Choquet Integration on Riesz Spaces and Dual Comonotonicity

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 367, Number 12, December 2015, Pages 8521–8542 http://dx.doi.org/10.1090/tran/6313 Article electronically published on May 13, 2015

CHOQUET INTEGRATION ON RIESZ SPACES AND DUAL COMONOTONICITY

SIMONE CERREIA-VIOGLIO, FABIO MACCHERONI, MASSIMO MARINACCI, AND LUIGI MONTRUCCHIO

Abstract. We give a general integral representation theorem for nonadditive functionals defined on an Archimedean Riesz space X with unit. Additivity is replaced by a weak form of modularity, or, equivalently, dual comonotonic additivity, and integrals are Choquet integrals. Those integrals are defined through the Kakutani isometric identification of X with a C (K) space. We further show that our notion of dual comonotonicity naturally generalizes and characterizes the notions of comonotonicity found in the literature when X is assumed to be a space of functions.

1. Introduction Consider the following classical Riesz-type integral representation theorem in the space of real valued, bounded, and F-measurable functions B (Ω, F), where F is a σ-algebra of sets.1 Theorem 1. Let X = B (Ω, F) and let V be a functional from X to R.The following statements are equivalent: (i) V is monotone and additive; (ii) there exists a unique finitely additive measure μ : F→[0, ∞) such that (1) V (x)= x (ω) dμ (ω) ∀x ∈ X. Ω This paper aims at extending the above Riesz representation result along the following lines: (1) the domain X of the functional will be a general Archimedean Riesz space with (order) unit; (2) the functional V : X → R will neither be assumed monotone nor additive. In turn, X is identified with a C (K) space and the integral in (1) is a Cho- quet integral (see Choquet [8]) with μ being a suitable nonadditive set function defined on a lattice of sets. Several partial extensions in this direction have already been established. For instance, when X = B (Ω, F), Schmeidler [23] introduced comonotonic additivity, for a functional, in place of additivity: a property actually

Received by the editors April 27, 2012 and, in revised form, September 29, 2013. 2010 Mathematics Subject Classiﬁcation. Primary 28A12, 28A25, 28C05, 46B40, 46B42, 46G12. The ﬁnancial support of ERC (Advanced Grant BRSCDP-TEA), AXA Research Fund, and MIUR (Grant PRIN 20103S5RN3 005) is gratefully acknowledged. 1Theorem 1 holds true also when F is an algebra; in such a case, B (Ω, F)mightnotbea vector space.

c 2015 American Mathematical Society 8521

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shared by Choquet integrals.2 Along the same lines Zhou [25] generalized X to be a Stone vector lattice, while Cerreia-Vioglio, Maccheroni, Marinacci, and Montruc- chio [7], inter alia, also removed the monotonicity assumption and considered Stone lattices.3 Theorem 2 (Schmeidler). Let X = B (Ω, F) and let V be a functional from X to R. The following statements are equivalent: (i) V is monotone and comonotonic additive; (ii) there exists a unique capacity μ : F→[0, ∞) such that (2) V (x)= x (ω) dμ (ω) ∀x ∈ X. Ω In the above result there are two new elements: comonotonicity and the Choquet integral in (2). Recall that two functions f and g from Ω to R are said to be comonotonic if and only if (3) (f (ω) − f (ω)) (g (ω) − g (ω)) ≥ 0 ∀ω, ω ∈ Ω. Comonotonic additivity, in Theorem 2, means that the functional V is additive when restricted to pairs of comonotonic functions. On the other hand, when μ is a capacity (that is a monotone set function) the Choquet integral (2) is defined as the sum of two improper Riemann integrals: ∞ 0 (4) x (ω) dμ (ω)= μ (x ≥ t) dt + [μ (x ≥ t) − μ (Ω)] dt ∀x ∈ X Ω 0 −∞ where (x ≥ t)={ω ∈ Ω:x (ω) ≥ t}. This definition can be easily extended from capacities μ to set functions of bounded variation (see [19], [18], and [7]). At the same time, also Theorem 2 has been extended to the nonmonotone case but always by retaining the assumption that X is a space of functions. To state our main result, we recall some definitions regarding Riesz spaces. Definition 3. An Archimedean Riesz space with unit is a real vector space X with a partial order ≥ such that: (i) (Ordered vector space) x ≥ y implies αx + z ≥ αy + z for all α ≥ 0andall z ∈ X; (ii) (Archimedean property) 0 ≤ nx ≤ y for all n ∈ N implies x =0; (iii) (Riesz property) X is a lattice with respect to ≥; (iv) (Existence of a unit) there exists an element e ∈ X\{0} such that X = {x ∈ X : |x|≤ne} . n∈N

An element e satisfying (iv) is said to be a unit. We deﬁne X+ = {x ∈ X : x ≥ 0}. It is then well known that · : X → [0, ∞) deﬁned by (5) x =min{λ ≥ 0:|x|≤λe}∀x ∈ X is an M-norm over X.4 When X is norm complete, X is said to be an AM-space. Given an Archimedean Riesz space with unit X, we endow X with the norm above and we denote by X∗ the norm dual. We endow X∗ and any of its subsets with

2Comonotonicity was already used under different names by many authors. See Denneberg [11] for more details as well as for different characterizations of comonotonicity. 3See Section 1.1 for some definitions, notation, and terminology. 4 That is, if |x|≤|y|,thenx≤y,andx ∨ y =max{x , y} for all x, y ∈ X+.

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the w*-topology. The map (x, ξ) −→ x, ξ for all (x, ξ) ∈ X × X∗ denotes the dual pairing. The positive unit sphere is (6) Δ = {ξ ∈ X∗ : e, ξ =1and x, ξ ≥0 for all x ≥ 0} . We denote by E the set of extreme points of the compact and convex set Δ. The set E is compact. By Krein-Milman’s Theorem, Δ is the closed convex hull of E. We also endow E with the Baire σ -algebra: Baire(E). Next, we list some special properties for functionals V : X → R. Deﬁnition 4. Let X be an Archimedean Riesz space with unit. The functional V : X → R is: (i) monotone if V (x) ≥ V (y) whenever x ≥ y; (ii) of bounded variation if for all x ∈ X+, Var (0,x) V n =sup |V (xi) − V (xi−1)| :0=x0 ≤ .... ≤ xn = x and n ∈ N < ∞; i=1 (iii) unit-additive if V (x + λe)=V (x)+V (λe) for all x ∈ X and all λ ≥ 0; (iv) unit-modular if V (x ∧ λe)+V (x ∨ λe)=V (x)+V (λe) for all x ∈ X and all λ ∈ R; (v) dual comonotonic additive if V (x + y)=V (x)+V (y) for all x, y ∈ X such that ( x, ξ − x, ξ )( y, ξ − y, ξ ) ≥ 0 ∀ξ,ξ ∈E; (vi) supermodular if V (x ∧ y)+V (x ∨ y) ≥ V (x)+V (y) for all x, y ∈ X; (vii) superadditive if V (x + y) ≥ V (x)+V (y) for all x, y ∈ X. The next statement is a Choquet-Bishop-DeLeeuw variant of Krein-Milman’s Theorem. Theorem 5. Let X be an Archimedean Riesz space with unit and V a functional from X to R. The following statements are equivalent: (i) V is monotone and additive; (ii) there exists a unique measure μ on Baire(E) such that (7) V (x)= x, ξ dμ (ξ) ∀x ∈ X. E This result can also be viewed as an abstract Gelfand’s integral representation theorem.5 The goal of this paper is to provide a nonadditive and nonmonotone version of the above representation result. The next theorem, our main result, achieves our goal. Theorem 6. Let X be an Archimedean Riesz space with unit and V a functional from X to R. The following statements are equivalent: (i) V is dual comonotonic additive and of bounded variation; (ii) V is unit-additive, unit-modular, and of bounded variation;

5Since E is compact, the existence of a measure μ in Theorem 5 when V ∈ Δ is merely a consequence of [20, Proposition 1.2]. By [20, Corollary 10.9] and since Δ is a Choquet simplex and E is an Fσ set, also the uniqueness of μ follows.

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(iii) there exists an outer continuous set function γ : U (E) → R of bounded variation such that V (x)= x, ξ dγ (ξ) ∀x ∈ X. E Moreover, γ is unique, while V is monotone if and only if γ is. Here U (E) denotes the lattice of upper level sets generating Baire(E), that is, U (E)={(f ≥ t):f ∈ C (E) ,t∈ R} . The close relation of Theorem 6 with Theorem 5 is evident. Theorem 5 can be seen as a corollary of Theorem 6, as detailed in Remark 15. It is also worth emphasizing that the equivalence between (i) and (ii) of Theorem 6 is novel even in the standard case X = B (Ω, F) where dual comonotonic additivity and comonotonic additivity coincide. This equivalence is obtained in Theorem 22 in the Appendix, which extends all the existing representation results in the literature when X is assumed to be a space of functions. Section 2 is entirely devoted to the proof of Theorem 6. Behind the definition of dual comonotonic additivity introduced in Definition 4, there is the property of two elements x and y in X being dually comonotonic, that is, such that (8) ( x, ξ − x, ξ )( y, ξ − y, ξ ) ≥ 0 ∀ξ,ξ ∈E. Section 3 provides a careful analysis of this notion and, inter alia, shows how dual comonotonicity naturally generalizes and characterizes the notions of comonotonicity found in the literature when X is assumed to be a space of functions. Section 4 is dedicated to the special case in which the functional V has the extra property of being either supermodular or superadditive. We study the close relation of these two properties in view of their integral representation. Moreover, we show that superadditivity is crucial in guaranteeing the uniqueness of γ in Theorem 6 also when γ is extended to Baire(E). Section 5 presents two applications of our main result to Mathematical Finance. Finally, we confine to the Appendix all our representation results when X is a space of functions.

1.1. Notation. We close this introductory section by adding some further deﬁni- tions used in the paper. If Σ is a lattice of subsets of a set Ω such that ∅, Ω ∈ Σ, a function γ :Σ→ R is a set function if γ (∅) = 0. In particular, with each subset of Ω being understood to be in Σ, a set function γ is: (i) monotone or a capacity if γ (A) ≤ γ (B) whenever A ⊆ B; (ii) supermodular if γ (A ∩ B)+γ (A ∪ B) ≥ γ (A)+γ (B) for all A and B; (iii) outer continuous at A ∈ Σ if limn γ (An)=γ (A) whenever An ↓ A; (iv) outer continuous if γ is outer continuous at each A; (v) of bounded variation if n sup |γ (Ai) − γ (Ai−1)| : ∅ = A0 ⊆ .... ⊆ An =Ωandn ∈ N < ∞; i=1 (vi) a nonadditive probability measure if γ is an outer continuous capacity such that γ (Ω) = 1.

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Submodularity and inner continuity are deﬁned similarly. We say that a set function is continuous (resp., modular) if and only if it is inner and outer continuous (resp., submodular and supermodular). The notion of set function of bounded variation goes back to Aumann and Shapley [4]. The space of set functions of bounded variation on Σ is denoted by bv (Σ).6 It is immediate to see that capacities are set functions of bounded variation. Given a subset A ⊆ Ω, we denote by χA the indicator function of A.AStone vector lattice is a Riesz space, L, of real valued and bounded functions on a set Ω which further contains χΩ. In this case, we denote by ΣL the lattice of upper level sets, that is, ΣL = {(f ≥ t):f ∈ L and t ∈ R}.

2. Proof of the main theorem Let X be an Archimedean Riesz space with unit endowed with the supnorm (5). Let E be the set {ξ ∈ Δ:ξ is an extreme point of Δ}. By [2, Theorems 3.14 and 4.28], we have that ∅ = E = {ξ ∈ Δ:ξ is a lattice homomorphism and e, ξ =1} . For this reason, E, endowed with the w*-topology, is Hausdorff and compact. We define by C (E) the space of all continuous functions over E and we endow it with the supnorm. Define T : X → C (E)by (9) x → T (x)=ˆx wherex ˆ (ξ)= x, ξ ∀ξ ∈E, ∀x ∈ X. By the classical Kakutani-Bohnenblust-M. Krein-S. Krein theorem (see for instance [2, Theorem 4.29]), T turns out to be an isometric lattice homomorphism such that 7,8 T (e)=χE .Thus,X is lattice isometric to C = T (X). By [1, Theorem 9.12], the latter is uniformly dense in C (E). Then, we consider T : X → C.Inthisway, T is an isometry, a lattice isomorphism, and the same applies for its inverse T −1. Both maps are positive. Proof of Theorem 6. (i) implies (ii). Let x ∈ X and λ ∈ R. Since ( x, ξ − x, ξ ) ( λe, ξ − λe, ξ ) = 0 for all ξ,ξ ∈E, x and λe are dually comonotonic. Since V is dual comonotonic additive, we have that (10) V (x + λe)=V (x)+V (λe) , proving, in particular, that V is unit-additive. Since T is a lattice isomorphism such that T (e)=χE ,wehavethatT (x ∧ λe)=ˆx ∧ λχE and T (x ∨ λe)=ˆx ∨ λχE . By [18, Lemma 4.6],x ˆ ∧ λχE andx ˆ ∨ λχE are comonotonic in C. This implies that x ∧ λe and x ∨ λe are dually comonotonic. By (10) and since V is dual comonotonic additive, it follows that V (x ∧ λe)+V (x ∨ λe)=V (x ∧ λe + x ∨ λe)=V (x + λe)=V (x)+V (λe) , proving that V is unit-modular. (ii) implies (iii). Suppose V is unit-additive, unit-modular, and of bounded variation. Given T as defined in (9), define Vˆ : C → R by Vˆ = V ◦ T −1.

6A detailed study of the space of set functions of bounded variation can be found in [7, Section 3]. 7From here on, all compact topological spaces are understood to be Hausdorﬀ. 8Some of the results cited in [2] apply to the case when X is an AM-space and, in particular, it is norm complete. Nevertheless, it is routine to check that they apply to an Archimedean Riesz space X with unit when endowed with the supnorm.

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The set C ⊆ C (E) is a Stone vector lattice. Since T −1 is a positive operator −1 such that T (χE )=e and V is unit-additive, we have that −1 −1 −1 Vˆ (f + λχE )=V T (f + λχE ) = V T (f)+λe = V T (f) + V (λe)

= Vˆ (f)+Vˆ (λχE ) ∀f ∈ C, ∀λ ≥ 0, proving that Vˆ is unit-additive. Since T −1 is a lattice isomorphism such that −1 T (χE )=e and V is unit-modular, we have that

Vˆ (f ∧ λχE )+Vˆ (f ∨ λχE ) −1 −1 = V T (f ∧ λχE ) + V T (f ∨ λχE ) = V T −1 (f) ∧ λe + V T −1 (f) ∨ λe −1 = V T (f) + V (λe)=Vˆ (f)+Vˆ (λχE ) ∀f ∈ C, ∀λ ∈ R, ˆ ≤ ∈ { }n proving that V is unit-modular. Finally, consider 0 f C and a chain fi i=0 ≤ ≤···≤ { }n −1 such that 0 = f0 f1 fn = f. Deﬁne x and xi i=0 by x = T (f)and −1 −1 xi = T (fi)fori ∈{0, ..., n}.SinceT is a positive operator, it follows that 0=x0 ≤ x1 ≤···≤xn = x and

n n Vˆ (fi) − Vˆ (fi−1) = |V (xi) − V (xi−1)|≤VarV (0,x) . i=1 i=1 Since V is of bounded variation, Vˆ is of bounded variation. By Lemma 21, we have that Vˆ : C → R is Lipschitz continuous. Since C is uniformly dense in C (E), it follows that Vˆ : C → R admits a unique Lipschitz continuous extension to C (E). We denote the extension by V¯ .SinceVˆ is unit-additive, unit-modular, and of bounded variation, it follows that V¯ shares the same properties. Finally, consider ∈ E { } ⊆ E ↓ f C ( ) and a sequence fn n∈N C ( ) such that fn f pointwise. By Dini’s { } ¯ Theorem, we have that fn n∈N converges in supnorm to f.SinceV is Lipschitz ¯ ¯ continuous, it follows that limn V (fn)=V (f). By Theorem 22, there exists an outer continuous γ ∈ bv Σ E = bv (U (E)) such that C( ) V¯ (f)= f (ξ) dγ (ξ) ∀f ∈ C (E) . E In particular, we have that V (x)= x, ξ dγ (ξ) ∀x ∈ X. E (iii) implies (i). Assume there exists an outer continuous γ ∈ bv (U (E)) such that (11) V (x)= x, ξ dγ (ξ) ∀x ∈ X. E We start by considering the case when γ is a capacity. Consider x and y in X such that x ≥ y.SinceT is positive, it follows thatx ˆ ≥ yˆ. By (11) and the monotonicity of the Choquet integral when γ is a capacity, we have that V (x)= xˆ (ξ) dγ (ξ) ≥ yˆ(ξ) dγ (ξ)=V (y) , E E proving that V is monotone. Now let x and y be dually comonotonic, that is, ( x, ξ − x, ξ )( y, ξ − y, ξ ) ≥ 0 ∀ξ,ξ ∈E.

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It follows that the two functions,x ˆ andy ˆ, are comonotonic. Since Choquet integrals are comonotonic additive, we have that V (x + y)= (ˆx (ξ)+ˆy (ξ)) dγ (ξ)= xˆ (ξ) dγ (ξ)+ yˆ(ξ) dγ (ξ)=V (x)+V (y) , E E E proving that V is dual comonotonic additive according to (v) of Deﬁnition 4. Next, consider the case when γ is of bounded variation. By [7, Proposition 7] and since U (E) is a lattice of sets, there exist two capacities γ1,γ2 : U (E) → [0, ∞) such that γ = γ1 − γ2. By (11), this implies that V = V1 − V2 where

Vi (x)= x, ξ dγi (ξ) ∀x ∈ X, i =1, 2. E

By the previous part of the proof, V1 and V2 are monotone and dual comonotonic additive. Consequently, V is dual comonotonic additive and of bounded variation. We are left to prove the uniqueness of γ and the characterization of monotonicity. We start by the uniqueness of γ. Consider two outer continuous set functions γ1,γ2 ∈ bv (U (E)) such that

V (x)= x, ξ dγi (ξ) ∀x ∈ X, i ∈{1, 2} . E

∈{ } ˆ → R ˆ ∈ For i 1, 2 deﬁne Vi : C by Vi (f)= E f (ξ) dγi (ξ) for all f C.By Theorem 22, it follows that γ1 (A)=γ2 (A) for all A ∈ ΣC ⊆U(E). On the other hand, if A ∈U(E), then there exist f ∈ C (E)andt ∈ R such that A =(f ≥ t). E ∈ { } ⊆ Since C is uniformly dense in C ( )andχE C, there exists a sequence fn n∈N C such that fn ≥ fn+1 ≥ f for all n ∈ N and fn − f→0. It follows that (fn ≥ t) ↓ (f ≥ t). Since γ1 and γ2 are outer continuous and they coincide on ΣC , we have that

γ1 (A)=γ1 (f ≥ t) = lim γ1 (fn ≥ t) = lim γ2 (fn ≥ t)=γ2 (f ≥ t)=γ2 (A) , n n proving uniqueness. Finally, we prove the characterization of monotonicity. Consider Vˆ = V ◦ T −1 : C → R and V¯ the Lipschitz continuous extension of Vˆ from C to C (E), as in (ii) implies (iii). Since T −1 is a positive operator, if V is further monotone, then Vˆ is also monotone and so is the extension V¯ . By Theorem 22, there exists a unique U E → ∞ ¯ outer continuous capacity γ : ( ) [0, ) such that V (f)= E f (ξ) dγ (ξ)for ∈ E ∈ all f C ( ). In particular, we have that V (x)= E x, ξ dγ (ξ) for all x X.On the other hand, if γ is a capacity that satisﬁes (11), then V is monotone, as proved in the initial part of (iii) implies (i).

3. Dual comonotonicity 3.1. Equivalence. In this section we study the dual comonotonicity relation contained in (8). Notice that two elements x and y in X are dually comonotonic if and only if the functionsx ˆ (·)andˆy (·)inC (E) are comonotonic in the usual sense. At the same time, to check that two elements are dually comonotonic, it suﬃces that the relation in (8) holds for all ξ,ξ ∈ Γ ⊆E where Γ is dense in E. When X is a space of functions we thus have two notions of comonotonicity: the traditional one and the dual one. We show that these notions coincide. This

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confirms that the new notion of dual comonotonicity we propose is a legitimate generalization of the standard notion of comonotonicity to Archimedean Riesz spaces with unit. We start by considering a σ-algebra F of subsets of a nonempty set Ω. The space B (Ω, F) is a Banach lattice with unit χΩ; its norm dual is lattice isometric to the set of bounded and finitely additive set functions ba (Ω, F). In this case, the dual −→ pairing is such that (f,μ) Ω fdμ. Δ is the set of finitely additive probabilities on F and E is the set of {0, 1}-valued probabilities. The next proposition shows the coincidence of comonotonicity and dual comonotonicity in this setting.

Proposition 7. The functions x, y ∈ B (Ω, F) are comonotonic if and only if they are dually comonotonic.

Proof. For each ω ∈ Ω deﬁne the Dirac measure δω on (Ω, F)by

1ifω ∈ A δ (A)= for all A ∈F. ω 0ifω ∈ A

The set Γ = {δω : ω ∈ Ω} is dense in E.Ifx and y are dually comonotonic, in particular, we have that

(x (ω) − x (ω)) (y (ω) − y (ω)) =( x, δω − x, δω )( y, δω − y, δω ) ≥ 0 ∀ω, ω ∈ Ω,

proving that x and y are comonotonic. Conversely, if x and y are comonotonic, then

0 ≤ (x (ω) − x (ω)) (y (ω) − y (ω)) =( x, δω − x, δω )( y, δω − y, δω ) ∀ω, ω ∈ Ω,

proving that the relation in (8) is true on the dense subset Γ of E. This implies that x and y are dually comonotonic.

In light of Proposition 7, given a comonotonic additive functional V : B (Ω, F) → R of bounded variation, we have two representations. The ﬁrst one is a speciﬁcation of Theorem 6 if X is assumed to be B (Ω, F). The second one is the direct representation according to Schmeidler’s result as generalized to the nonmonotone case by [19] and [18]. To understand their relationship, observe that if V : B (Ω, F) → R is comonotonic additive and of bounded variation, then there exists a unique set function η ∈ bv (F) such that (12) V (x)= x (ω) dη (ω) ∀x ∈ B (Ω, F) . Ω On the other hand, Theorem 6 implies that there exists a unique outer continuous set function γ ∈ bv (U (E)) such that (13) V (x)= xˆ (ξ) dγ (ξ) ∀x ∈ B (Ω, F) . E

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The next proposition clarifies the relationship among the objects E, C (E), η,and γ. By definition,x ˆ (ξ)= x, ξ for all ξ ∈E and x ∈ X. To this purpose, if A is an element of the σ-algebra F, define the following subset of E: Aˆ = {μ ∈E: μ (A)=1} . That is, A ˆ is the collection of {0, 1}-valued probabilities having the set A as carrier. Set Fˆ = Aˆ : A ∈F .

Proposition 8. For the representations (12) and (13), the following properties hold: (i) the space E can be identiﬁed with the compact, Hausdorﬀ, and totally disconnected Stone space of the Boolean algebra F; (ii) Fˆ is the σ-algebra of clopen sets of E which is a base of the topology in E; (iii) C (E)=B E, Fˆ and T (χ )=χ ; A Aˆ (iv) γ Aˆ = η (A) for all A ∈F.

Proof. Before starting we denote by τ1 the topology on E induced by the w*- topology and by τ2 the topology generated by Fˆ. (i) and (ii). Deﬁne the map h : F→Fˆ by A −→ Aˆ for all A ∈F.Itisroutine to check that it is a lattice isomorphism as in [21, Deﬁnition 1.4.4]. This implies

that Fˆ is an algebra. Next, consider Aˆn ⊆ Fˆ, it follows that there exists n∈N { } ⊆F ˆ { ∈E } ∈ N F An n∈N such that An = μ : μ (An)=1 for all n .Since is a ˆ ∈E ∈ Fˆ σ-algebra, it is immediate to see that n∈N An = μ : μ n∈N An =1 , proving that Fˆ is a σ-algebra. It is routine to check that each set in Fˆ is a τ1-clopen set. Since Fˆ is closed under intersection and complementation, Fˆ is a base for τ2 and each τ2-closed set is intersection of sets in Fˆ. It follows that τ2 ⊆ τ1.Consider the identity idE :(E,τ1) → (E,τ2). Since τ2 ⊆ τ1, idE is a continuous bijection. (E,τ1) is compact and Hausdorﬀ while (E,τ2) is Hausdorﬀ. In fact, consider μ1 and μ2 in E. By contradiction, assume that μ1 = μ2 and that for each C ∈ Fˆ

μ1 ∈ C =⇒ μ2 ∈ C.

Since μ1 and μ2 are {0, 1}-valued, this implies that μ1 (A)=μ2 (A) for all A ∈F, that is, μ1 = μ2, a contradiction. It follows that if μ1,μ2 ∈Eand μ1 = μ2, then there exists C ∈ Fˆ such that μ1 ∈ C and μ2 ∈ C.SinceFˆ is closed by c complementation, we have that μ1 ∈ C and μ2 ∈ C and τ2 is Hausdorﬀ. We can conclude that idE is a homeomorphism. Thus, τ1 and τ2 coincide, proving that (E,τ) is compact, Hausdorﬀ, totally disconnected, and Fˆ is a base for τ.Itis routine to check that any clopen set of (E,τ) belongs to Fˆ. ˆ ∈ Fˆ ∈ E (iii) Clearly, for each A we have that T (χA)= χAˆ C ( ). Since T is linear, this implies that B0 E, Fˆ ⊆ C (E) ⊆ B E, Fˆ .SinceC (E)isaBanach space, the statement follows. (iv) By (12) and (13), we have that for each x ∈ X, (14) x (ω) dη (ω)= xˆ (ξ) dγ (ξ) . Ω E

Setting x = χA in (14), we get the relation η (A)=γ Aˆ .

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The next result is interesting in itself and will be useful in the sequel.

Proposition 9. Let X and Y be two AM-spaces with units eX and eY and let J : X → Y be a lattice homomorphism such that J (eX )=eY . The following statements are true: (i) If x and y in X are dually comonotonic, then J (x) and J (y) in Y are dually comonotonic. (ii) If J is further injective, x and y are dually comonotonic if and only if J (x) and J (y) are.

Proof. Since X and Y are two AM-spaces, Kakutani’s maps TX : X → C (EX ) and TY : Y → C (EY ) are isomorphisms. Deﬁne the lattice homomorphism Jˆ : E → E ˆ ◦ ◦ −1 ˆ ˆ C ( X ) C ( Y )byJ = TY J TX . J is such that J (χEX )=χEY . It follows that Jˆ ◦ TX = TY ◦ J. From a general result on homomorphisms between spaces of continuous functions (see for instance [1, Theorem 14.23]), it follows that there exists a continuous map h : EY →EX such that

Jˆ(f)=f ◦ h ∀f ∈ C (EX ) . (i) By contradiction, assume x and y in X are dually comonotonic and that J (x) and J (y) are not. It follows that there exist ξ1 and ξ2 in EY such that 0 > [T (J (x)) (ξ ) − T (J (x)) (ξ )] [T (J (y)) (ξ ) − T (J (y)) (ξ )] Y 1 Y 2 Y 1 Y 2

= Jˆ(TX (x)) (ξ1) − Jˆ(TX (x)) (ξ2) Jˆ(TX (y)) (ξ1) − Jˆ(TX (y)) (ξ2)

= Jˆ(ˆx)(ξ1) − Jˆ(ˆx)(ξ2) Jˆ(ˆy)(ξ1) − Jˆ(ˆy)(ξ2)

=[ˆx (h (ξ1)) − xˆ (h (ξ2))] [ˆy (h (ξ1)) − yˆ(h (ξ2))] , a contradiction with the hypothesis that x and y are dually comonotonic. (ii) If J is further injective, then the map h : EY →EX is onto. Necessity follows from point (i). By contradiction, suppose that J (x)andJ (y) are dually comonotonic but x and y are not. It follows that there exist ξ1 and ξ2 in EX − − such that [ˆx (ξ1) xˆ (ξ2)] [ˆy (ξ1) yˆ(ξ2)] < 0. Since h is onto, there exist ξ1 and E ˆ ξ2 in Y such that ξ1 = h (ξ1)andξ2 = h (ξ2). Since J (ˆx)=TY (J (x)) and Jˆ(ˆy)=TY (J (y)), this implies that 0 > [ˆx (h (ξ )) − xˆ (h (ξ ))] [ˆy (h (ξ )) − yˆ(h (ξ ))] 1 2 1 2 ˆ − ˆ ˆ − ˆ = J (ˆx)(ξ1) J (ˆx)(ξ2) J (ˆy)(ξ1) J (ˆy)(ξ2) − − =[TY (J (x)) (ξ1) TY (J (x)) (ξ2)] [TY (J (y)) (ξ1) TY (J (y)) (ξ2)] , a contradiction with J (x)andJ (y) being dually comonotonic.

3.2. The AM-spaces Cb (S) and c. We give some immediate applications of Proposition 9. Let Cb (S) be the space of all bounded and continuous functions on a topological space S. Cb (S), endowed with the supnorm and the pointwise order, is a Banach lattice with unit χS.

Proposition 10. Two functions f and g in the AM-space Cb (S) are dually comonotonic if and only if they are comonotonic.

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Proof. Let BS be the Borel σ-algebra of S and consider the space B (S, BS). Every continuous function is BS -measurable. Thus, the inclusion map J : Cb (S) → B (S, BS) is an injective lattice homomorphism. By Propositions 7 and 9, the statement follows. Consider the subspace c of l∞ where the latter is the space of all bounded sequences and the former is the space of all convergent sequences

∞ c = x ∈ l : x∞ = lim xi exists in R . i→∞ Notice that l∞ = B (N, P (N)) where P (N)isthepowersetofN. We endow c with | | the supnorm x =supi∈N xi . Clearly, c is an AM-space with unit and its norm dual can be identiﬁed with the AL-space l1 ⊕R via the lattice isometry ξ −→ (y, λ) with y ∈ l1 and λ ∈ R such that (see also [1, Theorem 16.14]) ∞ x, ξ = λx∞ + yixi ∀x ∈ c. i=1 E { } ∪{ } ∈ N We have that = ei i∈N e∞ where ei,x = xi for all i and e∞,x = x∞. The elements x and y in c are functions deﬁned over N; thus, comonotonicity means

(15) [xi − xj ][yi − yj ] ≥ 0 ∀i, j ∈ N. By Propositions 7 and 9 and since the inclusion map J : c → l∞ is a lattice homomorphism, we have that x and y in c are dually comonotonic if and only if they are comonotonic. 3.3. The AM-space L∞ (Ω, F, N ). Let (Ω, F) be a measurable space where F is a σ-algebra. Let N⊆Fbe a proper σ-ideal.9 Deﬁne by L∞ (Ω, F, N )the space of all real valued F-measurable functions that are essentially bounded, that is, for each f ∈ L∞ (Ω, F, N ) there exists a scalar k such that (|f| >k) ∈N.The ∞ function ·∞ : L (Ω, F, N ) → [0, ∞) deﬁned by ∞ f∞ =inf{k ≥ 0:(|f| >k) ∈N} ∀f ∈ L (Ω, F, N ) is a seminorm. By introducing the equivalence relation

f ∼ g if and only if f − g∞ =0ifandonlyif(|f − g| > 0) ∈N the quotient space L∞ (Ω, F, N )=L∞ (Ω, F, N ) / ∼ is an AM-space with unit.10 By f we will denote an equivalence class in L∞ (Ω, F, N )andbyf arepre- sentative element of the class f. The norm dual of L∞ (Ω, F, N ) is lattice iso- morphic to ba (Ω, F, N ) and the extreme points E of Δ are the set of all ﬁnitely additive probabilities on (Ω, F)thatare{0, 1}-valued and take value zero on sets belonging to N .Ifweconsideraσ-additive probability measure P : F→[0, 1] and we deﬁne N = {E ∈F: P (E)=0},thenL∞ (Ω, F, N ) becomes the standard space L∞ (Ω, F,P). On the other hand, when N = {∅} we have that B (Ω, F)=L∞ (Ω, F, N )=L∞ (Ω, F, N ). ∈L∞ F N ˆ ∈ E If f (Ω, , ), then we denote by f C ( ) the image of f under Kaku- ˆ E→R ˆ ∈E11 tani’s map T .Thatis,f : is such that f (μ)= Ω fdμ for all μ . ∈ ∞ F N ∼ Given two functions f,g L (Ω, , ), f g if and only if Ω fdμ = Ω gdμ 9 ∈N F ⊆ ∈N ∈N ∞ ∈N { } That is, Ω / ; A B implies A ; n=1 An if An n∈N is a sequence in N . 10 ∞ For a thorough analysis of the space L (Ω, F, N ) and its norm dual see [21, pp. 137-140]. 11 ∈ ∈ F N Notice that if f1,f2 f,then Ω f1dμ = Ω f2dμ for all μ ba (Ω, , ).

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for all μ ∈Eif and only if f = g. Therefore, we can also deﬁne L∞ (Ω, F, N )= L∞ (Ω, F, N ) / ker T .

Lemma 11. Let ϕ : R → R be continuous. If f ∈L∞ (Ω, F,N ), then for each f ∈ f (16) (ϕ ◦ f) dμ = ϕ fdμ ∀μ ∈E. Ω Ω Proof. If f ∈L∞ (Ω, F, N ), then there exists a function f¯ ∈ B (Ω, F) such that f¯ ∼ f.Sinceϕ is continuous, ϕ ◦ f¯ ∼ ϕ ◦ f.Iff¯ is a simple function, that is, f¯ ∈ B0 (Ω, F), then it is immediate to see that (16) holds for f¯. On the other hand, ¯ ∈ F { } ⊆ F if f B (Ω, ), then there exists a sequence fn n∈N B0 (Ω, )thatconverges in supnorm to f¯. It follows that ¯ ϕ fdμ = ϕ fdμ = ϕ lim fndμ = lim ϕ fndμ Ω Ω n Ω n Ω = lim (ϕ ◦ fn) dμ = ϕ ◦ f¯ dμ = (ϕ ◦ f) dμ ∀μ ∈E, n Ω Ω Ω proving the statement.

Proposition 12. Let f, g ∈L∞ (Ω, F, N ). The following statements are equivalent: (i) f and g are dually comonotonic; (ii) there exist f ∈ f, g ∈ g, and N ∈N such that (17) [f (ω) − f (ω)] [g (ω) − g (ω)] ≥ 0 ∀ω, ω ∈ N c; (iii) for each f ∈ f and g ∈ g there exists N ∈N such that [f (ω) − f (ω)] [g (ω) − g (ω)] ≥ 0 ∀ω, ω ∈ N c.

Proof. The canonical mapping J : B (Ω, F) →L∞ (Ω, F, N ), such that f −→ f,is a lattice homomorphism. (i) implies (ii). Let f and g be dually comonotonic. This implies that fˆ andg ˆ are comonotonic. By [11, Proposition 4.5], there exist two continuous and monotone ˆ ˆ ˆ functions ϕ1,ϕ2 : R → R such that f = ϕ1 f +ˆg andg ˆ = ϕ2 f +ˆg .Set

h = f + g. By the previous lemma, we have that f ∼ ϕ1 ◦ h and g ∼ ϕ2 ◦ h. On the other hand, by [11, Proposition 4.5], the functions ϕ1 ◦ h and ϕ2 ◦ h are comonotonic. This implies that ϕ1 ◦ h and ϕ2 ◦ h satisfy (17) for N = ∅. (ii) implies (iii). If f ∈ f and g ∈ g, then there exists M ∈Nsuch that ∈ ∪ c c ∩ c f|M c = f|M c and g|M c = g|M c . It follows that for each ω, ω (N M) = N M , [f (ω) − f (ω)] [g (ω) − g (ω)] = [f (ω) − f (ω)] [g (ω) − g (ω)] ≥ 0 and N ∪ M ∈N. (iii) implies (i). Let f ∈ f, g ∈ g, and N ∈Nbe as in (iii). There exist ¯ k ∈ R and two functions f andg ¯ in B+ (Ω, F) that are further comonotonic in the ¯ standard sense and such that f ∼ f + kχΩ andg ¯ ∼ g + kχΩ. By Propositions 7 and 9, J f¯ = f + k and J (¯g)=g + k are dually comonotonic and so are f and g.

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4. Superadditivity When X = B (Ω, F), the relation between superadditivity and supermodularity is well known and the results are simple and nice (see, e.g., [18, Corollary 4.2]). We shall extend this result to our general setting. Moreover, superadditivity will be a key property in allowing us to provide a representation where the set function γ is defined and unique not only over the class U (E),butontheentireBaireσ-algebra, Baire(E), providing a complete generalization of Theorem 5. Proposition 13. Let X be an Archimedean Riesz space with unit and V afunc- tional from X to R. The following statements are equivalent: (i) V is monotone, dual comonotonic additive, and supermodular; (ii) V is monotone, dual comonotonic additive, and superadditive; (iii) there exists a unique supermodular and outer continuous capacity, γ : U (E) → [0, ∞), such that V (x)= x, ξ dγ (ξ) ∀x ∈ X. E Proof. Consider Vˆ : C → R and V¯ : C (E) → R as in the proof of Theorem 6. Recall that Vˆ = V ◦ T −1 where T is Kakutani’s map and V¯ is the Vˆ Lipschitz continuous extension to C (E). (i) implies (iii) (resp., (ii) implies (iii)). Since V is monotone, dual comonotonic additive, and supermodular (resp., superadditive), and T is a lattice isomorphism, we have that Vˆ is monotone, comonotonic additive, and supermodular (resp., superadditive). By Lipschitz continuity, the extension V¯ inherits the same properties. By Dini’s Theorem, V¯ is outer continuous. By Theorem 22, it follows that there exists an outer continuous supermodular capacity, γ : U (E) → [0, ∞), such that ¯ ∈ E V (f)= E f (ξ) dγ (ξ) for all f C ( ). It follows that V (x)= E x, ξ dγ (ξ)for all x ∈ X. Uniqueness follows from Theorem 6. (iii) implies (i) (resp., (iii) implies (ii)). Consider a supermodular and outer continuous capacity γ : U (E) → [0, ∞). Define γ∗ : Baire(E) → [0, ∞)by γ∗ (A)=sup{γ (B):U (E) B ⊆ A}∀A ∈ Baire(E) . ∗ It is immediate to see that γ is a supermodular capacity which further coincides U E E E → R −→ ∗ to γ on ( ). Next, define W : B ( ,Baire( )) by f E f (ξ) dγ (ξ). By [18, Corollary 4.2] and since γ∗ is supermodular, W is monotone, comonotonic additive, supermodular, and superadditive. Since V = W ◦ T and T is a lattice isomorphism, it follows that V is monotone, dual comonotonic additive, supermodular, and superadditive. In the next result we assume that V satisfies an extra property of continuity but is not necessarily monotone. Its proof is heavily dependent on the Daniell-Stone extension theorem given by [7]. Theorem 14. Let X be an Archimedean Riesz space with unit and V a functional from X to R. The following statements are equivalent: (i) V is dual comonotonic additive, superadditive, sequentially weakly continuous, and of bounded variation; (ii) V is dual comonotonic additive, superadditive, sequentially weakly continuous at x =0, and of bounded variation;

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(iii) there exists a unique supermodular and continuous γ ∈ bv (Baire(E)) such that V (x)= x, ξ dγ (ξ) ∀x ∈ X. E Moreover, V is monotone if and only if γ is a capacity. Proof. Again consider Vˆ : C → R as in the proof of Theorem 6. Recall that Vˆ = V ◦ T −1 where T is Kakutani’s map. (i) implies (ii). It is obvious. (ii) implies (iii). Since V is dual comonotonic additive, superadditive, and of bounded variation (resp., monotone), we have that Vˆ is comonotonic additive, superadditive, and of bounded variation (resp., monotone). Next, consider a uni- { } ⊆ −→ formly bounded sequence fn n∈N C such that fn 0 pointwise. It follows { } ⊆ −1 ∈ N that the sequence xn n∈N X, deﬁned by xn = T (fn) for all n ,is bounded and it weakly converges to 0. Since V is sequentially weakly continuous at x =0,wehavethatVˆ (0) = V (0) = limn V (xn) = limn Vˆ (ˆxn), proving that Vˆ is bounded pointwise continuous at 0. By [7, Theorem 22], it follows that there exists a continuous and supermodular γ ∈ bv (σ (Σ )) (resp., capacity) such that C Vˆ (f)= f (ξ) dγ (ξ) ∀f ∈ C. E

Since C is uniformly dense in C (E), the σ-algebra generated by ΣC coincides to Baire(E), that is, γ ∈ bv (Baire(E)). Finally, since V = Vˆ ◦ T and given the uniqueness of γ in [7, Theorem 22], the statement follows. (iii) implies (i). By Theorem 6, it follows that V is dual comonotonic additive and of bounded variation (resp., monotone if γ is a capacity). Deﬁne W : E E → R ∈ E E B ( ,Baire( )) by W (f)= E f (ξ) dγ (ξ) for all f B ( ,Baire( )). No- tice that V = W ◦ T . By [18, Corollary 4.2] and since γ is supermodular, we have that W is superadditive. Since T is a positive operator, V is superadditive. { } ⊆ Finally, consider a sequence xn n∈N X such that xn x. It follows that the { } ⊆ E E sequence , xˆn n∈N B ( ,Baire( )) is bounded and it converges pointwise tox ˆ. By [7, Theorem 22] and since γ is continuous, it follows that

V (x)=W (ˆx) = lim W (ˆxn) = lim V (xn) , n n proving that V is sequentially weakly continuous. Remark 15. We could have provided a version of Theorem 14 where superadditivity was replaced by subadditivity and supermodularity by submodularity. If V is both monotone and additive, then it is immediate to see that V ∈ X∗.Thus,V is sequentially weakly continuous. By the initial observation, it further follows that Theorem 14 guarantees the existence of a unique continuous and modular capacity γ on Baire(E). This delivers that γ is a σ-additive measure and that Theorem 5 is a corollary of Theorem 14. Since Theorem 14 is a reﬁnement of Theorem 6, this shows that Theorem 5 can be seen as a corollary of Theorem 6, as mentioned in the Introduction.

5. Two applications to mathematical finance 5.1. Put-call parity and the Fundamental Theorem of Finance. Starting with Kreps [16], ordered vector spaces have played an important role in modelling

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financial markets. The partial order ≥ on the vector space allowed Kreps to capture the essential behavioral assumption of the modern theory of arbitrage, that is, that agents prefer more to less, and the resulting property of monotonicity of the price functional V : X → R. Since important classes of derivative assets can be stated in terms of lattice operations, Brown and Ross [5] observed that the Riesz space structure is actually needed “... to express the basic derivative assets, i.e., calls and puts, as nonlinear functions of the stock, bond, and strike price. That is, if a stock, s, and a bond, b, are elements of a Riesz space X, then a call option on s with strike price λ is given by (s − λb)+ and a put on s with the same strike price is given by (λb − s)+...”. Thus, in this abstract setting, the basic financial identity Call − Put = Stock − Strike Price × Bond continuestoholdintheform: − (18) (s − λb)+ − (λb − s)+ =(s − λb)+ − (s − λb) = s − λb. The bond b is interpreted as the risk free bond. It is immediate to see that if V is linear then the put-call parity monetary property also holds (19) V (s − λb)+ + V − (λb − s)+ = V (s − λb)=V (s) − λV (b) . The property of linearity is interpreted as capturing the absence of frictions. On the other hand, the put-call parity monetary property only implies that the two financial strategies in (18) must have the same cost. As discussed in Cerreia- Vioglio, Maccheroni, and Marinacci [6], the put-call parity monetary property alone is clearly weaker than the assumption of absence of frictions. Thus, it is easier to verify and also to test empirically. From a mathematical point of view, when it is assumed that the bond is a unit for X,asin[5],12 it can be shown that the put-call parity monetary property implies unit-additivity and unit-modularity. Proposition 16. Let the financial market X be an Archimedean Riesz space with unit b.IfV is a price functional from X to R that satisfies the put-call parity monetary property for all s ∈ X and λ ≥ 0,thenV is unit-additive and unit- modular. Proof. The relation (19) for s =0andλ ≥ 0 delivers V (0) + V (−λb)=V (−λb)= V (0) − λV (b), that is, V (0) = 0 and V (−λb)=−λV (b) for all λ ≥ 0. For all x ∈ X and λ ≥ 0, the second part of (19) also implies V ((x + λb) − λb)= V (x + λb) − λV (b), that is, V (x + λb)=V (x)+λV (b). Since V (0) = 0, the choice of x = 0 implies V (λb)=λV (b). Summing up, V (x + λb)=V (x)+λV (b)=V (x)+V (λb) ∀ (x, λ) ∈ X × R. By (19), if λ =0,thenV (s ∨ 0) + V (s ∧ 0) = V (s+)+V (−s−)=V (s) for all s ∈ X.Thus,wehavethat V (x ∨ λb)+V (x ∧ λb)=V ([(x − λb) ∨ 0] + λb)+V ([(x − λb) ∧ 0] + λb) = V ([(x − λb) ∨ 0]) + V ([(x − λb) ∧ 0]) + 2λV (b) = V (x − λb)+2λV (b)=V (x)+λV (b)=V (x)+V (λb) for all x ∈ X and for all λ ∈ R, proving the statement.

12Brown and Ross [5] make the equivalent assumption that s belongs to the principal ideal generated by b.

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A direct application of our main Theorem 6 provides a novel and general version of the Fundamental Theorem of Finance. In a nutshell, the classic version of this result assumed the financial market X to be a space of functions and stated that in the market there are no frictions and no arbitrages, that is, the price functional V is linear and monotone, if and only if V can be rewritten as a discounted expected value.13 The following generalization instead shows that, even if frictions are considered, the put-call parity monetary property and monotonicity of V are equivalent to V being a Choquet discounted expected value. Theorem 17. Let the financial market X be an Archimedean Riesz space with unit b and V a nonzero price functional from X to R. The following statements are equivalent: (i) V is monotone and satisfies the put-call parity monetary property for all s ∈ X and λ ≥ 0; (ii) there exists a nonadditive probability measure π : U (E) → R andadiscount factor v>0 such that V (x)=v x, ξ dπ (ξ) ∀x ∈ X. E In this case, π is unique and so is v. The number v = V (b) is called discount factor since it is the price of the risk free bond, while the space E is interpreted as the implicit space of the states of the world, and π as a nonadditive risk neutral probability; see again [5] and [6]. The latter paper contains a complete discussion of different nonadditive versions of the Fundamental Theorem of Finance. The result presented above is new, but assumes directly that the financial market X is an Archimedean Riesz space with unit. Conversely, [6] starts from mark-to-market evaluations of securities and derives a Riesz space structure: with a little loss of generality which is compensated by a sounder economic foundation and a more direct financial interpretation.

5.2. Comonotone risk measures. The seminal paper of Artzner, Delbaen, Eber, and Heath [3] started the literature of capital requirements in mathematical ﬁnance.14 The object of study of this literature is a functional V on a space of functions X.Eachelementx of X is interpreted as representing a future contingent loss, while V (x) is interpreted as the capital requirement needed to make the risk x acceptable. Artzner, Delbaen, Eber, and Heath [3] developed their theory in Rn, while Delbaen [9] extended it to L∞ (Ω, F,P). In our paper, a risk measure is just a monotone functional V from the space L∞ (Ω, F,P)ofriskstoR that satisﬁes the monetary property

V (x − λ1Ω)=V (x) − λV (1Ω) for all x ∈L∞ (Ω, F,P)andforallλ ≥ 0. One of the most important and used risk measure is Value at Risk, also known as VaR. It is well known that VaR is comonotonic additive.15 This fact alone suﬃces to justify the study of comonotonic

13For an introduction to the topic see Dybvig and Ross [12], Ross [22], Föllmer and Schied [13], and Delbaen and Schachermayer [10]. 14For an introduction to the topic see Föllmer and Schied [13]. 15See Proposition 12 for a definition of comonotonicity in L∞ (Ω, F,P). See [13, Remark 4.85], for a proof that VaR is comonotonic additive.

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additive risk measures. Our main Theorem 6, together with a simple modiﬁcation of Proposition 16, delivers the following characterization. Proposition 18. AriskmeasureV : L∞ (Ω, F,P) → R is comonotonic additive if and only if (20) V (x)=V (x ∨ 0) + V (0 ∧ x) ∀x ∈ X. Given the relevance of risk measures in banking and insurance regulation, this simple result allows us to better understand the normative content of comonotonic additivity. For, clearly x ∨ 0 represents the loss corresponding to position x while 0 ∧ x represents the gain corresponding to such a position, since negative losses are gains. Therefore, the gain-loss decomposition (20) of risks has a clear interpretation for risk measurement, unlike comonotonic additivity.

6. Appendix In this appendix we provide a nonadditive integral representation theorem (The- orem 22) for functionals defined over a Stone vector lattice L.16 This result is ancillary in proving Theorem 6. Theorem 22 is closely related to Sipoˇˇ s’s approach [24] (see also Greco [14]). We denote by F a σ-algebra of sets of Ω such that L ⊆ B (Ω, F). In the sequel, with a small abuse of notation, given k ∈ R,we will denote by k both the real number and the constant function on Ω that takes value k.Moreover,wesetL+ = {f ∈ L : f ≥ 0}. In view of Definition 4, given a functional V : L → R and two points f,g ∈ L such that f ≤ g, we define n VarV (f,g)=sup |V (fi) − V (fi−1)|∈[0, ∞] , i=1

where the supremum is taken over all finite chains f = f0 ≤ f1 ≤···≤fn = g contained into L. The notions of monotonicity, unit-additivity, unit-modularity, bounded variation are the ones of Definition 4 applied to this setting where the unit is the function χΩ.Moreover,wesaythatV is outer continuous if limn V (fn)= V (f) whenever fn ↓ f and the convergence is pointwise. It is immediate to check that if V is the difference of monotone functionals, then V is of bounded variation. Unit-modularity is very close to the following property introduced by Sipoˇˇ s [24] (see also [14]): (21) V (f)=V (f ∧ β)+V (f ∨ β − β) ∀f ∈ L, ∀β ∈ R. We next report some simple relations among the above properties. Their proof follows from standard arguments. Proposition 19. Let V be a functional from L to R. The following statements are true: (i) If V is unit-additive, then V (0) = 0. (ii) If V is unit-additive, then V (α)=αV (1) for all α ∈ Q. (iii) If V is monotone and unit-additive, then V (α)=αV (1) for all α ∈ R and V is Lipschitz continuous.

16The main statement in Theorem 22 could be proven when L is a subset of B (Ω, P (Ω)) such that if f ∈ L,thenαf, f ∧ β, f ∨ β, f + β ∈ L for all α ∈ R+ and β ∈ R.

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(iv) If V is unit-additive, then V (f + α)=V (f)+V (α) holds for all f ∈ L and all α ∈ R. (v) Given V unit-additive, V is unit-modular if and only if it satisfies (21). The next proposition delivers a nonadditive integral representation theorem when the functional V is further monotone and it is fundamentally due to Greco [14]. It is ancillary to prove Theorem 22. The proof of Proposition 20 consists of checking that all our assumptions on V allow us to apply Greco’s result. We omit it. Proposition 20. Let V be a functional from L to R. The following statements are equivalent: (i) V is monotone, unit-additive, and unit-modular; (ii) there exists a capacity γ :ΣL → [0, ∞) such that (22) ∞ 0 V (f)= f (ω) dγ (ω)= γ (f ≥ t) dt + [γ (f ≥ t) − γ (Ω)] dt ∀f ∈ L. Ω 0 −∞ The next lemma is crucial to extend Proposition 20 to nonmonotone functionals. Lemma 21. Let V be a functional from L to R. The following statements are equivalent: (i) V : L → R is unit-additive, unit-modular, and of bounded variation; (ii) V is the difference, that is, V = V1 − V2, of two monotone, unit-additive, and unit-modular functionals V1,V2 : L → R.Inparticular,V is Lipschitz continuous. Proof. (ii) implies (i). The proof of this implication is trivial and the details are omitted. (i) implies (ii). We will proceed by Steps. We start by defining V1 : L+ → R by

V1 (f)=VarV (0,f) ∀f ∈ L+.

Step 1. V1 is monotone and unit-additive on L+. Proof of Step 1. The proof follows from the arguments contained in the proof of [7, Lemma 14].

Step 2. V1 is unit-modular on L+.

Proof of Step 2. Fix f ∈ L+ and α ∈ R+. It is immediate to see that 0 ≤ α ≤ f ∨α, 0 ≤ f ∧ α ≤ f,and

(23) VarV (0,f ∨ α) ≥ VarV (0,α)+VarV (α, f ∨ α) and

(24) VarV (0,f) ≥ VarV (0,f ∧ α)+VarV (f ∧ α, f) . Next, we show that the converse inequalities also hold implying that the above relations are indeed equalities. Let ξ

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Since V is unit-modular, we have that n n ξ< |V (fi) − V (fi−1)| = |[V (fi)+V (α)] − [V (fi−1)+V (α)]| i=1 i=1 n = |[V (fi ∨ α) − V (fi−1 ∨ α)] + [V (fi ∧ α) − V (fi−1 ∧ α)]| i=1 n n ≤ |V (fi ∨ α) − V (fi−1 ∨ α)| + |V (fi ∧ α) − V (fi−1 ∧ α)| i=1 i=1

≤ VarV (α, f ∨ α)+VarV (0,α) { ∨ }n ∨ where the last step follows from the fact that fi α i=0 is a chain from α to f α, { ∧ }n while fi α i=0 is a chain from 0 to α. This implies that

VarV (0,f ∨ α) ≤ VarV (0,α)+VarV (α, f ∨ α) . By (23), it follows that

(25) VarV (0,f ∨ α)=VarV (0,α)+VarV (α, f ∨ α) .

Next, we show that VarV (α, f ∨ α)=VarV (f ∧ α, f). First, consider a chain { }n ∧ ≤ ≤···≤ fi i=0 such that f α = f0 f1 fn = f.SinceV is unit-modular, it follows that n n |V (fi)−V (fi−1)|= |V (fi ∨ α)−V (fi−1 ∨ α)+V (fi ∧ α)−V (fi−1 ∧ α)| . i=1 i=1 ∧ ∧ ∈{ } ∨ n At the same time, we have that fi α = f α for all i 0, ..., n and (fi α)i=0 is a chain from α to f ∨ α. This implies that n n |V (fi) − V (fi−1)| = |V (fi ∨ α) − V (fi−1 ∨ α)|≤VarV (α, f ∨ α) , i=1 i=1

proving that VarV (f ∧ α, f) ≤ VarV (α, f ∨ α). The converse inequality follows from a similar argument. By the same methods, we obtain that

(26) VarV (0,f)=VarV (0,f ∧ α)+VarV (f ∧ α, f) . Finally, given (25) and (26), we have that

V1 (f ∨ α)+V1 (f ∧ α)

= VarV (0,f ∨ α)+VarV (0,f ∧ α)

= VarV (0,α)+VarV (α, f ∨ α)+VarV (0,f) − VarV (f ∧ α, f)

= V1 (α)+V1 (f)

and V1 is unit-modular over L+.

Step 3. The functional V2 : L+ → R, deﬁned by V2 = V1 − V , is monotone, unit-additive, and unit-modular. Moreover, V (f)=V1 (f) − V2 (f) for all f ∈ L+.

Proof of Step 3. Consider V2 = V1 − V . It is immediate to see that V = V1 − V2. If 0 ≤ f ≤ g,thenwehavethat

V (g) − V (f) ≤|V (g) − V (f)|≤VarV (f,g) ≤ V1 (g) − V1 (f) ,

proving that V2 is monotone. Consequently, by Step 1 and Step 2 and since V2 = V1 − V , the functional V2 is also unit-additive and unit-modular over L+.

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Step 4. V is the diﬀerence of two monotone, unit-additive, and unit-modular functionals V1,V2 : L → R. In particular, V is Lipschitz continuous.

Proof of Step 4. Given a monotone and unit-additive functional on L+,itisarou- tine argument to show that such a functional admits a unique monotone and unit- additive extension to L. Moreover, if such a functional is unit-modular, then the extension is also unit-modular. In light of this fact, we consider V1,V2 : L+ → R, as defined in the previous part of the proof, and, without loss of generality, we denote their unique monotone, unit-additive, and unit-modular extensions by the same symbols. It is routine to check that V = V1 − V2 over the entire space L and that V is Lipschitz continuous in light of Proposition 19. Step 4 concludes the proof. We can state the main result of the appendix: Theorem 22. Let V : L → R be a functional over a Stone vector lattice L.The following statements are equivalent: (i) V is comonotonic additive and of bounded variation; (ii) V is unit-additive, unit-modular, and of bounded variation; (iii) there exists γ ∈ bv (Σ ) such that L (27) V (f)= f (ω) dγ (ω) ∀f ∈ L. Ω Moreover, V is outer continuous if and only if γ can be chosen to be outer continuous. Given γ outer continuous, then (1) γ is unique. (2) V is monotone if and only if γ is a capacity. (3) V is supermodular if and only if γ is supermodular. (4) V is superadditive only if γ is supermodular. Proof. (i) implies (ii). Consider f ∈ L, α ∈ R,andβ ∈ R. It is immediate to check that f and α are comonotonic and also f ∧ β and f ∨ β are comonotonic (see also [18, Lemma 4.6]). Since V is comonotonic additive, this implies that V (f + α)=V (f)+V (α)andV (f ∧ β)+V (f ∨ β)=V (f + β)=V (f)+V (β) , proving that V is unit-additive and unit-modular. (ii) implies (iii). By Lemma 21, there exist V1,V2 : L → R where V1 and V2 are monotone, unit-additive, and unit-modular and such that V = V1 − V2.By Proposition 20, there exist γ1,γ2 :ΣL → [0, ∞) that represent, respectively, V1 and V2 as in (22). The set function γ = γ1 − γ2 represents V as in (27). (iii) implies (i). By [7, Proposition 7], γ is the difference of two capacities γ1 and ∗ F→ ∞ γ2 on ΣL. Define γi : [0, )by ∗ { ⊆ }∀∈F γi (A)=sup γi (B):ΣL B A A . ∗ ∈{ } ∗ Since γi is a capacity, γi is a capacity for i 1, 2 and γi (A)=γi (A) for all ∈ ∗ ∗ − ∗ F A ΣL. It follows that γ = γ1 γ2 belongs to bv ( ) and extends γ. Define F → R ∗ W : B (Ω, ) by W (f)= Ω f (ω) dγ (ω). By [19] or [18], it follows that V (f)=W (f) for all f ∈ L and the latter functional is comonotonic additive and of bounded variation, proving that also V is. Moreover, by [7, Theorem 13], V is outer continuous if and only if γ can be chosen to be outer continuous. Similarly, provided γ is outer continuous, points 1, 2, and 3

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follow from the same result. We only need to prove 4. Assume V is superadditive ∈ { } { } ⊆ and consider A, B ΣL. By [7, Lemma 16], there exists fn n∈N , gn n∈N L+ ↓ ↓ { } ⊆ such that fn 1A and gn 1B pointwise. It follows that fn + gn n∈N L+ and fn + gn ↓ 1A +1B. Moreover, we have that limn V (fn)=γ (A) and limn V (gn)= γ (B). Deﬁne k = f1 + g1 +2.Observethat(fn + gn ≥ t) ↓ (1A +1B ≥ t)for all t ∈ [0,k]. Since ΣL is a lattice, for each t ∈ [0,k],

(1A +1B ≥ t) ∈{∅,A∩ B,A ∪ B,Ω}⊆ΣL.

Deﬁne hn :[0,k] → R by

hn (t)=γ (fn + gn ≥ t) ∀t ∈ [0,k] , ∀n ∈ N. Since γ is outer continuous and of bounded variation, we have that for each t ∈ [0,k]

|hn (t)| = |γ (fn + gn ≥ t)| < ∞ and hn (t) → γ (1A +1B ≥ t) . By (27) and Arzel`a’s Theorem (see [17]), it follows that k k lim V (fn + gn) = lim hn (t) dt = lim hn (t) dt n n n 0 0 k = γ (1A +1B ≥ t) dt = γ (A ∩ B)+γ (A ∪ B) . 0

Since V is superadditive, it follows that V (fn + gn) ≥ V (fn)+V (gn) for all n ∈ N. Passing to the limit, we obtain that

γ (A ∪ B)+γ (A ∩ B) = lim V (fn + gn) ≥ lim V (fn) + lim V (gn)=γ (A)+γ (B) . n n n

Acknowledgement The authors thank Daniele Pennesi and an anonymous referee for helpful sug- gestions.

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Dipartimento di Scienze delle Decisioni and IGIER, Universita` Bocconi, via Sarfatti, 25 Milan, Italy

Collegio Carlo Alberto, Universita` di Torino, via Real Collegio 30, 10024 Mon- calieri Torino, Italy

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