The λ -additive Measure in a New Light The Q ν measure and its connections with belief, probability, plausibility, rough sets, multiattribute utility functions and fuzzy operators

The aim of this paper is twofold. On the one hand, the λ -additive measure (Sugeno λ -measure) is revisited and a state-of-the-art summary of its most important properties is provided. On the other hand, the so-called ν -additive measure as an alternatively parameterized λ -additive measure is introduced. Here, the advantages of the ν -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 ν -additive measure lies in the fact that its parameter ν ∈ ( 0 , 1 ) has an important semantic meaning as it is the ﬁx point of the complement operation. Here, by utilizing the ν -additive measure, some well-known results concerning the λ -additive measure are put into a new light and rephrased in more advantageous forms. It is discussed here how the ν -additive measure is connected with the belief, probability-and plausibility measures. Next, it is also shown that two ν -additive measures, with the parameters ν 1 and ν 2 , are a dual pair of belief-and plausibility measures if and only if ν 1 + ν 2 = 1. Furthermore, it is demonstrated how a ν -additive measure (or a λ -additive measure) can be transformed to a probability measure and vice versa. Lastly, it is discussed here how the ν -additive measures are connected with rough sets, multi-attribute utility functions and with certain operators of fuzzy logic.

The aim of the present study is twofold. On the one hand, we will revisit the λ -additive measure and give a state-ofthe-art summary of its most important properties. On the other hand, we will introduce the so-called ν-additive measure as an alternatively parameterized λ -additive measure, demonstrate the advantages of the ν-additive measure and point out that these two measures are closely related to various areas of science. The motivation for introducing the νadditive measure lies in the fact that its parameter ν ∈ (0, 1) has an important semantic meaning. Namely, ν is the fix point of the complement operation; that is, if the ν additive measure of a set has the value ν, then the ν-additive measure of its complement set has the value ν as well. It should be added that by utilizing the ν-additive measure, some wellknown results concerning the λ -additive measure can be put into a new light and rephrased in more advantageous forms. Here, we will discuss how the ν-additive measure is connected with the belief-, probability-and plausibility measures (see, e.g. Wang and Klir (2013); Höhle (1987); Dubois and Prade (1980); Spohn (2012); Feng et al. (2014)). Also, we will demonstrate that a ν-additive measure is a (1) belief measure if and only if 0 < ν ≤ 1/2 (2) probability measure if and only if ν = 1/2 (3) plausibility measure if and only if 1/2 ≤ ν < 1.
The rest of this paper is structured as follows. In Section 2, we give an overview of the monotone (fuzzy) measures including the belief-, probability-and plausibility measures. In Section 3, the ν-additive measure is introduced and its key properties are discussed. In Section 4, we demonstrate how the ν-additive measure is related to the belief-, probabilityand plausibility measures, and in Section 5, we show how a ν-additive measure can be transformed to a probability measure and vice versa. Section 6 reveals some areas of science which the ν-additive (λ -additive) measures are connected with. Lastly, in Section 7, we give a short summary of our findings and highlight our future research plans including the possible application of ν-additive measure in network science.
In this study, we will use the common notations ∩ and ∪ for the intersection and union operations over sets, respectively. Also, will use the notation A for the complement of set A.

Monotone measures
Now, we will introduce the monotone measures and give a short overview of them that covers the probability-, beliefand plausibility measures.
If X is a finite set, then the continuity requirement in Definition 1 can be disregarded and the monotone measure is defined as follows.
Note that the monotone measures given by Definition 1 and Definition 2 are known as fuzzy measures, which were originally defined by Choquet (Choquet 1954) and Sugeno (Sugeno 1974).
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 ∈ P(X), Pr(A ∪ B) = Pr(A) + Pr(B).

Belief measure and plausibility measure
Definition 5 The function Bl : P(X) → [0, 1] is a belief measure on the finite set X, iff Bl satisfies the following requirements: (1) Bl( / 0) = 0, Bl(X) = 1 (2) for any A 1 , A 2 , . . . , A n ∈ P(X), Here, Bl(A) is interpreted as a grade of belief in that a given element of X belongs to A.
Lemma 1 If Bl is a belief measure on the finite set X, then for any A ∈ P(X), Proof Noting Definition 5, we have The inequality Bl(A) + Bl(A) ≤ 1 means that a lack of belief in x ∈ A does not imply a strong belief in x ∈ A. In particular, total ignorance is modeled by the belief function 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 ∈ P(X) and B ⊆ A, then Bl(B) ≤ Bl(A).
Proof Let B ⊆ A. Hence, there exists C ∈ P(X) such that A = B ∪C and B ∩C = / 0. Now, by utilizing the definition of the belief measure and the fact that B ∩C = / 0, we get 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 was proven in Proposition 1; that is, Bl also satisfies criterion (2) in Definition 2.
Lemma 2 If Pl is a plausibility measure on the finite set X, then for any A ∈ P(X), Proof Noting Definition 6, we have This result can be interpreted so that the plausibility of x ∈ A does not imply a strong plausibility of x ∈ 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 ∈ P(X) and B ⊆ A, then Pl(B) ≤ Pl(A).
Proof Let B ⊆ A. Let C ∈ P(X) such that A ∩ C = B and A ∪ C = X. Now, by utilizing the definition of plausibility measure, and the facts that A ∪C = X and Pl(C) ≤ 1, we get 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 was proven in Proposition 2; that is, Pl also satisfies criterion (2) in Definition 2.
The plausibility of a subset A of the finite set X was defined by Shafer (Shafer 1976) as where Bl is a belief function. The following proposition states an interesting connection between the belief measure and the plausibility measure.
Proposition 3 Let X be a finite set and let µ 1 , µ 2 : P(X) → [0, 1] be two monotone measures on X such that holds for any A ∈ P(X). Then, either (1) µ 1 is a belief measure on X if and only if µ 2 is a plausibility measure on X, or (2) µ 1 is a plausibility measure on X if and only if µ 2 is a belief measure on X.
Proof We will prove case (1), and the proof of case (2) is similar. Firstly, we will show that if µ 1 is a belief measure on X and µ 2 (A) is given as µ 2 (A) = 1 − µ 1 (A) for any A ∈ P(X), then µ 2 is a plausibility measure on X.
Next, applying the inequality in Eq. (4) to the complement sets A 1 , A 2 , . . . , A n ∈ P(X) and utilizing the fact that Noting the fact that the previous inequality can be written as Now, applying the De Morgan law to the last inequality, we get which means that function µ 2 is a plausibility measure. Secondly, we will demonstrate that if µ 2 is a plausibility measure on X and µ 2 (A) is given as µ 2 (A) = 1 − µ 1 (A) for any A ∈ P(X), then µ 1 is a belief measure on X. Let µ 2 be a plausibility measure on X and µ 2 (A) = 1 − µ 1 (A) for any A ∈ P(X). These conditions trivially imply that µ 1 ( / 0) = 0 and µ 1 (X) = 1; that is, function µ 1 satisfies requirement (1) for a belief measure given in Definition 5. Next, because function µ 2 is a plausibility measure, the inequality holds for any A 1 , A 2 , . . . , A n ∈ P(X). Then, applying the inequality in Eq. (6) to the complement sets A 1 , A 2 , . . . , A n ∈ P(X) and utilizing the condition that µ 2 (A) = 1 − µ 1 (A), we get Again, taking into account Eq. (5), the previous inequality can be written as Now, applying the De Morgan law to the last inequality, we get Hence, µ 1 is a belief measure.
Later, we will use the concept of dual pair of belief-and plausibility measures.
Definition 7 Let 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 holds for any A ∈ P(X).
In the Dempster-Shafer theory of evidence, a belief mass is assigned to each element of the power set P(X), where X is a finite set. The belief mass is given by the so-called basic probability assignment m from P(X) to [0, 1] that is defined as follows.
Definition 8 The function m : P(X) → [0, 1] is a basic probability assignment (mass function) on the finite set X, iff m satisfies the following requirements: The subsets A of X for which m(A) > 0 are called the focal elements of m. Let x ∈ A and A ∈ P(X). Then, the mass m(A) can be interpreted as the probability of knowing x ∈ A given the available evidence. Utilizing a given basic probability assignment m, the belief Bl(A) for the set A is A basic probability assignment m can be represented by its belief function Bl as where B ∈ P(X). Here, m is the basic probability assignment of the belief measure Bl. Note that plausibility measures and belief functions were introduced by Dempster Dempster (1967) under the names upper and lower probabilities, induced by a probability measure by a multivalued mapping.
Remark 2 The monotonicity of the plausibility measure Pl can also be demonstrated by utilizing the duality Pl(A) = 1 − Bl(A) and the monotonicity of the belief measure Bl. Namely, if B ⊆ A, then A ⊆ B and so which means that 3 Introduction to the Q ν measure Relaxing the additivity property of the probability measure, the λ -additive measures were proposed by Sugeno in 1974(Sugeno 1974. Definition 9 The function Q λ : P(X) → [0, 1] is a λadditive measure (Sugeno λ -measure) on the finite set X, iff Q λ satisfies the following requirements: (1) Q λ (X) = 1 (2) for any A, B ∈ P(X) and A ∩ B = / 0, where λ ∈ (−1, ∞).
Note that if X is an infinite set, then the continuity of function Q λ is also required. Here, we will show that the λ -additive measures are monotone measures as well.
Proposition 4 Every λ -additive measure is a monotone measure.
Proof Let Q λ be a λ -additive measure on the set X. Then Q λ (X) = 1 holds by definition. Next, by utilizing Eq. (7), we Thus, Q λ satisfies criterion (1) of a monotone measure given in Definition 2.
Next, let A, B ∈ P(X) and let B ⊆ A. Then there exists a C ∈ P(X) such that A = B ∪ C and B ∩ C = / 0. Now, by utilizing Eq. (7) and the fact that λ > −1, we get It means that Q λ also satisfies the monotonicity criterion of a monotone measure.
Remark 3 The requirement λ ≥ −1 instead of the requirement λ > −1 would be sufficient to ensure the monotonicity of Q λ (see Proposition 4). The requirement λ ≥ −1 also ensures that for any A, B ∈ P(X) and holds, and so if λ ≥ −1, then However, the requirement λ > −1 is given in the definition of λ -additive measures. Later, we will see that certain properties of λ -additive measures hold only if λ > −1.

The λ -additive complement and the Dombi form of negation
Proposition 5 If X is a finite set and Q λ is a λ -additive measure on X, then for any A ∈ P(X) the Q λ measure of the complement set A = X \ A is Proof Since A ∩ A = / 0, we can write from which we get Remark 4 For any A ∈ P(X), we have It can be seen from Eq. (9) that We have shown in Proposition 5 that if X is a finite set and Q λ is a λ -additive measure on X, then for any A ∈ P(X) the Q λ measure of the complement set A = X \ A is Now, let us assume that 0 ≤ Q(A) < 1. Then, Eq. (10) can be written as In continuous-valued logic, the Dombi form of negation with the neutral value ν ∈ (0, 1) is given by the operator n ν : [0, 1] → [0, 1] as follows: where x ∈ [0, 1] is a continuous-valued logic variable (Dombi 2008). Note that the Dombi form of negation is the unique Sugeno's negation (Sugeno 1993) with the fix point ν ∈ (0, 1). Also, for Q λ (A) ∈ [0, 1), the formula of λadditive measure of Q λ (A) in Eq. (11) is the same as the formula of the Dombi form of negation in Eq. (12) with x = Q λ (A) and Based on the definition of λ -additive measures, λ > −1, and since is a bijection between (0, 1) and (−1, ∞), the λ -additive measure of the complement set A can be alternatively redefined as Following this line of thinking, here, we will introduce the ν-additive measure and state some of its properties.
Note that if X is an infinite set, then the continuity of function Q ν is also required. Here, we state a key proposition that we will frequently utilize later on.
Proposition 6 Let X be a finite set, and let Q λ and Q ν be a λ -additive and a ν-additive measure on X, respectively. Then, for any A ∈ P(X), if and only if where λ > −1, ν ∈ (0, 1).
Proof This proposition immediately follows from the definitions of the λ -additive measure and ν-additive measure.
If Q ν is a ν-additive measure on the finite set X, then, by utilizing Eq. (13), the Q ν measure of the complement set A is Moreover, as the ν parameter is the neutral value of the Dombi negation operator (see Eq. (12)), the following property of the ν-additive measure holds as well.
Proposition 7 Let X be a finite set, Q ν a ν-additive measure on X and let the set A ν be given as where ν ∈ (0, 1). Then for any A ∈ A ν the Q ν measure of the complement set A is equal to ν; that is, Q ν (A) = ν.
Proof If A ∈ A ν , then Q ν (A) = ν and utilizing the ν-additive negation given by Eq. (17), we have This result means that the ν-additive complement operation may be viewed as a complement operation characterized by its fix point ν.

Main properties of the ν-additive (λ -additive) measures
It is worth mentioning that the definition of the ν-additive measure is the same as that of the λ -additive measure with an alternative parametrization. Thus, utilizing the fact that any ν-additive measure is a λ -additive measure with λ = 1−ν ν 2 − 1, some of the properties of λ -additive measures can be expressed in terms of ν-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 λ -additive form to demonstrate some properties and then we will state them in terms of the ν-additive measure as well. We will follow this approach from now on, and Q λ will always denote a λ -additive measure with the parameter λ ∈ (−1, ∞) and Q ν will always denote a ν-additive measure with the parameter ν ∈ (0, 1).

ν-additive (λ -additive) measure of collection of disjoint sets
Here, we will outline the computation of the ν-additive (λadditive) measure of collection of pairwise disjoint sets.
(1) In this case, the proposition trivially follows from the definition of the λ -additive measures.
With this notation, G n+1 = G n (1 + λ Q λ (A n+1 )), and the equality that we seek to prove is By utilizing the definition of the λ -additive measures and the associativity of the union operation over sets, we get Now, utilizing the inductive condition, the last equation can be written as Remark 5 Note that in Eq. (18), the case λ = 0 may be viewed as a special case of λ > −1 and λ = 0. Namely, the right hand side of Eq. (19) can be written as Proposition 8 can be stated in terms of the ν-additive measure as follows.
Proposition 9 If X is a finite set, Q ν is a ν-additive measure on X and A 1 , A 2 , . . . , A n ∈ P(X) are pairwise disjoint sets, then where ν ∈ (0, 1), Proof Recalling Proposition 6, this proposition directly follows from Proposition 8.

General forms for the ν-additive (λ -additive) measure of union and intersection of two sets
The calculations of the λ -additive measure and ν-additive measure of two disjoint sets are given in Definition 9 and Definition 10, respectively. Here, we will show how the νadditive (λ -additive) measure of two sets can be computed when these sets are not disjoint. We will also discuss how the ν-additive (λ -additive) measure of intersection of two sets can be computed.
Proposition 10 If X is a finite set and Q λ is a λ -additive measure on X, then for any A, B ∈ P(X), Now, by expressing Q λ (A ∩ B) in terms of (22), we get and substituting this into (21), we get Hence, we have the general form of the λ -additive measure of the union of two sets.
Remark 6 Notice that if λ = 0, then Eq. (23) reduces to , which has the same form as the probability measure of union of two sets. Later, we will discuss how the λ -additive (ν-additive) measure is related to the probability measure.
Remark 7 Note that Eq. (23) can be written in the following equivalent forms: Corollary 3 If X is a finite set and Q λ is a λ -additive measure on X, then for any A, B ∈ P(X), Proof By expressing Q λ (A ∩ B) in Eq. (23), we get Eq. (24).

Other properties of the ν-additive (λ -additive) measure of union and the intersection of two sets
The following results are related to the ν-additive measure of union and the intersection of two sets.
Proposition 11 Let X be a finite set, Q ν be a ν-additive measure on X and let A, B ∈ P(X). Then And by using the equation (2) Since A ∩B = / 0, applying the definition of the ν-additive measure gives Note that the term ν Eq, (25) may be regarded as the corrective term of the in-

Characterization by independent variables
We have demonstrated (see Proposition 8) that if X is a finite set, Q λ is a λ -additive measure on X, λ > −1, λ = 0, A 1 , A 2 , . . . , A n ∈ P(X) are pairwise disjoint sets, and It means that the value of Q λ (A) can be readily calculated from the independent values Q λ (A i ), where i = 1, 2, . . . , n. If The following proposition demonstrates that Eq. (27) has only one root in the interval (−1, 0) ∪ (0, ∞).
Proof This proof is based on the proof of a theorem connected with the multiplicative utility functions described by Keeney in (Keeney 1974, Appendix B). Since λ = 0, Eq. (28) can be written as where z i = Q λ (A i ), i = 1, 2, . . . , n. Now, let S = ∑ n i=1 z i and let the polynomial f (q) be given as where −1 ≤ q < ∞. From Eq. (29) and Eq. (30), we get the following results: The first derivative of function f is from which we can see that f (q) is decreasing (with respect to q) in the interval (−1, ∞), and Here, we will distinguish three cases: (1) S < 1; (2) S = 1; (3) S > 1.

Dual ν-additive (λ -additive) measures and their properties
Later, we will utilize the concept of the dual pair of λadditive measures and the concept of the dual pair of νadditive measures.
Definition 11 Let Q λ 1 and Q λ 2 be two λ -additive measures on the finite set X. Then, Q λ 1 and Q λ 2 are said to be a dual pair of λ -additive measures iff Q λ 1 (A) + Q λ 2 (A) = 1 holds for any A ∈ P(X).
Definition 12 Let Q ν 1 and Q ν 2 be two ν-additive measures on the finite set X. Then, Q ν 1 and Q ν 2 are said to be a dual pair of ν-additive measures iff holds for any A ∈ P(X).
Later, we will utilize the following proposition.
Proposition 13 Let Q λ 1 and Q λ 2 be two λ -additive measures on the finite set X and let Then, for any A ∈ P(X) if and only if Proof Firstly, we will show that if λ 2 = − λ 1 1+λ 1 and Q λ 2 (A) > 1 − Q λ 1 (A) holds for any A ∈ P(X), then Q λ 2 (A) < 1 − Q λ 1 (A) holds as well. By utilizing the formula for the λ -additive measure of complementer set given by Eq. (8), we get for any A ∈ P(X). Next, based on the condition Q λ 2 (A) > 1 − Q λ 1 (A), we have the following inequality: From Eq. (36), via simple calculations, we get From the condition λ 2 = − λ 1 1+λ 1 , we have the equation λ 1 + λ 2 + λ 1 λ 2 = 0, and so the inequality relation in Eq. (37) can be written as which is equivalent to that stated in Eq. (35).
Here, we will demonstrate some key properties of the ν-additive (λ -additive) measure related to a dual pair of νadditive (λ -additive) measures.
Proposition 14 Let Q λ 1 and Q λ 2 be two λ -additive measures on the finite set X. Then Q λ 1 and Q λ 2 are a dual pair of λ -additive measures if and only if Proof Firstly, we will show that if Q λ 1 and Q λ 2 are a dual pair of λ -additive measures on the finite set X, then λ 2 = − λ 1 1+λ 1 . Let Q λ 1 and Q λ 2 be a dual pair of λ -additive measures on X. It means that Q λ 2 (A) = 1 − Q λ 1 (A) holds for any A ∈ P(X). Next, let A, B ∈ P(X) such that A∩B = / 0. Then, the formula for the λ -additive measure of the intersection of two sets given by Eq. (24) and the fact that Moreover, since Q λ 2 is a λ -additive measure and A ∩ B = / 0, the equation holds. Thus, from Eq. (40) and Eq. (41) we get that λ 2 = − λ 1 1+λ 1 .
Proposition 14 can be stated in terms of the ν-additive measure as follows.
Proposition 15 Let Q ν 1 and Q ν 2 be two ν-additive measures on the finite set X. Then, Q ν 1 and Q ν 2 are a dual pair of ν-additive measures if and only if ν 1 + ν 2 = 1.
Proof Utilizing Proposition 6, this proposition immediately follows from Proposition 14.
Utilizing the definition of the dual pair of λ -additive measures, the following corollary can be stated.
Corollary 4 can be stated in terms of the ν-additive measure as follows.
Proof Taking into account Proposition 6, this corollary immediately follows from Corollary 4.
It should be mentioned here that one of the λ parameters of a dual pair of λ -additive measures is always in the unbounded interval [0, ∞). At the same time, the ν parameters of a dual pair of ν-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).

The decomposition property of the λ -additive measure
The following proposition reveals an interesting property of the λ -additive measures.

Connection with belief-, probability-and plausibility measures
Here, we will discuss some important properties of the νadditive (λ -additive) measure and how it is connected to the belief-, probability-and plausibility measures.
Proposition 17 Let X be a finite set and let Q λ be a λadditive measure on X. Then, on set X, Q λ is a (1) plausibility measure if and only if −1 < λ ≤ 0 (2) probability measure if and only if λ = 0 (3) belief measure if and only if λ ≥ 0.
Proof See Dubois and Prade (1980) and Banon (1978) Note that in terms of the ν-additive measure, Proposition 17 can be stated as follows.
Proposition 18 Let X be a finite set and let Q ν be a νadditive measure on X. Then, on set X, Q ν is a (1) belief measure if and only if 0 < ν ≤ 1/2 (2) probability measure if and only if ν = 1/2 (3) plausibility measure if and only if 1/2 ≤ ν < 1.
Proof Taking into account Proposition 6, this proposition immediately follows from Proposition 17.  Figure 1 shows the connection between Q ν (A) and Q ν (A) for various values of parameter ν of the ν-additive measure Q ν . From this figure, in accordance with Proposition 18, we notice the following. If ν = 1/2, then Q ν is a probability measure and so Q ν (A) = 1 − Q ν (A). If 0 < ν ≤ 1/2, then Q ν is a belief measure and Q ν (A) ≤ 1 − Q ν (A). If 1/2 ≤ ν < 1, then Q ν is a plausibility measure and Q ν (A) ≥ 1 − Q ν (A). Moreover, in accordance with Eq. (17), for a given set A, Q ν (A) increases with the value of parameter ν. That is, the smaller the value of parameter ν, the stronger the complement operation is. 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 λ 1 and Q λ 2 be two λ -additive measures on the finite set X. Then, Q λ 1 and Q λ 2 are a dual pair of belief-and plausibility measures on X if and only if they are a dual pair of λ -additive measures on X.
Proof Firstly, we will show that if the condition of the proposition is satisfied and Q λ 1 and Q λ 2 are a dual pair of belief-and plausibility measures on X, then Q λ 1 and Q λ 2 are a dual pair of λ -additive measures on X. Let Q λ 1 and Q λ 2 be a dual pair of belief-and plausibility measures on X. Since, Q λ 1 and Q λ 2 are a dual pair; that is, Q λ 2 (A) = 1 − Q λ 1 (A) holds for any A ∈ P(X), and Q λ 1 and Q λ 2 are λ -additive measures on X, they are also a dual pair of λ -additive measures on X.
Secondly, we will show that if Q λ 1 and Q λ 2 are a dual pair of λ -additive measures on X, then Q λ 1 and Q λ 2 are a dual pair of belief-and plausibility measures on X. Let Q λ 1 and Q λ 2 be a dual pair of λ -additive measures on X. Then, based on Corollary 4, either λ 1 ∈ (−1, 0] and λ 2 ∈ [0, ∞), or λ 1 ∈ [0, ∞) and λ 2 ∈ (−1, 0] holds. Now, utilizing Proposition 17, we get that either Q λ 1 is a plausibility measure and Q λ 2 is a belief measure, or Q λ 1 is a belief measure and Q λ 2 is a plausibility measure. Thus, noting that Q λ 1 and Q λ 2 are a dual pair of λ -additive measures on X, we may conclude that they are also a dual pair of belief-and plausibility measures on X.
Proposition 19 can be stated in terms of the ν-additive measure as follows.
Proposition 20 Let Q ν 1 and Q ν 2 be two ν-additive measures on the finite set X. Then, Q ν 1 and Q ν 2 are a dual pair of belief-and plausibility measures on X if and only if they are a dual pair of ν-additive measures on X.
Proof Taking into account Proposition 6, this proposition directly follows from Proposition 19.
Proposition 21 Let Q λ 1 and Q λ 2 be two λ -additive measures on the finite set X. Then, Q λ 1 and Q λ 2 are a dual pair of belief-and plausibility measures on X if and only if Proof Following Proposition 19, if Q λ 1 and Q λ 2 are two λadditive measures on the finite set X, then Q λ 1 and Q λ 2 are a dual pair of belief-and plausibility measures on X if and only if they are a dual pair of λ -additive measures on X. Furthermore, based on Proposition 14, if Q λ 1 and Q λ 2 are two λ -additive measures on the finite set X, then Q λ 1 and Q λ 2 are a dual pair of λ -additive measures if and only if λ 2 = − λ 1 1+λ 1 . Hence, this proposition follows from Proposition 19 and Proposition 14.
Proposition 21 can be stated in terms of the ν-additive measure as follows.
Proposition 22 Let Q ν 1 and Q ν 2 be two ν-additive measures on the finite set X. Then Q ν 1 and Q ν 2 are a dual pair of belief-and plausibility measures on X if and only if ν 1 + ν 2 = 1.
Proof Based on Proposition 6, this proposition immediately follows from Proposition 21.
It should be added here that a ν-additive measure may be supermodular or submodular depending on the value of its parameter ν.

Definition 13
The set function f : P(X) → R on the finite set X is said to be submodular if holds for any A, B ∈ P(X).

Definition 14
The set function f : P(X) → R on the finite set X is said to be supermodular if holds for any A, B ∈ P(X).
Proof Since every belief measure is supermodular and every plausibility measure is submodular, this corollary immediately follows from Proposition 18.

A transformation between a ν-additive (λ -additive) measure and a probability measure
Here, we will demonstrate that the ν-additive (λ -additive) measures can be utilized for generating probability measures; and, conversely, ν-additive (λ -additive) measures can be generated from probability measures.
Proposition 24 Let Σ be a σ -algebra over the set X and let Q λ and P λ be two continuous functions on the space (X, Σ ) such that holds for any A ∈ Σ , λ > −1, λ = 0. Then, P λ is a probability measure on (X, Σ ) if and only if Q λ is a λ -additive measure on (X, Σ ).
Proof Firstly, we will show that if Eq. (48) holds and Q λ is a λ -additive measure on (X, Σ ), then P λ is a probability measure on (X, Σ ). Since ∀A ∈ Σ : P λ (A) =Q λ ,c (A) with c = 1/ ln(1 + λ ), based on Proposition 23, P λ is a measure. Moreover, as Q λ (X) = 1, P λ (X) = 1 holds as well; and so the function P λ satisfies all the requirements of a probability measure given by Definition 4. Secondly, we will show that if Eq. (48) holds and P λ is a probability measure on (X, Σ ), then Q λ is a λ -additive measure on (X, Σ ). Let P λ be a probability measure on (X, Σ ). From Eq. (48) we have for any A ∈ Σ . Since P λ is a probability measure on (X, Σ ), P λ (X) = 1; and so from Eq. (49) we get Q λ (X) = 1. That is, Q λ satisfies requirement (1) of the λ -additive measures given by Definition 9. Now, let A, B ∈ Σ such that A ∩ B = / 0. Then, as P λ is a probability measure on (X, Σ ), the equation holds. Utilizing Eq. (49) and Eq. (50), Q λ (A ∪ B) can be written as It means that Q λ satisfies requirement (2) of the λ -additive measures given in Definition 9 as well; that is, Q λ meets all te requirements of a λ -additive measure.
Utilizing the definition of the ν-additive measure, Proposition 24 can be stated as follows.
Proposition 25 Let Σ be a σ -algebra over the set X and let Q ν and P ν be two continuous functions on the space (X, Σ ) such that P ν (A) = 1 2 holds for any A ∈ Σ , ν ∈ (0, 1), ν = 1/2. Then, P ν is a probability measure on (X, Σ ) if and only if Q ν is a ν-additive measure on (X, Σ ).
Proof Taking into account Proposition 6, this corollary immediately follows from Proposition 24.
Based on the result of Proposition 25, the formula in Eq. (51) may be viewed as a transformation between probability measures and ν-additive measures.

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 ν-additive (λ -additive) measures are connected with the rough set theory.
Definition 16 Let X be a finite set, and let R ⊆ X × X be a binary equivalence relation on X. The pair (R(A), R(A)) is said to be the the rough set of A ⊆ X in the approximation space (X, R) if The concept of a rough set was introduced by Pawlak (Pawlak 1982). for any A ⊆ X. Skowron (Skowron 1989(Skowron , 1990 showed that the functions q and q are a dual pair of belief-and plausibility measures and the corresponding basic probability assignment is m(A * ) = |A * |/|X| for all A * ∈ X/R, and 0 otherwise. Furthermore, Yao and Lingras (Yao and Lingras 1998)  for any A ⊆ X Wu et al. (2002). Based on these results and on our proposition findings, we will establish some connections between rough sets and ν-additive measures by using the following propositions.
Proposition 26 Let Q ν 1 and Q ν 2 be two ν-additive measures on the finite set X, and let R ⊆ X × X be a binary equivalence relation on X. Furthermore, let (R(A), R(A)) be the rough set of A ∈ P(X) with respect to the approximation space (X, R) and let the functions q, q : P(X) → [0, 1] be given by where R(A) and R(A) are the lower-and upper approximations of A, respectively, for any A ∈ P(X). Then, if the equations hold for any A ∈ P(X), then Q ν 1 and Q ν 2 are a dual pair of ν-additive measures on X with ν 1 ∈ (0, 1/2], ν 2 ∈ [1/2, 1).
Proof Based on Skowron's results in (Skowron 1989(Skowron , 1990, if the conditions of this proposition are satisfied, then the functions q and q are a dual pair of belief-and plausibility measures on X. Hence, the conditions that (i) Q ν 1 (A) = q(A), Q ν 2 (A) = q(A) hold for any A ∈ P(X) (ii) Q ν 1 and Q ν 2 are two ν-additive measures on X and the fact that q and q are a dual pair of belief-and plausibility measures on X together imply that Q ν 1 and Q ν 2 are also a dual pair of ν-additive measures on X. Furthermore, as q is a belief measure and q is a plausibility measure, based on Proposition 18, ν 1 ∈ (0, 1/2] and ν 2 ∈ [1/2, 1) hold as well.
Proposition 27 If Q ν 1 and Q ν 2 are a dual pair of ν-additive measures on the finite set X with ν 1 ∈ (0, 1/2], ν 2 ∈ [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 for any A ∈ P(X). Therefore, it is sufficient to show that if the conditions of our proposition are satisfied, then Q ν 1 is a belief measure on X, Q ν 2 is a plausibility measure on X, and m is the basic probability assignment of the belief measure Q ν 1 . Let us assume that the conditions of this proposition are satisfied. Then, since Q ν 1 and Q ν 2 are a dual pair of ν-additive measures on the finite set X, based on Proposition 20, Q ν 1 and Q ν 2 are a dual pair of belief-and plausibility measures on X. Furthermore, as ν 1 ∈ (0, 1/2] and ν 2 ∈ [1/2, 1), based on Proposition 18, Q ν 1 is a belief measure on X and Q ν 2 is a plausibility measure on X, and so condition (3) means that m is the basic probability assignment of the belief measure Q ν 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 (Yao and Lingras 1998) are satisfied as well.
6.2 The λ -additive measure and the multi-attribute utility function Here we will state interesting analogies between the λadditive measure and the multi-attribute utility function. Let X 1 , X 2 , . . . , X n be attributes, where each X i may be either a scalar attribute or a vector of scalar attributes (i = 1, 2, . . . , 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 , . . . , x n ), where x i is a particular value of the attribute X i (i = 1, 2, . . . , n). The utility function u : X → R, which is assumed to be continuous, assigns a utility value to the consequence (x 1 , x 2 , . . . , x n ); that is, the utility of consequence (x 1 , x 2 , . . . , x n ) is u(x 1 , x 2 , . . . , 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 (Keeney and Raiffa 1993) proved the following proposition which states that the mutual utility independence of attributes implies a multiplicative multiattribute utility function.
We can see that the right hand side of Eq. (18) with λ > −1, λ = 0 has the same form as the right hand side of Eq. (53). It means that there is an interesting connection between the λ -additive measures and the multi-attribute multiplicative utility function. Namely, a λ -additive measure with λ = 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.
Here, the formula in Eq. (53) can be written as from which lim k→0 u M (x 1 , x 2 , . . . , Note that is the so-called multi-attribute additive utility function (Keeney 1974). We can get Eq. (56) from Eq. (55) by allowing for k = 0.
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 × 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 , . . . , 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 λ = 0 has the same form as the right hand side of Eq. (56). It means that a λ -additive measure with λ = 0 of the union of n pairwise disjoint sets is computed in the same way as the multiattribute utility of n additive independent attributes. Table 1 summarizes the analogies between the λadditive measures and the multi-attribute utility functions. Table 1 λ -additive measure of union of pairwise disjoint sets and utility value of consequence (x 1 , x 2 , . . . , x n ) (1 + λ Q λ (A i )) − 1 multi-attribute utility u(x 1 , x 2 , . . . , x n ); k > −1 (1 + kk i u i (x i )) − 1 6.3 The λ -additive measure and some operators of continuous-valued logic Here, we will state a formal connection between the λadditive measure and certain operators of continuous-valued logic.
It can be shown that if α > 0, then o (α) GD,γ is a conjunction operator, and if α < 0, then o (α) GD,γ is a disjunction operator (see Dombi (2008)). Moreover, the operator o (α) GD,γ is general because depending on its parameter values it can cover a range of familiar fuzzy conjunction and disjunction operators including the Dombi operators (Dombi 1982), the product operators (Dombi 2008), the Einstein operators (Wang and Liu 2012), the Hamacher operators (Hamacher 1978), the drastic operators (Zimmermann 2013) and the min-max operators (Zadeh 1965). Table 2 summarizes the operators that the generalized Dombi operator class can cover.
From Eq. (59) and Eq. (60), we notice an interesting analogy. Namely, a λ -additive measure with λ = 0 of the union of 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 continuousvalued logic variables. It should be added that this analogy is just a formal one since g(x i ) ∈ (0, ∞) and Q λ (A i ) ∈ [0, 1], and g(x i ) and Q λ (A i ) have different meanings.

Summary and future plans
In our study, we introduced the ν-additive measure as an alternatively parameterized λ -additive measure. Here, we will summarize our main findings concerning the ν-additive (λadditive) measures.
(2) Two ν-additive measures are a dual pair if and only if the sum of their parameters equals 1.
(3) A ν-additive measure is a (a) belief measure if and only if 0 < ν ≤ 1/2 (b) probability measure if and only if ν = 1/2 (c) plausibility measure if and only if 1/2 ≤ ν < 1. (4) Two ν-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 ν-additive (λ -additive) measure into a probability measure; and conversely, this transformation can be utilized for transforming a probability measure into a ν-additive (λ -additive) measure. (6) Dual pairs of ν-additive measures are strongly associated with the lower-and upper approximation pairs of rough sets. (7) There are interesting formal connections between the λ -additive measures and the multi-attribute utility functions. Namely, (a) if λ = 0, then the λ -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 λ > −1 and λ = 0, then the λ -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. (8) There is an interesting formal connection between the λ -additive measure and certain operators of continuousvalued logic. Namely, if λ > −1 and λ = 0, then the computation method of λ -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.
As part of our future research plans, we would like to formulate a calculus of the ν-additive measure and generalize the Bayes theorem and the Poincaré formula for νadditive measures. We also plan to study how the ν-additive measure can be utilized in the fields of computer science, engineering and economics. Especially, we aim to investigate the potential application of ν-additive measures in network science.
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.