45), from Eq. (43) we can further derive the result$$\begin{aligned} Q_{\lambda _2} ({\overline{B}}) < 1- Q_{\lambda _1} (B). \end{aligned}$$(45)On the one hand, utilizing \(\lambda _2 = - \frac{\lambda _1}{1+\lambda _1}\), from Eqs. (42), (43) and (46), we get$$\begin{aligned} \begin{aligned}&1- \frac{Q_{\lambda _2}(\overline{A}) + Q_{\lambda _2}(\overline{B}) + \lambda _2 Q_{\lambda _2}(\overline{A}) Q_{\lambda _2}(\overline{B}) - 1}{1+ \lambda _2} \\&\quad > 1- \frac{1-Q_{\lambda _1}(A) + 1-Q_{\lambda _1}(B)}{1+ \lambda _2} \\&\qquad - \frac{\lambda _2 (1-Q_{\lambda _1}(A))(1-Q_{\lambda _1}(B)) - 1}{1+ \lambda _2} \\&\quad = Q_{\lambda _1}(A) + Q_{\lambda _1}(B) - \frac{\lambda _2}{1+\lambda _2} Q_{\lambda _1}(A) Q_{\lambda _1}(B). \end{aligned} \end{aligned}$$(46)On the other hand, as \(A \cap B=\emptyset \) and \(Q_{\lambda _1}\) is a \(\lambda \)-additive measure, we have$$\begin{aligned} Q_{\lambda _1}(A \cup B) > Q_{\lambda _1}(A) + Q_{\lambda _1}(B) + \lambda _1 Q_{\lambda _1}(A) Q_{\lambda _1}(B). \end{aligned}$$Thus, the assumption that \(\lambda _2 = - \frac{\lambda _1}{1+\lambda _1}\) and \(Q_{\lambda _2} ({A}) > 1 - Q_{\lambda _1}(\overline{A})\) leads to a contradiction.$$\begin{aligned} Q_{\lambda _1}(A \cup B) = Q_{\lambda _1}(A) + Q_{\lambda _1}(B) + \lambda _1 Q_{\lambda _1}(A) Q_{\lambda _1}(B). \end{aligned}$$ - (2)Following the same steps as in case (1), the assumption that \(\lambda _2 = - \frac{\lambda _1}{1+\lambda _1}\) and \(Q_{\lambda _2} ({A}) < 1 - Q_{\lambda _1}(\overline{A})\) leads to the inequalitywhich contradicts the fact that \(Q_{\lambda _1}\) is a \(\lambda \)-additive measure.$$\begin{aligned} Q_{\lambda _1}(A \cup B) < Q_{\lambda _1}(A) + Q_{\lambda _1}(B) + \lambda _1 Q_{\lambda _1}(A) Q_{\lambda _1}(B), \end{aligned}$$

*X*. \(\square \)

Proposition 14 can be stated in terms of the \(\nu \)-additive measure as follows.

### Proposition 15

*X*. Then, \(Q_{\nu _1}\) and \(Q_{\nu _2}\) are a dual pair of \(\nu \)-additive measures if and only if

### Proof

Utilizing Proposition 6, this proposition immediately follows from Proposition 14. \(\square \)

Utilizing the definition of the dual pair of \(\lambda \)-additive measures, the following corollary can be stated.

### Corollary 4

Let \(Q_{\lambda _1}\) and \(Q_{\lambda _2}\) be a dual pair of \(\lambda \)-additive measures on the finite set *X*. Then, \(\lambda _1 \in (-\,1, 0]\) if and only if \(\lambda _2 \in [ 0, \infty )\).

### Proof

Since \(\lambda _2=-\frac{\lambda _1}{1+\lambda _1}\) is a bijection from \((-\,1, 0]\) to \([0, \infty )\), this corollary follows from Proposition 14. \(\square \)

Corollary 4 can be stated in terms of the \(\nu \)-additive measure as follows.

### Corollary 5

Let \(Q_{\nu _1}\) and \(Q_{\nu _2}\) be a dual pair of \(\nu \)-additive measures on the finite set *X*. Then, \(\nu _1 \in [ 1/2, 1)\) if and only if \(\nu _2 \in (0, 1/2]\).

### Proof

Taking into account Proposition 6, this corollary immediately follows from Corollary 4. \(\square \)

It should be mentioned here that one of the \(\lambda \) parameters of a dual pair of \(\lambda \)-additive measures is always in the unbounded interval \([0,\infty )\). At the same time, the \(\nu \) parameters of a dual pair of \(\nu \)-additive measures are both in a bounded interval; namely, one of them is in the interval (0, 1 / 2] and the other one is in the interval [1 / 2, 1).

#### 3.3.1 The decomposition property of the \(\lambda \)-additive measure

The following proposition reveals an interesting property of the \(\lambda \)-additive measures.

### Proposition 16

*X*is a finite set and \(Q_{\lambda }\) is a \(\lambda \)-additive measure on

*X*, \(A_1, \ldots , A_n, B_1, \ldots , B_m \in \mathscr {P}(X)\), \(A_i \cap A_j=\emptyset \), \(B_i \cap B_j=\emptyset \) for all \(i \ne j\), \(A_i \cap B_j=\emptyset \) for all

*i*,

*j*, \(\lambda >-\,1\), \(\lambda \ne 0\) and

### Proof

*A*and

*B*are two disjoint sets and \(Q_{\lambda }\) is a \(\lambda \)-additive measure,

## 4 Connection with belief-, probability- and plausibility measures

Here, we will discuss some important properties of the \(\nu \)-additive (\(\lambda \)-additive) measure and how it is connected to the belief-, probability- and plausibility measures.

### Proposition 17

*X*be a finite set and let \(Q_{\lambda }\) be a \(\lambda \)-additive measure on

*X*. Then, on set

*X*, \(Q_{\lambda }\) is a

- (1)
plausibility measure if and only if \(-\,1< \lambda \le 0\)

- (2)
probability measure if and only if \(\lambda =0\)

- (3)
belief measure if and only if \(\lambda \ge 0\).

Note that in terms of the \(\nu \)-additive measure, Proposition 17 can be stated as follows.

### Proposition 18

*X*be a finite set and let \(Q_{\nu }\) be a \(\nu \)-additive measure on

*X*. Then, on set

*X*, \(Q_{\nu }\) is a

- (1)
belief measure if and only if \(0<\nu \le 1/2\)

- (2)
probability measure if and only if \(\nu =1/2\)

- (3)
plausibility measure if and only if \(1/2 \le \nu <1\).

### Proof

Taking into account Proposition 6, this proposition immediately follows from Proposition 17. \(\square \)

Figure 1 shows the connection between \(Q_{\nu }(\overline{A})\) and \(Q_{\nu }(A)\) for various values of parameter \(\nu \) of the \(\nu \)-additive measure \(Q_{\nu }\). From this figure, in accordance with Proposition 18, we notice the following. If \(\nu =1/2\), then \(Q_{\nu }\) is a probability measure and so \(Q_{\nu }(\overline{A}) = 1- Q_{\nu }(A)\). If \(0<\nu \le 1/2\), then \(Q_{\nu }\) is a belief measure and \(Q_{\nu }(\overline{A}) \le 1- Q_{\nu }(A)\). If \(1/2 \le \nu <1\), then \(Q_{\nu }\) is a plausibility measure and \(Q_{\nu }(\overline{A}) \ge 1- Q_{\nu }(A)\). Moreover, in accordance with Eq. (17), for a given set *A*, \(Q_{\nu }(\overline{A})\) increases with the value of parameter \(\nu \). That is, the smaller the value of parameter \(\nu \), the stronger the complement operation. It also means that any belief measure of a complement set is always less than or equal to any plausibility measure of the same complement set.

### Proposition 19

Let \(Q_{\lambda _1}\) and \(Q_{\lambda _2}\) be two \(\lambda \)-additive measures on the finite set *X*. Then, \(Q_{\lambda _1}\) and \(Q_{\lambda _2}\) are a dual pair of belief- and plausibility measures on *X* if and only if they are a dual pair of \(\lambda \)-additive measures on *X*.

### Proof

Firstly, we will show that if the condition of the proposition is satisfied and \(Q_{\lambda _1}\) and \(Q_{\lambda _2}\) are a dual pair of belief- and plausibility measures on *X*, then \(Q_{\lambda _1}\) and \(Q_{\lambda _2}\) are a dual pair of \(\lambda \)-additive measures on *X*. Let \(Q_{\lambda _1}\) and \(Q_{\lambda _2}\) be a dual pair of belief- and plausibility measures on *X*. Since, \(Q_{\lambda _1}\) and \(Q_{\lambda _2}\) are a dual pair; that is, \(Q_{\lambda _2}(A) = 1-Q_{\lambda _1}(\overline{A})\) holds for any \(A \in \mathscr {P}(X)\), and \(Q_{\lambda _1}\) and \(Q_{\lambda _2}\) are \(\lambda \)-additive measures on *X*, they are also a dual pair of \(\lambda \)-additive measures on *X*.

Secondly, we will show that if \(Q_{\lambda _1}\) and \(Q_{\lambda _2}\) are a dual pair of \(\lambda \)-additive measures on *X*, then \(Q_{\lambda _1}\) and \(Q_{\lambda _2}\) are a dual pair of belief- and plausibility measures on *X*. Let \(Q_{\lambda _1}\) and \(Q_{\lambda _2}\) be a dual pair of \(\lambda \)-additive measures on *X*. Then, based on Corollary 4, either \(\lambda _1 \in (-\,1, 0]\) and \(\lambda _2 \in [ 0, \infty )\), or \(\lambda _1 \in [ 0, \infty )\) and \(\lambda _2 \in (-\,1, 0]\) holds. Now, utilizing Proposition 17, we get that either \(Q_{\lambda _1}\) is a plausibility measure and \(Q_{\lambda _2}\) is a belief measure, or \(Q_{\lambda _1}\) is a belief measure and \(Q_{\lambda _2}\) is a plausibility measure. Thus, noting that \(Q_{\lambda _1}\) and \(Q_{\lambda _2}\) are a dual pair of \(\lambda \)-additive measures on *X*, we may conclude that they are also a dual pair of belief- and plausibility measures on *X*. \(\square \)

Proposition 19 can be stated in terms of the \(\nu \)-additive measure as follows.

### Proposition 20

Let \(Q_{\nu _1}\) and \(Q_{\nu _2}\) be two \(\nu \)-additive measures on the finite set *X*. Then, \(Q_{\nu _1}\) and \(Q_{\nu _2}\) are a dual pair of belief- and plausibility measures on *X* if and only if they are a dual pair of \(\nu \)-additive measures on *X*.

### Proof

Taking into account Proposition 6, this proposition directly follows from Proposition 19. \(\square \)

### Proposition 21

*X*. Then, \(Q_{\lambda _1}\) and \(Q_{\lambda _2}\) are a dual pair of belief- and plausibility measures on

*X*if and only if

### Proof

Following Proposition 19, if \(Q_{\lambda _1}\) and \(Q_{\lambda _2}\) are two \(\lambda \)-additive measures on the finite set *X*, then \(Q_{\lambda _1}\) and \(Q_{\lambda _2}\) are a dual pair of belief- and plausibility measures on *X* if and only if they are a dual pair of \(\lambda \)-additive measures on *X*. Furthermore, based on Proposition 14, if \(Q_{\lambda _1}\) and \(Q_{\lambda _2}\) are two \(\lambda \)-additive measures on the finite set *X*, then \(Q_{\lambda _1}\) and \(Q_{\lambda _2}\) are a dual pair of \(\lambda \)-additive measures if and only if \(\lambda _2 = - \frac{\lambda _1}{1+\lambda _1}\). Hence, this proposition follows from Propositions 19 and 14. \(\square \)

Proposition 21 can be stated in terms of the \(\nu \)-additive measure as follows.

### Proposition 22

*X*. Then, \(Q_{\nu _1}\) and \(Q_{\nu _2}\) are a dual pair of belief- and plausibility measures on

*X*if and only if

It should be added here that a \(\nu \)-additive measure may be supermodular or submodular depending on the value of its parameter \(\nu \).

### Definition 13

*X*is said to be submodular if

### Definition 14

*X*is said to be supermodular if

### Corollary 6

A \(\nu \)-additive measure is supermodular if \(\nu \in (0,1/2]\), and it is submodular if \(\nu \in [1/2,1)\).

### Proof

Since every belief measure is supermodular and every plausibility measure is submodular, this corollary immediately follows from Proposition 18. \(\square \)

## 5 A transformation between a \(\nu \)-additive (\(\lambda \)-additive) measure and a probability measure

Here, we will demonstrate that the \(\nu \)-additive (\(\lambda \)-additive) measures can be utilized for generating probability measures, and conversely, \(\nu \)-additive (\(\lambda \)-additive) measures can be generated from probability measures.

### Definition 15

*X*. Then, the function \(\mu : \varSigma \rightarrow [0,\infty )\) is a measure on the space \((X, \varSigma )\) iff \(\mu \) satisfies the following requirements:

- (1)
\(\forall A \in \varSigma : \mu (A) \ge 0\)

- (2)
\(\mu (\emptyset ) = 0\)

- (3)\(\forall A_{1}, A_{2}, \ldots \in \varSigma \), if \(A_i \cap A_j = \emptyset , \forall i\ne j\), then$$\begin{aligned} \mu \left( \bigcup \limits _{i=1}^{\infty } A_i \right) =\sum \limits _{i=1}^{\infty }\mu (A_i). \end{aligned}$$

### Proposition 23

*X*, \(Q_{\lambda }\) is a \(\lambda \)-additive measure, which satisfies the continuity property of monotone measures, on the space \((X, \varSigma )\), \(\lambda >-\,1\), \(\lambda \ne 0\), \(c>0\) and the function \(\hat{Q}_{\lambda ,c}: \varSigma \rightarrow [0, \infty )\) is given by

### Proof

### Proposition 24

*X*and let \(Q_{\lambda }\) and \(P_{\lambda }\) be two continuous functions on the space \((X, \varSigma )\) such that

### Proof

Firstly, we will show that if Eq. (48) holds and \(Q_{\lambda }\) is a \(\lambda \)-additive measure on \((X, \varSigma )\), then \(P_{\lambda }\) is a probability measure on \((X, \varSigma )\). Since \(\forall A \in \varSigma : P_{\lambda }(A) = \hat{Q}_{\lambda ,c}(A)\) with \(c= 1/\ln (1+\lambda )\), based on Proposition 23, \(P_{\lambda }\) is a measure. Moreover, as \(Q_{\lambda }(X)=1\), \(P_{\lambda }(X)=1\) holds as well, and so the function \(P_{\lambda }\) satisfies all the requirements of a probability measure given by Definition 4.

### Remark 8

Utilizing the definition of the \(\nu \)-additive measure, Proposition 24 can be stated as follows.

### Proposition 25

*X*and let \(Q_{\nu }\) and \(P_{\nu }\) be two continuous functions on the space \((X, \varSigma )\) such that

### Proof

Taking into account Proposition 6, this corollary immediately follows from Proposition 24. \(\square \)

Based on the result of Proposition 25, the formula in Eq. (51) may be viewed as a transformation between probability measures and \(\nu \)-additive measures.

## 6 Connections of \(\nu \)-additive (\(\lambda \)-additive) measures with other areas

### 6.1 Connection with rough sets

It is a well-known fact that the belief- and plausibility measures are connected with the rough set theory (see Dubois and Prade 1990; Yao and Lingras 1998; Wu et al. 2002). Here, we will show how the \(\nu \)-additive (\(\lambda \)-additive) measures are connected with the rough set theory.

### Definition 16

*X*be a finite set, and let \(R \subseteq X \times X\) be a binary equivalence relation on

*X*. The pair \((\underline{R}(A), \overline{R}(A))\) is said to be the rough set of \(A \subseteq X\) in the approximation space (

*X*,

*R*) if

*R*-equivalence class containing

*x*.

*A*by the pair of lower and upper approximations \((\underline{R}(A), \overline{R}(A))\). The lower approximation \(\underline{R}(A)\) is the union of all elementary sets that are subsets of

*A*, and the upper approximation \(\overline{R}(A)\) is the union of all elementary sets that have a non-empty intersection with

*A*. Note that the definitions of \(\underline{R}(A)\) and \(\overline{R}(A)\) are equivalent to the following statement: an element of

*X*necessarily belongs to

*A*if all of its equivalent elements belong to

*A*, while an element of

*X*possibly belongs to

*A*if at least one of its equivalent elements belongs to

*A*(Wu et al. 2002). Let the functions \(\underline{q}, \overline{q}: \mathscr {P}(X) \rightarrow [0,1]\) be given as follows:

*Pl*and

*Bl*are a dual pair of plausibility and belief functions on

*X*and

*m*is the basic probability assignment of

*Bl*satisfying the conditions: (1) the set of focal elements of

*m*is a partition of

*X*, (2) \(m(A^{*}) = \vert A^{*} \vert / \vert X \vert \) for every focal element \(A^{*}\) of

*m*, then there exists an equivalence relation

*R*on the set

*X*, such that the induced qualities of upper and lower approximations satisfy

Based on these results and on our proposition findings, we will establish some connections between rough sets and \(\nu \)-additive measures by using the following propositions.

### Proposition 26

*X*, and let \(R \subseteq X \times X\) be a binary equivalence relation on

*X*. Furthermore, let \((\underline{R}(A), \overline{R}(A))\) be the rough set of \(A \in \mathscr {P}(X)\) with respect to the approximation space (

*X*,

*R*), and let the functions \(\underline{q}, \overline{q}: \mathscr {P}(X) \rightarrow [0,1]\) be given by

*A*, respectively, for any \(A \in \mathscr {P}(X)\). Then, if the equations

*X*with \(\nu _1 \in (0,1/2]\), \(\nu _2 \in [1/2,1)\).

### Proof

*X*. Hence, the conditions that

- (i)
\(Q_{\nu _1}(A) = \underline{q}(A)\), \(Q_{\nu _2}(A) = \overline{q}(A)\) hold for any \(A \in \mathscr {P}(X)\)

- (ii)
\(Q_{\nu _1}\) and \(Q_{\nu _2}\) are two \(\nu \)-additive measures on

*X*

*X*together imply that \(Q_{\nu _1}\) and \(Q_{\nu _2}\) are also a dual pair of \(\nu \)-additive measures on

*X*. Furthermore, as \(\underline{q}\) is a belief measure and \(\overline{q}\) is a plausibility measure, based on Proposition 18, \(\nu _1 \in (0,1/2]\) and \(\nu _2 \in [1/2,1)\) hold as well. \(\square \)

### Proposition 27

*X*with \(\nu _1 \in (0,1/2]\), \(\nu _2 \in [1/2,1)\) and

*m*is a basic probability assignment that satisfies the conditions:

- (1)
The set of focal elements of

*m*is a partition of*X* - (2)
\(m(A^{*}) = \vert A^{*} \vert / \vert X \vert \) for every focal element \(A^{*}\) of

*m* - (3)
\(m(A^{*}) = \sum \limits _{B \subseteq A^{*}} (-1)^{\vert A^{*} \setminus B \vert } Q_{\nu _1}(B)\) for any \(A^{*} \in \mathscr {P}(X)\),

*R*on the set

*X*, such that the equations

*A*with respect to the approximation space (

*X*,

*R*), \(\underline{q}, \overline{q}: \mathscr {P}(X) \rightarrow [0,1]\) are given as

*A*, respectively.

### Proof

*Pl*and

*Bl*are a dual pair of plausibility and belief functions on

*X*and

*m*is the basic probability assignment of

*Bl*satisfying the conditions: (i) the set of focal elements of

*m*is a partition of

*X*, (ii) \(m(A^{*}) = \vert A^{*} \vert / \vert X \vert \) for every focal element \(A^{*}\) of

*m*, then there exists an equivalence relation

*R*on the set

*X*, such that the induced qualities of upper and lower approximations satisfy

*X*, \(Q_{\nu _2}\) is a plausibility measure on

*X*, and

*m*is the basic probability assignment of the belief measure \(Q_{\nu _1}\).

Let us assume that the conditions of this proposition are satisfied. Then, since \(Q_{\nu _1}\) and \(Q_{\nu _2}\) are a dual pair of \(\nu \)-additive measures on the finite set *X*, based on Proposition 20, \(Q_{\nu _1}\) and \(Q_{\nu _2}\) are a dual pair of belief- and plausibility measures on *X*. Furthermore, as \(\nu _1 \in (0,1/2]\) and \(\nu _2 \in [1/2,1)\), based on Proposition 18, \(Q_{\nu _1}\) is a belief measure on *X* and \(Q_{\nu _2}\) is a plausibility measure on *X*, and so condition (3) means that *m* is the basic probability assignment of the belief measure \(Q_{\nu _1}\). That is, we have shown that if the conditions of this proposition are satisfied, then all the conditions that are required to apply the result of Yao and Lingras (1998) are satisfied as well. \(\square \)

### 6.2 The \(\lambda \)-additive measure and the multi-attribute utility function

Here we will state interesting analogies between the \(\lambda \)-additive measure and the multi-attribute utility function. Let \(X_1, X_2, \ldots , X_n\) be attributes, where each \(X_i\) may be either a scalar attribute or a vector of scalar attributes \((i=1,2, \ldots , n)\). Furthermore, let the consequence space *X* be a rectangular subset of the *n*-dimensional Euclidean space. Then, a specific consequence may be given by a vector \((x_1, x_2, \ldots , x_n)\), where \(x_i\) is a particular value of the attribute \(X_i\)\((i=1,2, \ldots , n)\). The utility function \(u: X \rightarrow \mathbb {R}\), which is assumed to be continuous, assigns a utility value to the consequence \((x_1, x_2, \ldots , x_n)\); that is, the utility of consequence \((x_1, x_2, \ldots , x_n)\) is \(u(x_1, x_2, \ldots , x_n)\) (Keeney 1974). Here, we will utilize the concept of the utility independence of attributes (see, e.g., Keeney and Raiffa 1993).

### Definition 17

Attribute \(X_i\) is utility independent of attribute \(X_j\) if conditional preferences for lotteries over \(X_i\) given a fixed value for \(X_j\) do not depend on the particular value of \(X_j\).

Keeney and Raiffa (1993) proved the following proposition which states that the mutual utility independence of attributes implies a multiplicative multi-attribute utility function.

### Proposition 28

### Proof

See Keeney and Raiffa (1993). \(\square \)

*k*is positive in Eq. (54), then \(u^{*}(x_1, x_2, \ldots , x_n)= 1+k u_M(x_1, x_2, \ldots , x_n)\) is a multi-attribute utility function, \(u_i^{*}(x_i)=1+kk_iu_i(x_i)\) are utility functions and \(u^{*}(x_1, x_2, \ldots , x_n) = \prod _{i=1}^{n} u_i^{*}(x_i)\), where \(i=1,2, \ldots , n\). Similarly, if

*k*is negative in Eq. (54), then \(u^{*}(x_1, x_2, \ldots , x_n)= -(1+k u_M(x_1, x_2, \ldots , x_n))\) is a multi-attribute utility function, \(u_i^{*}(x_i)=-(1+kk_iu_i(x_i))\) are utility functions and \(-u^{*}(x_1, x_2, \ldots , x_n) = (-1)^{n} \prod _{i=1}^{n} u_i^{*}(x_i)\), where \(i=1,2, \ldots , n\). That is, Eq. (54) describes a multiplicative relationship between the multi-attribute utility function and the individual univariate utility functions. Hence, Eq. (53) is referred to as the multi-attribute multiplicative utility function.

We can see that the right-hand side of Eq. (18) with \(\lambda >-1\), \(\lambda \ne 0\) has the same form as the right-hand side of Eq. (53). It means that there is an interesting connection between the \(\lambda \)-additive measures and the multi-attribute multiplicative utility function. Namely, a \(\lambda \)-additive measure with \(\lambda \ne 0\) of the union of *n* pairwise disjoint sets is computed in the same way as the multi-attribute utility of *n* mutually utility independent attributes.

### Definition 18

Two attributes \(X_i\) and \(X_j\) are additive independent if the paired preference comparison of any two lotteries, defined by two joint probability distributions on \(X_i \times X_j \), depends only on their marginal distributions.

It can be shown that if and only if the preferences over lotteries on attributes \(X_1, X_2, \ldots , X_n\) depend only on their marginal probability distributions (i.e., the attributes are additive independent), then the *n*-attribute utility function is additive (Keeney and Raiffa 1993).

Notice that the right-hand side of Eq. (18) with \(\lambda =0\) has the same form as the right-hand side of Eq. (56). It means that a \(\lambda \)-additive measure with \(\lambda = 0\) of the union of *n* pairwise disjoint sets is computed in the same way as the multi-attribute utility of *n* additive independent attributes.

\(\lambda \)-Additive measure of union of pairwise disjoint sets and utility value of consequence \((x_1, x_2, \ldots , x_n)\)

\(Q_{\lambda }\)\(\lambda \)-additive measure of \(\bigcup \limits _{i=1}^n A_i\); \(\lambda >-1\) | |

\(\lambda =0\): | \(\sum \limits _{i=1}^{n} Q_{\lambda }(A_i)\) |

\(\lambda \ne 0\): | \(\frac{1}{\lambda } \left( \prod \limits _{i=1}^{n} (1+\lambda Q_{\lambda }( A_i))-1 \right) \) |

Multi-attribute utility \(u(x_1,x_2, \ldots ,x_n)\); \(k>-1\) | |

\(k=0\): | \(\sum \limits _{i=1}^{n} k_i u_i(x_i)\) |

\(k \ne 0:\) | \(\frac{1}{k} \left( \prod \limits _{i=1}^{n} \left( 1+kk_iu_i(x_i)\right) -1 \right) \) |

### 6.3 The \(\lambda \)-additive measure and some operators of continuous-valued logic

Here, we will state a formal connection between the \(\lambda \)-additive measure and certain operators of continuous-valued logic.

### Definition 19

Operators covered by the generalized Dombi operator class

Operator | \(\gamma \) | Conjunction | Disjunction |
---|---|---|---|

Value of \(\alpha \) | |||

Dombi | 0 | \(\alpha >0\) | \(\alpha <0\) |

Product | 1 | 1 | − 1 |

Einstein | 2 | 1 | − 1 |

Hamacher | \(\gamma \in (0,\infty )\) | 1 | − 1 |

Drastic | \(\infty \) | \(\alpha >0\) | \(\alpha <0\) |

Min–max | 0 | \(\infty \) | \(-\,\infty \) |

*X*is a finite set, \(Q_{\lambda }\) is a \(\lambda \)-additive measure on

*X*, \(\lambda >-1\), \(\lambda \ne 0\) and \(A_1, A_2, \ldots , A_n \in \mathscr {P}(X)\) are pairwise disjoint sets, then

*n*pairwise disjoint sets is computed in the same way as the value of the generator function of Dombi operator for the value of the generalized Dombi operation over

*n*continuous-valued logic variables. It should be added that this analogy is just a formal one since \(g(x_i) \in (0, \infty )\) and \(Q_{\lambda }(A_i) \in [0,1]\), and \(g(x_i)\) and \(Q_{\lambda }(A_i)\) have different meanings.

## 7 Summary and future plans

- (1)A \(\nu \)-additive measure and a \(\lambda \)-additive measure (Sugeno \(\lambda \)-measure) are identical if and only ifwhere \(\lambda \in (-1,\infty )\), \(\nu \in (0,1)\).$$\begin{aligned} \lambda = \left( \frac{1 - \nu }{\nu } \right) ^{2} -1, \end{aligned}$$
- (2)
Two \(\nu \)-additive measures are a dual pair if and only if the sum of their parameters equals 1.

- (3)A \(\nu \)-additive measure is a
- (a)
belief measure if and only if \(0<\nu \le 1/2\)

- (b)
probability measure if and only if \(\nu =1/2\)

- (c)
plausibility measure if and only if \(1/2 \le \nu <1\).

- (a)
- (4)
Two \(\nu \)-additive measures are a dual pair of belief- and plausibility measures if and only if the sum of their parameters equals 1.

- (5)
There exists a transformation that can be utilized for transforming a \(\nu \)-additive (\(\lambda \)-additive) measure into a probability measure; conversely, this transformation can be utilized for transforming a probability measure into a \(\nu \)-additive (\(\lambda \)-additive) measure.

- (6)
Dual pairs of \(\nu \)-additive measures are strongly associated with the lower- and upper approximation pairs of rough sets.

- (7)There are interesting formal connections between the \(\lambda \)-additive measures and the multi-attribute utility functions. Namely,
- (a)
if \(\lambda = 0\), then the \(\lambda \)-additive measure of the union of

*n*pairwise disjoint sets is computed in the same way as the multi-attribute utility of*n*additive independent attributes - (b)
if \(\lambda >-1\) and \(\lambda \ne 0\), then the \(\lambda \)-additive measure of the union of

*n*pairwise disjoint sets is computed in the same way as the multi-attribute utility of*n*mutually utility independent attributes.

- (a)
- (8)
There is an interesting formal connection between the \(\lambda \)-additive measure and certain operators of continuous-valued logic. Namely, if \(\lambda >-1\) and \(\lambda \ne 0\), then the computation method of \(\lambda \)-additive measure of union of

*n*pairwise disjoint sets is identical with that of the generator function of the Dombi operator at the value of the generalized Dombi operation over*n*continuous-valued logic variables.

## Notes

### Acknowledgements

Open access funding provided by Eötvös Loránd University (ELTE).

### Compliance with ethical standards

### Conflict of interest

József Dombi (author) declares that he has no conflict of interest. Tamás Jónás (author) declares that he has no conflict of interest.

### Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

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