Quasi-Arithmetic Type Mean Generated by the Generalized Choquet Integral

It is known that the quasi-arithmetic means can be characterized in various ways, with an essential role of a symmetry property. In the expected utility theory, the quasi-arithmetic mean is called the certainty equivalent and it is applied, e.g., in a utility-based insurance contracts pricing. In this paper, we introduce and study the quasi-arithmetic type mean in a more general setting, namely with the expected value being replaced by the generalized Choquet integral. We show that a functional that is defined in this way is a mean. Furthermore, we characterize the equality, positive homogeneity, and translativity in this class of means.


Introduction
The notion of quasi-arithmetic mean, playing an important role in several branches of mathematics and its applications, was introduced in the book by Hardy, Littlewood, and Pólya [1]. Recall that, if I ⊆ R is an interval and u : I → R is a strictly monotone continuous function, then the quasi-arithmetic mean M u : n∈N I n → I, which is generated by u, is given by u(x i ) for x = (x 1 , ..., x n ) ∈ I n , n ∈ N.
Various axiomatic characterizations of the quasi-arithmetic mean have been independently established by de Finetti [2], Kolmogorov [3], and Nagumo [4]. For these characterizations, the symmetry property of the mean is essential. Quasi-arithmetic means have been generalized in several directions. In particular, Bajraktarević [5] introduced quasi-arithmetic means that were weighted by a weight function. Bajraktarević means were characterized by Páles [6]. A further generalization of the quasi-arithmetic means has been developed by Daróczy [7,8], who introduced deviation means. The notion of the quasi-arithmetic mean can be considered in more general settings. In particular, if X : S → R is an F −measurable essentially bounded function on a given probability space (S, F , P), then the quasi-arithmetic mean of X, generated by a strictly monotone continuous function u : R → R, is defined as follows where E P denotes the expected value with respect to P.
In the expected utility theory developed by von Neumann and Morgenstern [9], the quasi-arithmetic mean, as given by (1), is known as the certainty equivalent. It establishes a symmetry in preferences between a risk that is represented by X and a deterministic payoff M (u,P) (X), in a where E µ is the Choquet integral with respect to µ (see Section 2). It turns out that, for any F −measurable function X : S → R,μ−essentially bounded from below, and µ−essentially bounded from above M (u,µ) (X) is a well-defined mean (see Sections 2 and 3 for details). The functional M u,µ,ν , as given by (2), refers to the certainty equivalent under the model of Schmeidler [10]. Some particular cases of M (u,µ) have been investigated in [11]. Inspired by the results in [11], in this paper, we study the properties of the functional defined on a family of all F −measurable functions X : S → R, which are ν−essentially bounded from below and µ−essentially bounded from above. Here E µν stands for the generalized Choquet integral with respect to capacities µ and ν on (S, F ). We show that M (u,µ,ν) is well-defined and it is a mean, which is inf Furthermore, we establish the characterizations of the equality, positive homogeneity, and translativity in the class of means defined by (3). In the whole paper (S, F ) stands for a measurable space.

Generalized Choquet Integral
A set function µ : F → [0, 1] is called the capacity on (S, F ) provided that it satisfies the following conditions: Obviously, every probability measure on (S, F ) is a capacity on (S, F ). Furthermore, simple calculations show that, if µ is a capacity on (S, F ), then so is the set functionμ of the form It is called the conjugation of µ.
If µ is a capacity on (S, F ), then an F -measurable function X : S → R is called µ-essentially bounded from above provided that there exists t ∈ R, such that µ({s ∈ S : X(s) > t}) = 0. In such a case the µ-essential supremum of X is defined in the following way µ − ess sup X := inf{t ∈ R : µ({s ∈ S : X(s) > t}) = 0}.
An F -measurable function X : S → R is said to be a µ-essentially bounded from below if there exists t ∈ R, such that µ({s ∈ S : X(s) < t}) = 0. In that case, µ − ess inf X := sup{t ∈ R : µ({s ∈ S : X(s) < t}) = 0} is called the µ-essential infimum of X.

Remark 1.
Assume that µ is a capacity on (S, F ) and X : S → R is an F -measurable function. Note that: • if X is µ-essentially bounded from above, then −∞ ≤ µ − ess sup X < ∞; • if X is µ-essentially bounded from below then −∞ < µ − ess inf X ≤ ∞.
The following example shows that, in general, µ − ess sup X and µ − ess inf X need not be finite.
Obviously, µ is a capacity on (S, F ). Furthermore, X = id R is F -measurable and, for every t ∈ R, we have Hence, X is µ-essentially bounded, but µ − ess sup X = −∞ and µ − ess inf X = ∞.

Lemma 1.
Assume that µ is a capacity on (S, F ) and X : S → R is an F -measurable function µ-essentially bounded from above andμ-essentially bounded from below. Thenμ − ess inf X and µ − ess sup X are finite and Proof. Suppose that which yields a contradiction.

Remark 2.
Assume that X : S → R is an F -measurable function. Subsequently, for every capacity µ on Therefore: (i) if X is bounded from below (above) then X is µ-essentially bounded from below (above) for every capacity µ on (S, F ); and, (ii) if, for some capacity µ on (S, F ), X is µ-essentially bounded from below then inf X ≤ µ − ess inf X ≤ sup X; (iii) if, for some capacity µ on (S, F ), X is µ-essentially bounded from above, then inf X ≤ µ − ess sup X ≤ sup X.

Lemma 2.
Assume that µ is a capacity on (S, F ) and u : R → R is a strictly increasing continuous function.

(i)
If X : S → R is an F -measurable function µ-essentially bounded from above, then so is u • X. Furthermore, µ − ess sup u • X = u(µ − ess sup X).
(ii) If X : S → R is an F -measurable function µ-essentially bounded from below, then so is u • X. Moreover, Proof. Let X : S → R be an F -measurable function µ-essentially bounded from above. Because u is continuous, u • X is F -measurable. Moreover, as u is strictly increasing, we have Therefore, because X is µ-essentially bounded from above, so is u • X. We show that Suppose that (10) does not hold and fix t ∈ R, such that Thus, µ({u • X > t}) > 0, hence t < u(∞). Consequently t = u(s) for some s ∈ R and so, we obtain Hence, s ≤ µ − ess sup X, that is t = u(s) ≤ u(µ − ess sup X), which yields a contradiction and proves (10). Now, we prove that Suppose that (11) is not true and fix t ∈ R, such that µ − ess sup u • X < t < u(µ − ess sup X).
Subsequently, µ({u • X > t}) = 0, which implies that t > u(−∞). Hence, t = u(s) for some s ∈ R and we obtain that is s ≥ µ − ess sup X. Thus t = u(s) ≥ u(µ − ess sup X), which gives a contradiction and proves (11). In this way, we have proved (i). A proof of (ii) is similar.
If µ is a capacity on (S, F ) and X : S → R is an F -measurable function µ-essentially bounded from above andμ-essentially bounded from below, then the Choquet integral of X is defined in following way (cf. [12]) where the integrals on the right-hand side of (12) are the Riemann integrals. Note that both of the integrals on the right-hand side of (12) are finite. Several details concerning various properties of Choquet integral can be found in [13]. In particular, in [13] (Proposition 5.1), it is proved that the Choquet integral is positively homogeneous, translative, and monotonic. Moreover, it is asymmetric, which is, for every F -measurable function X : S → R µ-essentially bounded from above and µ-essentially bounded from below, it holds Remark 3. Note that, for all capacities µ and ν on (S, F ) and any F -measurable function X : S → R µ-essentially bounded from above and ν-essentially bounded from below, we have and In particular, taking ν =μ, in view of (12), we obtain Proposition 1. Assume that µ is a capacity on (S, F ). Subsequently, for every F -measurable function X : S → R µ-essentially bounded from above andμ-essentially bounded from below, it holds Proof. Assume that X : S → R is an F -measurable function µ-essentially bounded from above andμ-essentially bounded from below and put m =μ − ess inf X and M = µ − ess sup X.
Furthermore, from Corollary 1, we derive that m and M are finite and m ≤ M. Therefore, the following three cases are possible: In the first case, using the monotonicity of µ, for every x ∈ (−∞, m), we get On the other hand, applying Remark 3 with ν =μ, we obtain Therefore, (17) holds.
In the second case, for every x ∈ (M, ∞), we obtain Furthermore, while taking Remark 3 with ν =μ into account, we conclude that Hence, (17) is valid.
In the third case, while using the monotonicity of Choquet integral, in view of Remark 3, we get which yields the assertion. Now, assume that µ and ν are capacities on (S, F ) and X : S → R is an F -measurable µ-essentially bounded from above and ν-essentially bounded from below function. The generalized Choquet integral of X is given by where X + and X − denote the positive and negative part of X, respectively, which is X + := max{X, 0} and X − := max{−X, 0}. Note that X + and X − are F -measurable, X + is µ-essentially bounded from above andμ-essentially bounded from below (cf. Remark 2 (i)), while X − is ν-essentially bounded from belowν-essentially bounded from above. Thus, the generalized Choquet integral is well-defined by (18). Generalized Choquet integral was introduced in [14] for discrete random variables. (16)

Remark 4. It follows from
for F -measurable function X : S → R µ-essentially bounded from above andμ-essentially bounded from below. Hence, the Choquet integral is a particular case of the generalized Choquet integral. Another important particular case of the generalized Choquet integral is the Šipoš integral, which refers to case µ = ν.

Remark 5.
Because the Choquet integral is positively homogeneous and monotonic, so is the generalized Choquet integral. Moreover, in view of (13), for every capacities µ and ν on (S, F ) F -measurable function X : S → R µ-essentially bounded from above and ν-essentially bounded from below, we obtain Remark 6. Assume that µ and ν are capacities on (S, F ) and X : S → R is an F -measurable function µ-essentially bounded from above and ν-essentially bounded from below. Let m = ν − ess inf X and M = µ − ess sup X. Subsequently, µ({X > x}) = 0 for x ∈ (M, ∞) and µ({X < x}) = 0 for x ∈ (−∞, m). Therefore, making use of (14), (15) and (18), we obtain that:

Remark 7.
Assume that µ and ν are capacities on (S, F ). Subsequently, in view of Remark 6, we have for every F -measurable function X : S → R µ-essentially bounded from above and ν-essentially bounded from below.
We complete with this section with the following result.

Proposition 2.
Assume that µ and ν are the capacities on (S, F ) and X : S → R is of the form with some A ∈ F and x 1 , x 2 ∈ R, such that x 1 < x 2 . Subsequently, X is F -measurable µ-essentially bounded from above and ν-essentially bounded from below. Furthermore, Proof. An F -measurability of X is obvious. Note that ν( , hence X is µ-essentially bounded from above and ν-essentially bounded from below. Furthermore, we have Therefore, while taking into account (14), (15) and (18), after a standard computation, we obtain the assertion.

Quasi-Arithmetic Type Mean Generated by the Generalized Choquet Integral
In this section, we prove that M (u,µ,ν) defined on a family of all F −measurable function X : S → R ν−essentially bounded from below and µ−essentially bounded from above, as given by (3), is well-defined and it is a mean. Theorem 1. Assume that µ and ν are capacities on (S, F ) and u : R → R be a strictly increasing continuous function. Subsequently, for every F -measurable function X : S → R µ-essentially bounded from above and ν-essentially bounded from below, there exists a unique M (u,µ,ν) (X) ∈ R, such that Furthermore, M (u,µ,ν) is a mean, which is (4) holds for every F -measurable function X : S → R µ-essentially bounded from above and ν-essentially bounded from below.
Proof. Assume that X : S → R is an F -measurable function µ-essentially bounded from above and ν-essentially bounded from below. Subsequently, according to Lemma 2, so is u • X. Let M = µ − ess sup X and m = ν − ess inf X. If there exist s 1 , s 2 ∈ S, such that (u • X)(s 1 ) < 0 < (u • X)(s 2 ) then, applying Lemma 2 and Corollary 7, we obtain Because u is continuous, this means that (26) holds with some M (u,µ,ν) (X) ∈ R. Furthermore, as u is strictly increasing, such an M (u,µ,ν) (X) is unique and it satisfies (4). Now, assume that u • X takes only non-positive values. Afterwards, according to Remark 2 (ii), (iii), we have ν − ess inf u • X ≤ 0 and µ − ess sup u • X ≤ 0. Hence, applying Remark 6, in view of (13), we have Hence, applying Proposition 1, Lemma 2, and Remark 2, we obtain Thus, arguing as previously, we conclude that there exists a unique M (u,µ,ν) (X) ∈ R, such that (26) holds and, moreover, (4) is valid. If u • X takes only non-negative values, applying Remark 2 (ii), (iii), we have ν − ess inf u • X ≥ 0 and µ − ess sup u • X ≥ 0. Thus, making use of Remark 6, we conclude that hence, applying Proposition 1, Lemma 2, and Remark 2, we get Therefore, as u is continuous and strictly increasing, we have the assertion.

Remark 8.
It follows from (26) that, for arbitrary capacities µ and ν on (S, F ) and for every continuous strictly increasing function u : R → R, the functional that is given by (3) is a well-defined mean on the family of all F -measurable functions X : S → R µ−essentially bounded from above and ν−essentially bounded from below. In particular, while taking (19) into account, we conclude that (2) defines a mean on a family of all F -measurable functions X : S → R µ−essentially bounded from above andμ−essentially bounded from below.

Remark 9.
Assume that u : R → R is a strictly increasing continuous function. It follows from (3) and (23)-(25) that, if X : S → R is of the form (22) with some A ∈ F and x, y ∈ R, such that x < y, then if u(y) ≤ 0. (29)

Main Properties of the Mean
In this section, we are going to investigate some of the properties of means defined by (3). From now on, we assume that µ and ν are capacities on (S, F ). By X , we denote a family of all F -measurable function X : S → R µ-essentially bounded from above and ν-essentially bounded from below. A subfamily of X of functions X : S → R of the form (22), where A ∈ F and x, y ∈ R, such that x < y, will be denoted by X (2) .
We begin with a characterization of the equality in this class of means. The following result will play an essential role in our considerations.
Furthermore, for every x ∈ (−∞, z) and y ∈ (z, ∞) sufficiently close to z, we have Therefore, applying (45), from (46), we derive that for every x ∈ (−∞, z) and y ∈ (z, ∞) sufficiently close to z. Hence, in view of (32), we get α + = α − and, so, while taking into account (45), we obtain (35) with α := α + = α − and β = 0. Now, suppose that (37) does not hold. Subsequently, as u and v are continuous, there exists a non-degenerate interval J ⊆ R, such that u(x)v(x) < 0 for x ∈ J. In the first case, in view of (29), we obtain for every B ∈ F and x, y ∈ J, such that x < y. Hence, replacing x by u −1 (x) and y by u −1 (y), for every B ∈ F and x, y ∈ u(J) with x < y, we get where f : u(J) → R is given by (40). Accordingly, as f is continuous, applying Lemma 3, we conclude that ν(B) = 1 − µ(S\B) for B ∈ F , which is ν =μ. Using the similar arguments, in the second case, for every B ∈ F and x, y ∈ J, such that x < y, we obtain Hence, according to Lemma 3, we obtain µ(B) = 1 − ν(S\B) for B ∈ F , which again gives ν =μ. Because ν =μ, from (29) and (33), we derive that for every x, y ∈ R, such that x < y. Thus, for every x, y ∈ R with x < y, where f is given by (40). Hence, taking (32) into account and applying Lemma 3, we obtain that there exist α ∈ (0, ∞) and β ∈ R, such that (43) holds. Hence , in view of (40), we get (35). In this way, we proved the implication (i) ⇒ (iii). Now, we prove that (iii) ⇒ (ii). Assume that (iii) holds. Subsequently, by (35), we have If (37) is valid, then, as we have already noted, either I − = R or I + = R or I − = (−∞, z) and I + = (z, ∞) for some z ∈ R. In the first case, in view of (13) and (18), for every X ∈ X , we have (27) and E µν [v(X)] = Eν[v(X)]. Thus, using the fact that Choquet integral is positively homogeneous and translative, taking into account (3), (35), and (47), for every X ∈ X , we obtain In the second case, in view of (18), for every X ∈ X , we have (28) and Hence, arguing, as previously, we obtain (34). In the third case from the continuity of u and v, we derive that u(z) = v(z) = 0. Hence, in view of (35), we get β = 0 and, so, as the generalized Choquet integral is positively homogeneous, making use of (47), for every X ∈ X , we obtain If (37) does not hold then, according to (36), we have ν =μ. Therefore, taking (19), (35), and (47) into account, for every X ∈ X , we have obtain This proves that (iii) ⇒ (ii). The implication (ii) ⇒ (i) is obvious.
Applying Theorem 2, we are going to characterize positive homogeneity and translativity in the class of means given by (3). Theorem 3. Let u : R → R be a strictly increasing continuous function. Assume that there exists a set A ∈ F , such that (32) holds. Subsequently, the following conditions are equivalent: (iii) there exist α, β, r ∈ (0, ∞) and δ ∈ R, such that and ν =μ whenever δ = 0.

Conclusions
We have introduced a new class of quasi-arithmetic type means generated by the generalized Choquet Integral. Our approach is motivated by recent applications of this type of means in the theory of decision making under risk and in the theory of insurance premium principles. Using the methods of functional equations, we established characterizations of some important properties in the considered class of means. It is remarkable that the aforementioned characterizations are expressed not only in terms of the relations between functions generating means, but they involve also the properties of capacities under consideration.