Choquet operators associated to vector capacities
aa r X i v : . [ m a t h . F A ] S e p CHOQUET OPERATORS ASSOCIATED TO VECTORCAPACITIES
SORIN G. GAL AND CONSTANTIN P. NICULESCU
Abstract.
The integral representation of Choquet operators defined on aspace C ( X ) is established by using the Choquet-Bochner integral of a real-valued function with respect to a vector capacity. Introduction
Choquet’s theory of integrability (as described by Denneberg [10] and Wang andKlir [30]) leads to a new class of nonlinear operators called in [15]
Choquet opera-tors because they are defined by a mix of conditions representative for Choquet’sintegral. Its technical definition is detailed as follows.Given a Hausdorff topological space X, we will denote by F ( X ) the vector latticeof all real-valued functions defined on X endowed with the pointwise ordering. Twoimportant vector sublattices of it are C ( X ) = { f ∈ F ( X ) : f continuous } and C b ( X ) = { f ∈ F ( X ) : f continuous and bounded } . With respect to the sup norm, C b ( X ) becomes a Banach lattice. See the nextsection for details concerning the ordered Banch spaces.As is well known, all norms on the N -dimensional real vector space R N areequivalent. See Bhatia [3], Theorem 13, p. 16. When endowed with the sup normand the coordinate wise ordering, R N can be identified (algebraically, isometricallyand in order) with the space C ( { , ..., N } ), where { , ..., N } carries the discretetopology.Suppose that X and Y are two Hausdorff topological spaces and E and F arerespectively ordered vector subspaces of F ( X ) and F ( Y ) . An operator T : E → F is said to be a Choquet operator (respectively a
Choquet functional when F = R )if it satisfies the following three conditions:(Ch1) ( Sublinearity ) T is subadditive and positively homogeneous, that is, T ( f + g ) ≤ T ( f ) + T ( g ) and T ( af ) = aT ( f )for all f, g in E and a ≥ Comonotonic additivity ) T ( f + g ) = T ( f ) + T ( g ) whenever the functions f, g ∈ E are comonotone in the sense that( f ( s ) − f ( t )) · ( g ( s ) − g ( t )) ≥ s, t ∈ X ; Date : May 28, 2020.2000
Mathematics Subject Classification.
Key words and phrases.
Choquet integral, Choquet-Bochner integral, comonotonic additivity,monotonic operator, vector capacity. (Ch3) (
Monotonicity ) f ≤ g in E implies T ( f ) ≤ T ( g ) . The linear Choquet operators acting on ordered Banach spaces are nothing butthe linear and positive operators acting on these spaces; see Corollary 1. Whilethey are omnipresent in the various fields of mathematics, the nonlinear Choquetoperators are less visible, their study beginning with the seminal papers of Schmei-dler [28], [29] in the 80’s. An important step ahead was done by the contributions ofZhou [31], Marinacci and Montrucchio [17] and Cerreia-Vioglio, Maccheroni, Mari-nacci and Montrucchio [4], [5], which led to the study of vector-valued Choquetoperators in their own. See [14] and [15].Interestingly, the condition of comonotonic additivity (the substitute for addi-tivity) lies at the core of many results concerning the real analysis. Indeed, itsmeaning in the context of real numbers, can be easily understood by identifyingeach real number x with the affine function α x ( t ) = tx , t ∈ R . As a consequence,two real numbers x and y are comonotone if and only if the functions α x and α y are comonotone, equivalently, if either both x and y are nonnegative or both arenonpositive. This yields the simplest example of Choquet functional from R intoitself which is not linear, the function x → x + . At the same time one can indicatea large family of nonlinear Choquet operators from C ([ − , E,T ϕ,U ( f ) = U (cid:18)Z − f + ( tx ) ϕ ( x )d x (cid:19) , where ϕ ∈ C ([ − , ϕ (0) = 0 and U : C ([ − , → E is any monotonic linear operator.Based on previous work done by Zhou [31], Cerreia-Vioglio, Maccheroni, Mari-nacci and Montrucchio [4], proved that a larger class of functionals defined on aspace C ( X ) (where X is a Hausdorff compact space) admit a Choquet analogue ofthe Riesz representation theorem. The aim of our paper is to further extend theirresults to the case of operators by developing a Choquet-Bochner theory of inte-gration relative to monotone set functions taking values in ordered Banach spaces.Section 2 is devoted to a quick review of some basic facts from the theory of or-dered Banach spaces. While the particular case of Banach lattices is nicely coveredby a series of textbooks such as those by Meyer-Nieberg [18] and Schaefer [26], thegeneral theory of ordered Banach spaces is still waiting to become the subject ofan authoritative book.In Section 3 we develop the theory of Choquet-Bochner integral associated toa vector capacity (that is, to a monotone set function µ taking values in an or-dered Banach space such that µ ( ∅ ) = 0) . As is shown in Theorem 1, this integralhas all nice features of the Choquet integral: monotonicity, positive homogeneityand comonotonic additivity. The transfer of properties from vector capacities totheir integrals also works in a number of important cases such as the upper/lowercontinuity and submodularity. See Theorem 1. In the case of submodular vectorcapacities with values in a Banach lattice, the integral analogue of the modulusinequality also holds. See Theorem 2.Section 4 deals with the integral representation of the Choquet operators de-fined on spaces C ( X ) ( X being compact and Hausdorff) and taking values in aBanach lattice with order continuous norm. The main result, Theorem 3, showsthat each such operator is the Choquet-Bochner integral associated to a suitable upper continuous vector capacity. In Section 5, this representation is generalizedto the framework of comonotonic additive operators with bounded variation. SeeTheorem 4. The basic ingredient is Lemma 12, which shows that every comono-tonic additive operator with bounded variation can be written as the difference oftwo positively homogeneous, translation invariant and monotone operators.The paper ends with a short list of open problems.2. Preliminaries on ordered Banach spaces An ordered vector space is a real vector space E endowed with an order relation ≤ such that the following two conditions are verified: x ≤ y implies x + z ≤ y + z for all x, y, z ∈ E ; and x ≤ y implies λx ≤ λy for x, y ∈ E and λ ∈ R + = [0 , ∞ ) . In this case the set E + = { x ∈ E : x ≥ } is a convex cone, called the positivecone . A real Banach space endowed with an order relation that makes it an orderedvector space is called an ordered Banach space if the norm is monotone on thepositive cone, that is, 0 ≤ x ≤ y implies k x k ≤ k y k . Note that in this paper we will consider only ordered Banch spaces E whose positivecones are closed (in the norm topology), proper ( − E + ∩ E + = { } ) and generating ( E = E + − E + ).A convenient way to emphasize the properties of ordered Banach spaces is thatdescribed by Davies in [8]. According to Davies, a real Banach space E endowedwith a closed and generating cone E + such that k x k = inf {k y k : y ∈ E, − y ≤ x ≤ y } for all x ∈ E, is called a regularly ordered Banach space . Examples are the Banach lattices andsome other spaces such as Sym( n, R ), the ordered Banach space of all n × n -dimensional symmetric matrices with real coefficients. The norm of a symmetricmatrix A is defined by the formula k A k = sup k x k≤ |h Ax, x i| , and the positive cone Sym + ( n, R ) of Sym( n, R ) consists of all symmetric matrices A such that h Ax, x i ≥ x. Lemma 1.
Every ordered Banach space can be renormed by an equivalent norm tobecome a regularly ordered Banach space.
For details, see Namioka [19]. Some other useful properties of ordered Banachspaces are listed below.
Lemma 2.
Suppose that E is a regularly ordered Banach space. Then: ( a ) There exists a constant
C > such that every element x ∈ E admits adecomposition of the form x = u − v where u, v ∈ E + and k u k , k v k ≤ C k x k . ( b ) The dual space of
E, E ∗ , when endowed with the dual cone E ∗ + = { x ∗ ∈ E ∗ : x ∗ ( x ) ≥ for all x ∈ E + } is a regularly ordered Banach space. ( c ) x ≤ y in E is equivalent to x ∗ ( x ) ≤ x ∗ ( y ) for all x ∗ ∈ E ∗ + . ( d ) k x k = sup (cid:8) x ∗ ( x ) : x ∗ ∈ E ∗ + , k x ∗ k ≤ (cid:9) for all x ∈ E + . SORIN G. GAL AND CONSTANTIN P. NICULESCU ( e ) If ( x n ) n is a decreasing sequence of positive elements of E which convergesweakly to , then k x n k → . The assertion ( e ) is a generalization of Dini’s lemma in real analysis; see [7], p.173. Proof.
The assertion ( a ) follows immediately from Lemma 1. For ( b ), see Davies[8], Lemma 2.4. The assertion ( c ) is an easy consequence of the Hahn-Banachseparation theorem; see [24], Theorem 2.5.3, p. 100.The assertion ( d ) is also a consequence of the Hahn-Banach separation theorem;see [27], Theorem 4.3, p. 223. (cid:3) Corollary 1.
Every ordered Banach space E can be embedded into a space C ( X ) ,where X is a suitable compact space.Proof. According to the Alaoglu theorem, the set X = (cid:8) x ∗ ∈ E ∗ + : k x ∗ k ≤ (cid:9) iscompact relative to the w ∗ topology. Taking into account the assertions ( c ) and( d ) of Lemma 2 one can easily conclude that E embeds into C ( X ) (algebraically,isometrically and in order) via the mapΦ : E → C ( X ) , (Φ( x )) ( x ∗ ) = x ∗ ( x ) . (cid:3) The following important result is due to V. Klee [16]. A simple proof of it isavailable in [23].
Lemma 3.
Every positive linear operator T : E → F acting on ordered Banachspaces is continuous. Sometimes, spaces with a richer structure are necessary.A vector lattice is any ordered vector space E such that sup { x, y } and inf { x, y } exist for all x, y ∈ E. In this case for each x ∈ E we can define x + = sup { x, } (thepositive part of x ), x − = sup {− x, } (the negative part of x ) and | x | = sup {− x, x } (the modulus of x ). We have x = x + − x − and | x | = x + + x − . A vector latticeendowed with a norm k·k such that | x | ≤ | y | implies x ≤ k y k is called a normed vector lattice; it is called a Banach lattice when in addition it ismetric complete.Examples of Banach lattice are numerous: the discrete spaces R n , c , c and ℓ p for 1 ≤ p ≤ ∞ (endowed with the coordinate-wise order), and the function spaces C ( K ) (for K a compact Hausdorff space) and L p ( µ ) with 1 ≤ p ≤ ∞ (endowedwith pointwise order). Of a special interest are the Banach lattices with ordercontinuous norm , that is, the Banach lattices for which every monotone and orderbounded sequence is convergent in the norm topology. So are R n , c and L p ( µ ) for1 ≤ p < ∞ . Lemma 4.
Every monotone and order bounded sequence of elements in a Banachlattice E with order continuous norm admits a supremum and an infimum and allclosed order intervals in E are weakly compact. For details, see Meyer-Nieberg [18], Theorem 2.4.2, p. 86. The Choquet-Bochner Integral
This section is devoted to the extension of Choquet’s theory of integrability tothe framework with respect to a monotone set function with values in the positivecone of a regularly ordered Banach space E . This draws a parallel to the real-valuedcase already treated in full details by Denneberg [10] and Wang and Klir [30].Given a nonempty set X, by a lattice of subsets of X we mean any collection Σ ofsubsets that contains ∅ and X and is closed under finite intersections and unions. Alattice Σ is an algebra if in addition it is closed under complementation. An algebrawhich is closed under countable unions and intersections is called a σ -algebra.Of a special interest is the case where X is a compact Hausdorff space and Σ iseither the lattice Σ + up ( X ) , of all upper contour closed sets S = { x ∈ X : f ( x ) ≥ t } , or the lattice Σ − up ( X ) of all upper contour open sets S = { x ∈ X : f ( x ) > t } , asso-ciated to pairs f ∈ C ( X ) and t ∈ R . When X is a compact metrizable space, Σ + up ( X ) coincides with the lattice of allclosed subsets of X (and Σ − up ( X ) coincides with the lattice of all open subsets of X ).In what follows Σ denotes a lattice of subsets of an abstract set X and E is aregularly ordered Banach space. Definition 1.
A set function µ : Σ → E + is called a vector capacity if it verifiesthe following two conditions: ( C µ ( ∅ ) = 0; and ( C µ ( A ) ≤ µ ( B ) for all A, B ∈ Σ with A ⊂ B . Notice that any vector capacity µ is positive and takes values in the order interval[0 , µ ( X )] . An important class of vector capacities is that of additive (respectively σ - additive)vector measures with positive values , that is, of capacities µ : Σ → E + with theproperty µ (cid:16)[ n A n (cid:17) = X ∞ n =1 µ ( A n ) , for every finite (respectively infinite) sequence A , A , A , ... of disjoint sets belong-ing to Σ such that ∪ n A n ∈ Σ . Some other classes of capacities exhibiting various extensions of the property ofadditivity are listed below.A vector capacity µ : Σ → E + is called submodular if(3.1) µ ( A ∪ B ) + µ ( A ∩ B ) ≤ µ ( A ) + µ ( B ) for all A, B ∈ Σand it is called supermodular when the inequality (3.1) works in the reversed way.Every additive measure taking values in E + is both submodular and supermodular.A vector capacity µ : Σ → E + is called lower continuous (or continuous byascending sequences) if lim n →∞ µ ( A n ) = µ ( [ ∞ n =1 A n )for every nondecreasing sequence ( A n ) n of sets in Σ such that ∪ ∞ n =1 A n ∈ Σ; µ iscalled upper continuous (or continuous by descending sequences) iflim n →∞ µ ( A n ) = µ ( ∩ ∞ n =1 A n ) SORIN G. GAL AND CONSTANTIN P. NICULESCU for every nonincreasing sequence ( A n ) n of sets in Σ such that ∩ ∞ n =1 A n ∈ Σ . If µ is an additive capacity defined on a σ -algebra, then its upper/lower continuity isequivalent to the property of σ -additivity.When Σ is an algebra of subsets of X, then to each vector capacity µ defined onΣ, one can attach a new vector capacity µ , the dual of µ , which is defined by theformula µ ( A ) = µ ( X ) − µ ( X \ A ) . Notice that ( µ ) = µ . The dual of a submodular (supermodular) capacity is a supermodular (submod-ular) capacity. Also, dual of a lower continuous (upper continuous) capacity is anupper continuous (lower continuous) capacity.
Example 1.
There are several standard procedures to attach to a σ -additive vectormeasure µ : Σ → E + certain not necessarily additive capacities. So is the case of distorted measures, ν ( A ) = T ( µ ( A )) , obtained from µ by applying to it a continuousnondecreasing distortion T : [0 , µ ( X )] → [0 , µ ( X )] . The vector capacities ν soobtained are both upper and lower continuous.For example, this is the case when E = Sym( n, R ) , µ ( X ) = I ( the identity of R n ) , and T : [0 , I] → [0 , I] is the distortion defined by the formula T ( A ) = A . Taking into account the assertions ( c ) and ( d ) of Lemma 2, some aspects (butnot all) of the theory of vector capacities are straightforward consequences of thetheory of R + -valued capacities. Lemma 5.
A set function µ : Σ → E + is a submodular ( supermodular, lowercontinuous, upper continuous ) vector capacity if and only if x ∗ ◦ µ is a submodular ( supermodular, lower continuous, upper continuous ) R + -valued capacity whenever x ∗ ∈ E ∗ + . When E = R n , this assertion can be formulated via the components µ k = pr k ◦ µ of µ. In what follows the term of (upper) measurable function refers to any function f : X → R whose all upper contour sets { x ∈ X : f ( x ) ≥ t } belong to Σ. When Σis a σ -algebra, this notion of measurability is equivalent to the Borel measurability.We will denote by B (Σ) the set of all bounded measurable functions f : X → R . In general B (Σ) is not a vector space (unless the case when Σ is a σ -algebra).However, even when Σ is only an algebra, the set B (Σ) plays some nice propertiesof stability: if f, g ∈ B (Σ) and α, β ∈ R , theninf { f, g } , sup { f, g } and α + βf also belong to B (Σ) . See [17], Proposition 15.Given a capacity µ : Σ → R + , the Choquet integral of a measurable function f : X → R on a set A ∈ Σ is defined as the sum of two Riemann improperintegrals, (C) Z A f d µ = Z + ∞ µ ( { x ∈ A : f ( x ) ≥ t } ) d t (3.2) + Z −∞ [ µ ( { x ∈ A : f ( x ) ≥ t } ) − µ ( A )] d t. Accordingly, f is said to be Choquet integrable on A if both integrals above arefinite. See the seminal paper of Choquet [6]. If f ≥
0, then the last integral in the formula (3.2) is 0. When Σ is a σ -algebra,the inequality sign ≥ in the above two integrands can be replaced by > ; see [30],Theorem 11.1, p. 226.Every bounded measurable function is Choquet integrable. The Choquet integralcoincides with the Lebesgue integral when the underlying set function µ is a σ -additive measure defined on a σ -algebra.The theory of Choquet integral is available from numerous sources including thebooks of Denneberg [10], and Wang and Klir [30].The concept of integrability of a measurable function with respect to a vectorcapacity µ : Σ → E + can be introduced by a fusion between the Choquet integraland the Bochner theory of integration of vector-valued functions.Recall that a function ψ : R → E is Bochner integrable with respect to theLebesgue measure on R if there exists a sequence of step functions ψ n : R → E such that lim n →∞ ψ m ( t ) = ψ ( t ) almost everywhere and Z R k ψ − ψ n k d t → . In this case, the (Bochner) integral of ψ is defined by Z R ψ d t = lim n →∞ Z R ψ n d t. Notice that if T : E → F is a bounded linear operator, then(3.3) T (cid:18)Z R ψ d t (cid:19) = Z R T ◦ ψ d t. Details about Bochner integral are available in the books of Diestel and Uhl [11]and Dinculeanu [12].
Definition 2.
A measurable function f : X → R is called Choquet-Bochner inte-grable with respect to the vector capacity µ : Σ → E + on the set A ∈ Σ if for every A ∈ Σ the functions t → µ ( { x ∈ A : f ( x ) ≥ t } ) and t → µ ( { x ∈ A : f ( x ) ≥ t } ) − µ ( A ) are Bochner integrable respectively on [0 , ∞ ) and ( −∞ , . Under these cir-cumstances, the Choquet-Bochner integral over A is defined by the formula (CB) Z A f d µ = Z + ∞ µ ( { x ∈ A : f ( x ) ≥ t } ) d t + Z −∞ [ µ ( { x ∈ A : f ( x ) ≥ t } ) − µ ( A )] d t. According to the formula (3.3), if f is Choquet-Bochner integrable, then(3.4) x ∗ (cid:18) (CB) Z A f d µ (cid:19) = (C) Z A f d( x ∗ ◦ µ ) , for every positive linear functional x ∗ ∈ E ∗ . A large class of Choquet-Bochner integrable is indicated below.
Lemma 6. If f : X → R is a bounded measurable function, then it is Choquet-Bochner integrable on every set A ∈ Σ .Proof. Suppose that f takes values in the interval [0 , M ] . Then the function ϕ ( t ) = µ ( { x ∈ A : f ( x ) ≥ t } ) is positive and nonincreasing on the interval [0 , M ] and null SORIN G. GAL AND CONSTANTIN P. NICULESCU outside this interval. As a consequence, the function ϕ is the uniform limit of thesequence of step functions defined as follows: ϕ n ( t ) = 0 if t / ∈ [0 , M ] ϕ n ( t ) = ϕ ( t ) if t = M and ϕ n ( t ) = ϕ ( kn M )if t ∈ [ kn M, k +1 n M ) and k = 0 , ..., n − . A simple computation shows that Z + ∞ k ϕ ( t ) − ϕ n ( t ) k d t = Z M k ϕ ( t ) − ϕ n ( t ) k d t ≤ X n − l =0 Z t k +1 t k k ϕ ( t ) − ϕ n ( t ) k d t ≤ Mn µ ( X ) → n → ∞ , which means the Bochner integrability of the function ϕ. See [11], p.44.The other cases, when f takes values in the interval [ M,
0] or in an interval[ m, M ] with m < < M, can be treated in a similar way. (cid:3) The next lemma collects a number of simple (but important) properties of theChoquet-Bochner integral.
Lemma 7.
Suppose that E is an ordered Banach space, µ : Σ → E + is a vectorcapacity and A ∈ Σ . ( a ) If f and g are Choquet-Bochner integrable functions, then f ≥ implies (CB) Z A f d µ ≥ positivity ) f ≤ g implies (CB) Z A f d µ ≤ (CB) Z A g d µ ( monotonicity )(CB) Z A af d µ = a · (CB) Z A f d µ for all a ≥ positive homogeneity )(CB) Z A · d µ = µ ( A ) ( calibration ).( b ) In general, the Choquet-Bochner integral is not additive but if f and g areChoquet-Bochner integrable functions and also comonotonic in the sense of Del-lacherie [9]) ( that is, ( f ( ω ) − f ( ω ′ )) · ( g ( ω ) − g ( ω ′ )) ≥ , for all ω, ω ′ ∈ X ), then (CB) Z A ( f + g )d µ = (CB) Z A f d µ + (CB) Z A g d µ . In particular, the Choquet-Bochner integral is translation invariant, (CB) Z A ( f + c )d µ = (CB) Z A f d µ + c µ ( A ) , for all Choquet-Bochner integrable functions f , all sets A ∈ Σ and all numbers c ∈ R . Proof.
According to Lemma 2 ( c ), and formula (3.3), applied in the case of anarbitrary functional x ∗ ∈ E ∗ + , the proof of both assertions ( a ) and ( b ) reduce to thecase of capacities with values in R + , already covered by Proposition 5.1 in [10], pp.64-65. (cid:3) Corollary 2.
The equality (CB) Z A ( αf + c )d µ = α (cid:18) (CB) Z A f d µ (cid:19) + c · µ ( A ) , holds for all Choquet-Bochner integrable functions f , all sets A ∈ Σ and all numbers α ∈ R + and c ∈ R . The next result describes how the special properties of a vector capacity transferto the Choquet-Bochner integral.
Theorem 1. ( a ) If µ is an upper continuous capacity, then the Choquet-Bochnerintegral is an upper continuous operator, that is, lim n →∞ (cid:13)(cid:13)(cid:13)(cid:13) (CB) Z A f n d µ − (CB) Z A f d µ (cid:13)(cid:13)(cid:13)(cid:13) = 0 , whenever ( f n ) n is a nonincreasing sequence of Choquet-Bochner integrable functionsthat converges pointwise to the Choquet-Bochner integrable function f and A ∈ Σ . ( b ) If µ is a lower continuous capacity, then the Choquet-Bochner integral islower continuous in the sense that lim n →∞ (cid:13)(cid:13)(cid:13)(cid:13) (CB) Z A f n d µ − (CB) Z A f d µ (cid:13)(cid:13)(cid:13)(cid:13) = 0 whenever ( f n ) n is a nondecreasing sequence of Choquet-Bochner integrable functionsthat converges pointwise to the Choquet-Bochner integrable function f and A ∈ Σ . ( c ) If Σ is an algebra and µ : Σ → E + is a submodular capacity, then theChoquet-Bochner integral is a submodular operator in the sense that (CB) Z A sup { f, g } d µ + (CB) Z A inf { f, g } d µ ≤ (CB) Z A f d µ + (CB) Z A g d µ whenever f and g are Choquet-Bochner integrable and A ∈ Σ .Proof. ( a ) Since µ is an upper continuous capacity and ( f n ) n is a nonincreasing se-quence of measurable functions that converges pointwise to the measurable function f it follows that µ ( { x ∈ A : f n ( x ) ≥ t } ) ց µ ( { x ∈ A : f ( x ) ≥ t } )in the norm topology. Taking into account the property of monotonicity of theChoquet-Bochner integral (already noticed in Lemma 7 ( a )) we have(CB) Z A f n d µ ≥ (CB) Z A f n d µ ≥ · · · ≥ (CB) Z A f d µ , so by Bepo Levi’s monotone convergence theorem from the theory of Lebesgueintegral (see [12], Theorem 2, p. 133) it follows that x ∗ (cid:18) (CB) Z A f n d µ (cid:19) = (C) Z A f n d µ ∗ → (C) Z A f d µ ∗ = x ∗ (cid:18) (CB) Z A f d µ (cid:19) for all x ∗ ∈ E ∗ + . The conclusion of the assertion ( c ) is now a direct consequence ofthe generalized Dini’s lemma (see Lemma 2 ( e )). ( b ) The argument is similar to that used to prove the assertion ( a ).( c ) Since Σ is an algebra, both functions inf { f, g } and sup { f, g } are measurable.The fact that µ is submodular implies µ ( { x : sup { f, g } ( x ) ≥ t } ) + µ ( { x : inf { f, g } ( x ) ≥ t } )= µ ( { x : f ( x ) ≥ t } ∪ { x : g ( x ) ≥ t } ) + µ ( { x : f ( x ) ≥ t } ∩ { x : g ( x ) ≥ t } ) ≤ µ ( { x : f ( x ) ≥ t } ) + µ ( { x : g ( x ) ≥ t } ) , and the same works when µ is replaced by µ ∗ = x ∗ ◦ µ , where x ∗ ∈ E ∗ + is arbitrarilyfixed. Integrating side by side the last inequality it follows that(C) Z A sup { f, g } d µ ∗ + (C) Z A inf { f, g } d µ ∗ ≤ (C) Z A f d µ ∗ + (C) Z A g d µ ∗ , which yields (via Lemma 2 ( c )) the submodularity of the Choquet-Bochner integral. (cid:3) The property of subadditivity of the Choquet-Bochner integral makes the objec-tive of the following result:
Theorem 2. (The Subadditivity Theorem) If µ is a submodular capacity, then theassociated Choquet-Bochner integral is subadditive, that is, (CB) Z A ( f + g )d µ ≤ (CB) Z A f d µ + (CB) Z A g d µ whenever f , g and f + g are Choquet-Bochner integrable functions and A ∈ Σ .In addition, when E is a Banach lattice and f , g, f − g and g − f are Choquet-Bochner integrable functions, then the following integral analog of the modulus in-equality holds true: (cid:12)(cid:12)(cid:12)(cid:12) (CB) Z A f d µ − (CB) Z A g d µ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (CB) Z A | f − g | d µ for all A ∈ Σ . In particular, (cid:12)(cid:12)(cid:12)(cid:12) (CB) Z A f d µ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (CB Z A | f | d µ whenever f and − f are Choquet-Bochner integrable functions and A ∈ Σ . Proof.
According to Lemma 5, if µ is submodular, then every real-valued capacity µ ∗ = x ∗ ◦ µ also is submodular, whenever x ∗ ∈ E ∗ + . Then(C) Z A f d µ ∗ + (C) Z A g d µ ∗ ≤ (C) Z A ( f + g )d µ ∗ , as a consequence of Theorem 6.3, p. 75, in [10]. Therefore the first inequality inthe statement of Theorem 2 is now a direct consequence of Lemma 2 ( c ) and theformula (3.4).For the second inequality, notice that the subadditivity property implies(CB) Z A f d µ = (CB) Z A ( f − g + g ) d µ ≤ (CB) Z A ( f − g ) d µ + (CB) Z A g d µ, and taking into account that f − g ≤ | f − g | we infer that(CB) Z A f d µ − (CB) Z A g d µ ≤ (CB) Z A | f − g | d µ . Interchanging f and g we also obtain ± ((CB) Z A f d µ − (CB) Z A g d µ ) ≤ (CB) Z A | f − g | d µ and the proof is done. (cid:3) The integral representation of Choquet operators defined on aspace C ( X )The special case when X is a compact Hausdorff space and Σ = B ( X ) , the σ -algebra of all Borel subsets of X, allows us to shift the entire discussion concerningthe Choquet-Bochner integral from the vector space B (Σ) (of all real-valued Borelmeasurable functions) to the Banach lattice C ( X ) (of all real-valued continuousfunctions defined on X . Indeed, in this case C ( X ) is a subspace of B (Σ) and allnice properties stated in Lemma 7 and in Theorem 2 remain true when restrictingthe integral to the space C ( X ) . As a consequence the integral operator(4.1) CB µ : C ( X ) → E, CB µ ( f ) = (CB) Z X f d µ, associated to a vector-valued submodular capacity µ : B ( X ) → E, is a Choquetoperator.The linear and positive functionals defined on C ( X ) can be represented eitheras integrals with respect to a unique regular Borel measure (the Riesz-Kakutanirepresentation theorem) or as Choquet integrals (see Epstein and Wang [13]). Example 2.
The space c of all convergent sequences of real numbers can be iden-tified with the space of continuous functions on ˆ N = N ∪ {∞} ( the one-point com-pactification of the discrete space N ) . The functional I : c → R , I ( x ) = lim n →∞ x ( n ) is linear and monotone ( therefore continuous by Lemma and its Riesz represen-tation is I ( x ) = Z ˆ N x ( n )d δ ∞ ( n ) , where δ ∞ is the Dirac measure concentrated at ∞ , that is, δ ∞ ( A ) = 1 if A ∈ P ( ˆ N ) and {∞} ∈ A and δ ∞ ( A ) = 0 if {∞} / ∈ A. Meantime, I admits the Choquetrepresentation I ( x ) = (C) Z x ( n ) dµ ( n ) where µ is the capacity defined on the power set P ( ˆ N ) by µ ( A ) = 0 if A is finiteand µ ( A ) = 1 otherwise. As the Riesz-Kakutani representation theorem also holds in the case of linear andpositive operators T : C ( X ) → E , it seems natural to search for an analogue in thecase of Choquet operators. The answer is provided by the following representationtheorem that extend a result due to Epstein and Wang [13] from functionals tooperators: Theorem 3.
Let X be a compact Hausdorff space and E be a Banach lattice withorder continuous norm. Then for every comonotonic additive and monotone oper-ator I : C ( X ) → E with I (1) > there exists a unique upper continuous vectorcapacity µ : Σ + up ( X ) → E + such that (4.2) I ( f ) = (CB) Z X f d µ for all f ∈ C ( X ) . Moreover, µ admits a unique extension to B ( X ) ( also denoted µ ) that fulfils thefollowing two properties of regularity: ( R µ ( A ) = sup { µ ( K ) : K closed, K ⊂ A } for all A ∈ B ( X );( R µ ( K ) = inf { µ ( O ) : O open, O ⊃ K } for all closed sets K. Recall that Σ + up ( X ) represents the lattice of upper contour sets associated tothe continuous functions defined on X. This lattice coincides with the lattice of allclosed subsets of X when X is compact and metrizable.The proof of Theorem 3 needs several auxiliary results and will be detailed atthe end of this section. Lemma 8.
Let E be an ordered Banach space. Then every monotone and transla-tion invariant operator I : C ( X ) → E is Lipschitz continuous.Proof. Given f, g ∈ C ( X ) , one can choose a decreasing sequence ( α n ) n of positivenumbers such that α n ↓ k f − g k . Then f ≤ g + k f − g k ≤ g + α n · , which implies I ( f ) ≤ I ( g + α n ·
1) = I ( g ) + α n I (1)due to the properties of monotonicity and translation invariance and positive ho-mogeneity of I. Since the role of f and g is symmetric, this leads to the fact that | I ( f ) − I ( g ) | ≤ I (1) · α n for all n, whence by passing to the limit as n → ∞ , weconclude that | I ( f ) − I ( g ) | ≤ I (1) · k f − g k . (cid:3) Lemma 9.
Let E be an ordered Banach space. Then every comonotonic additiveand monotone operator I : C ( X ) → E is positively homogeneous.Proof. Let f ∈ C ( X ) + . Since I is comonotonic additive, we get I (0) = I (0 + 0) =2 · I (0) , which implies I (0) = 0. As a consequence, I (0 · f ) = 0 = 0 · I ( f ) . Thenthe same argument shows that I (2 f ) = I ( f + f ) = 2 I ( f ) and by mathematicalinduction we infer that I ( pf ) = pI ( f ) for all p ∈ N .Now, consider the case of positive rational numbers r = p/q, where p, q ∈ N .Then I ( f ) = I (cid:16) q · q f (cid:17) = qI (cid:16) q f (cid:17) , which implies I (cid:16) q f (cid:17) = q · I ( f ). Therefore I (cid:18) pq f (cid:19) = pI (cid:18) q f (cid:19) = pq · I ( f ) . Passing to the case of an arbitrary positive number α, let us choose a decreasingsequence ( r n ) n of rationals converging to α . Then r n f ≥ αf for all n, which yields r n I ( f ) = I ( r n f ) ≥ I ( αf ) n ∈ N . Passing here to limit (in the norm of E ) it follows αI ( f ) ≥ I ( αf ). On the other hand, considering a sequence of positive rationalnumbers s n ր α and reasoning as above, we easily obtain αI ( f ) ≤ I ( αf ), whichcombined with the previous inequality proves the assertion of Lemma 9 in the caseof nonnegative functions. When f ∈ C ( X ) is arbitrary, one can choose a positive number λ such that f + λ ≥ . By the above reasoning and the property of comonotonic additivity, forall α ≥ , αI ( f ) + αλI (1) = αI ( f + λ ) = I ( α ( f + λ ))= I ( αf + αλ ) = I ( αf ) + αλI (1) , which ends the proof of Lemma 9. (cid:3) Lemma 10.
Let E be a Banach lattice with order continuous norm. Then everymonotone, positively homogeneous and translation invariant operator I : C ( X ) → E is weakly compact and upper continuous.Proof. The weak compactness of I follows from the fact that I maps the closedorder interval in C ( X ) (that is, all closed balls) into closed order intervals in E andall such intervals are weakly compact in E . See Lemma 3.For the property of upper continuity, let ( f n ) n be any nonincreasing sequenceof functions in C ( X ), which converges pointwise to a continuous function f. ByDini’s lemma, the sequence ( f n ) n is convergent to f in the norm topology of C ( X ) . Therefore, for each ε > N such that f ≤ f n < f + ε for all n ≥ N. Since I is monotone it follows that(4.3) I ( f ) ≤ I ( f n ) ≤ I ( f + ε ) = I ( f ) + ε · I (1) for all n ≥ N, where 1 is the unit of C ( X ) . Taking into account that E has order continuousnorm and I ( f n ) ≥ I ( f n +1 ) ≥ I ( f ) for all n it follows that the limit lim n →∞ I ( f n )exists in E and lim n →∞ I ( f n ) ≥ I ( f ). Combining this fact with (4.3) we inferthat I ( f ) ≤ lim n →∞ I ( f n ) ≤ I ( f ) + εI (1) . Since ε > n →∞ I ( f n ) = I ( f ) . (cid:3) The next result, was stated for real-valued functionals in [31], Lemma 1 (andattributed by him to Masimo Marinacci). For the convenience of the reader weinclude here the details.
Lemma 11.
Let E be an ordered Banach space. ( a ) Suppose that I : C ( X ) → E is a monotone, positively homogeneous andtranslation invariant operator. The following two properties are equivalent: ( a ) lim n →∞ I ( f n ) = I ( f ) for any nonincreasing sequence ( f n ) n in C ( X ) thatconverges pointwise to a function f also in C ( X );( a ) lim n →∞ I ( f n ) ≤ I ( f ) for any nonincreasing sequence ( f n ) n in C ( X ) andany f in C ( X ) such that for each x ∈ X there is an index n x ∈ N such that f n ( x ) ≤ f ( x ) whenever n ≥ n x . ( b ) For any vector capacity µ : B ( X ) → E + , the following two properties areequivalent : ( b ) lim n →∞ µ ( A n ) = µ ( A ) , for any nonincreasing sequence ( A n ) n of sets in B ( X ) such that A = ∩ ∞ n =1 A n ;( b ) lim n →∞ µ ( A n ) ≤ µ ( A ) , for any nonincreasing sequence ( A n ) n of sets in B ( X ) and any A ∈ B ( X ) such that ∩ ∞ n =1 A n ⊂ A .All the limits above are considered in the norm topology of E .Proof. ( a ) ⇒ ( a ). Let an arbitrary sequence ( f n ) n in C ( X ) and f ∈ C ( X ), besuch that ( f n ) n is nonincreasing and, for all x ∈ X , there is an n x with f n ( x ) ≤ f ( x ) for all n ≥ n x . Since the sequence (max { f n , f } ) is also a nonincreasing sequence in C ( X ) and lim n →∞ max { f n ( x ) , f ( x ) } = f ( x ) for all x ∈ X , ( a i ) implieslim n →∞ I (max { f n , f } ) = I ( f ). By the monotonicity of I it follows lim n →∞ I ( f n ) ≤ lim n →∞ I (max { f n , f } ) = I ( f ).( a ) ⇒ ( a ). Let f n , f ∈ C ( X ), n ∈ N , be such that ( f n ) n is nonincreasing andlim n →∞ f n ( x ) = f ( x ), for all x ∈ X . Fix ε > x ∈ X ,there exists n x ∈ N such that f n ( x ) ≤ f ( x ) + ε , for all n ≥ n x , ( a ii ) implies (frommonotonicity, positive homogeneity and translation invariance) lim n →∞ I ( f n ) ≤ I ( f + ε ·
1) = I ( f ) + εI (1). Passing with ε → n →∞ I ( f n ) ≤ I ( f ). Butsince ( f n ) n is nonincreasing and I is monotone, we also have lim n →∞ I ( f n ) ≥ I ( f ),which combined with the previous inequality implies lim n →∞ I ( f n ) = I ( f ).The equivalence ( b ) ⇔ ( b ) can be proved in a similar way. (cid:3) Recall that in the case of compact Hausdorff space X the lattice Σ + up ( X ) repre-sents the lattice of upper contour sets associated to the continuous functions definedon X. This lattice coincides with the lattice of all closed subsets of X when X iscompact and metrizable; indeed, if d is the metric of X, then every closed subset A ⊂ X admits the representation A = { x : − d ( x, A ) ≥ } . Proof of Theorem . Notice first that according to Lemma 9 and Lemma 10 theoperator I is also positively homogeneous and upper continuous.Every set K ∈ Σ + up ( X ) admits a representation of the form K = { x : f ( x ) ≥ α } , for suitable f ∈ C ( X ) and α ∈ R . As a consequence its characteristic function χ K is the pointwise limit of a nonincreasing sequence ( f Kn ) n of continuous andnonnegative functions. For example, one may choose(4.4) f Kn ( x ) = 1 − inf (cid:8) , n ( α − f ) + (cid:9) = f ( x ) ≤ α − /n ∈ (0 ,
1) if α − /n < ϕ ( x ) < α f ( x ) ≥ α (i.e., x ∈ K ) . See [31], p. 1814. Since I is monotone, the sequence ( I ( f Kn )) n is also nonin-creasing and bounded from below by 0, which implies (due to the order continuityof the norm of E ), that it is also convergent in the norm topology of E. This allows us to define µ on the sets K ∈ Σ + up ( X ) by the formula(4.5) µ ( K ) = lim n →∞ I ( f Kn ) . The definition of µ ( K ) is independent of the particular sequence ( f Kn ) n with theaforementioned properties. Indeed, if ( g Kn ) n is another such sequence, fix a positiveinteger m , and infer from Lemma 11 ( a ) that lim n →∞ I ( g An ) ≤ I ( f Am ). Taking thelimit as m → ∞ on the right-hand side we getlim n →∞ I ( g An ) ≤ lim m →∞ I ( f Am ) . Then, interchanging ( f Kn ) n and ( g Kn ) n we conclude that actually equality holds.Clearly, the set function µ : Σ + up ( X ) → E + is a vector capacity and it takesvalues in the order interval [0 , I (1)] . We next show that µ is upper continuous on Σ + up ( X ), that is, µ ( K ) = lim n →∞ µ ( K n ) whenever ( K n ) n is a nonincreasing sequence of sets in Σ + up ( X ) such that K = ∩ ∞ n =1 K n . Indeed, using formula (4.4) one can choose a nonincreasing sequence ofcontinuous function g n such that g n ≥ χ K n , g n = 0 outside the neighborhood ofradius 1 /n of K n and I ( g n ) − µ ( K n ) →
0. See formula (4.4) and using analogousreasonings with those concerning relations (7)-(9) in [31], pp. 1814-1815. Then µ ( K ) = lim n →∞ I ( g n ) , which implies the equality µ ( K ) = lim n →∞ µ ( K n ) . The next goal is the representation formula (4.2). For this, let x ∗ ∈ E ∗ + bearbitrarily fixed and consider the comonotonic additive and monotone functional x ∗ ◦ I : C ( X ) → R . It verifies I ∗ (1) = x ∗ ( I (1)) > , so by Theorem 1 in [31] there is a unique uppercontinuous capacity ν ∗ : Σ + up ( X ) → [0 , I ∗ (1)] such that( x ∗ ◦ I ) ( f ) = ( C ) Z X f d ν ∗ for all f ∈ C ( X ) . The capacity ν ∗ is obtained via an approximation process similar to (4.5). Therefore ν ∗ ( K ) = lim n →∞ x ∗ ( I ( f Kn )) = x ∗ ( lim n →∞ [ I ( f Kn )) = x ∗ ( µ ( K ))for all K ∈ Σ + up ( X ) , which implies x ∗ ( I ( f )) = ( C ) Z X f d( x ∗ ◦ µ ) . Since x ∗ ∈ E ∗ + was arbitrarily fixed, an appeal to Lemma 2 ( c ) easily yields theequality I ( f ) = (CB) Z X f d µ . For the second part of Theorem 3,since µ takes values in an order boundedinterval, one can extend it to all Borel subsets of X via the formula µ ( A ) = sup { µ ( K ) : K closed, K ⊂ A } , A ∈ B ( X ) . The fact that the resulting set function µ is a vector capacity is immediate.This set function µ also verifies the regularity condition ( R K , we can consider the sequence of open sets O n = { x ∈ X : d ( x, K ) < /n } . Clearly, µ ( K ) ≤ µ ( O n ) ≤ µ ( K n ) where K n = { x ∈ X : d ( x, K ) ≤ /n } . Since µ is upper continuous on closed sets, it follows that lim n →∞ µ ( K n ) = µ ( K ), whence lim n →∞ µ ( O n ) = µ ( K ) . Therefore µ verifies the regularity condition( R µ to B ( X ) is motivated by the condition( R . (cid:3) Remark 1.
If the operator I is submodular, that is, I (sup { f, g } ) + I (inf { f, g } ) ≤ I ( f ) + I ( g ) for all f, g ∈ C ( X ) , then the vector capacity µ : Σ + up ( X ) → E + stated by Theorem is submodular.This is a consequence of Theorem
13 ( c ) in [4] . For the convenience of the readerwe will recall here the argument. Let A, B ∈ Σ + up ( X ) and consider the sequences ( f An ( x )) n and ( f Bn ( x )) n of continuous functions associated respectively to A and B by the formula (4.5) . Then µ ( A ) = lim n →∞ I ( f An ) , µ ( B ) = lim n →∞ I ( f Bn ) , µ ( A ∪ B ) = lim n →∞ I (sup (cid:8) f An , f Bn (cid:9) and µ ( A ∩ B ) = lim n →∞ I (inf (cid:8) f An , f Bn (cid:9) . Since I is submodular, it follows that µ ( A ∪ B ) + µ ( A ∩ B ) ≤ µ ( A ) + µ ( B ) , and the proof is done. The case of operators with bounded variation
The representation Theorem 3 can be extended outside the framework of mono-tone operators by considering the class of operators with bounded variation.As above, X is a compact Hausdorff space and E is a Banach lattice with ordercontinuous norm. Definition 3.
An operator I : C ( X ) → E has bounded variation over an orderinterval [ f, g ] if (5.1) ∨ gf I = sup X nk =0 | I ( f k ) − I ( f k − ) | exists in E, the supremum being taken over all finite chains f = f ≤ f ≤ · · · ≤ f n = g of functions in the Banach lattice C ( X ) . The operator I is said to have boundedvariation if it has bounded variations on all order intervals [ f, g ] in C ( X ) . Clearly, if I is monotone, then ∨ gf I = I ( g ) − I ( f ) for all f ≤ g in C ( X ) and thus I has bounded variation.More generally, every operator I : C ( X ) → E which can be represented asthe difference I = I − I of two monotone operators I , I : C ( X ) → E hasbounded variation. This follows from the order completeness of E and the modulusinequality, which provides an upper bound for the sums appearing in formula (5.1): X nk =0 | I ( f k ) − I ( f k − ) | ≤ I ( g ) − I ( f ) + I ( g ) − I ( f ) . Remarkably, the converse also holds. The basic ingredient is the following result.
Lemma 12.
Suppose that I : C ( X ) → E is a comonotonic additive operatorwith bounded variation. Then there exist two positively homogeneous, translationinvariant and monotone operators I , I : C ( X ) → E such that I = I − I . Moreover, if I is upper continuous, then both operators I and I can be chosento be upper continuous.Proof. The proof is done in the footsteps of Lemma 14 in [4] by noticing first thefollowing four facts:( a ) According to our hypotheses, I ( αf + β ) = αI ( f ) + I ( β )for all f ∈ C ( X ) , α ∈ R + and β ∈ R . ( b ) ∨ f + α I = ∨ f − α I for all f ∈ C ( X ) and α ∈ R with f + α ≥ . This follows from the definition of the variation.( c ) ∨ αf I = α ∨ f I for all f ∈ C ( X ) + and α ∈ R + . Indeed for every ε ∈ E, ε > , there exists a chain 0 = f ≤ f ≤ · · · ≤ f n = f such that X nk =0 | I ( f k ) − I ( f k − ) | ≥ ∨ f I − ε. According to fact ( a ) the chain 0 = αf ≤ αf ≤ · · · ≤ αf n = αf verifies ∨ αf I ≥ X nk =0 | I ( αf k ) − I ( αf k − ) | ≥ α ∨ f I − αε. As ε > ∨ αf I ≥ α ∨ f I. By replacing α to 1 /α and then f by αf , one obtains the reverse inequality, ∨ αf I ≤ α ∨ f I. ( d ) ∨ f − α I = ∨ − α I + ∨ f I = ∨ α I + ∨ f I for all f ∈ C ( X ) + and α ∈ R + . The fact that ∨ f − α I ≥ ∨ − α I + ∨ f I = ∨ α I + ∨ f I is a direct consequence of thedefinition of variation. For the other inequality, fix arbitrarily ε > E andchoose a chain − α = f ≤ f ≤ · · · ≤ f n = f such that X nk =0 | I ( f k ) − I ( f k − ) | ≥ ∨ f − α I − ε. Then − α = − f − ≤ − f − ≤ · · · ≤ − f − n = 0 and 0 = f +0 ≤ f +1 ≤ · · · ≤ f + n = f. Since all pairs − f − k , f + k are comonotonic,we have ∨ − α I + ∨ f I ≥ X nk =0 (cid:12)(cid:12) I ( − f − k ) − I ( − f − k − ) (cid:12)(cid:12) + X nk =0 (cid:12)(cid:12) I ( f + k ) − I ( f + k − ) (cid:12)(cid:12) ≥ X nk =0 (cid:12)(cid:12) I ( − f − k ) − I ( − f − k − ) + I ( f + k ) − I ( f + k − ) (cid:12)(cid:12) = X nk =0 | I ( f k ) − I ( f k − ) | ≥ ∨ f − α I − ε and it remains to take the supremum over ε > . Now we can proceed to the choice of the operators I and I . By definition, I ( f ) = ∨ f I if f ∈ C ( X ) + , while if f ∈ C ( X ) is an arbitrary function, one choose α ∈ R + such that f + α ≥ I ( f ) = ∨ f + α I − α ∨ I. The fact that I is well-defined (that is, independent of α ) can be proved as follows.Suppose that α, β > f + α ≥ f + β ≥ . Without loss of generality we may assume that α < β.
Indeed, according to thefacts ( b ) − ( d ), we have ∨ f + β I − β ∨ I = ∨ f + α +( β − α )0 I − β ∨ I fact ( b ) = ∨ f + α − ( β − α ) I − β ∨ I fact ( d ) = ∨ − ( β − α ) I + ∨ f + α I − β ∨ I fact ( b ) = ∨ β − α I + ∨ f + α I − β ∨ I fact ( c ) = ( β − α ) ∨ I + ∨ f + α I − β ∨ I = ∨ f + α I − α ∨ I. By definition, I = I − I . Let f, g be two functions in C ( X ) such that f ≤ g and let α > f + α ≥ . Since I is monotonic and has bounded variation,0 ≤ I ( g ) − I ( f ) = I ( g + α ) − I ( f + α ) ≤ ∨ g + αf + α I ≤ ∨ g + α I − ∨ f + α I = I ( g ) − I ( f ) , whence we infer that both operators I and I are monotonic. Due to the fact ( c ) , the operator I is positively homogeneous. Therefore thesame is true for I . For the property of translation invariance, let f ∈ C ( X ), β ∈ R and choose α > f + β + α ≥ β + α ≥ . Then I ( f + β ) = ∨ f + β + α I − α ∨ I = I ( f + β + α ) − α ∨ I while from facts ( b )&( d ) we infer that I ( f ) = I ( f + β + α ) − ( β + α ) ∨ I. Therefore I ( f + β ) = I ( f ) + I ( β )which proves that indeed I is translation invariant. The same holds for I = I − I . As concerns the second part of Lemma 12, it suffices to prove that I is uppercontinuous when I has this property.Let ( f n ) n be a decreasing sequence in C ( X ) which converges pointwise to afunction f also in C ( X ) . By Dini’s lemma, k f n − f k →
0. Since I is a Lipschitzoperator (see Lemma 8) we conclude that k I ( f n ) − I ( f ) k → . This ends the proof of Lemma 12. (cid:3)
Now it is clear that the representation Theorem 3 can be extended to the frame-work of comonotonic additive operators I : C ( X ) → E with bounded variation byconsidering Choquet-Bochner integrals associated to differences of vector capaci-ties (that is, to set functions with bounded variation in the sense of Aumann andShapley [1]).Let Σ be a lattice of sets of X and µ : ± → E a set function taking values in aBanach lattice E with order continuous norm.The variation of the set function µ is the set function | µ | defined by the formula | µ | ( A ) = sup X nk =0 | µ ( A k ) − µ ( A k − ) | for A ∈ Σ , where the supremum is taken over all finite chains A = ∅ ⊂ A ⊂ · · · ⊂ A n ⊂ A ofsets in the lattice Σ . The space of set functions with bounded variation,bv(Σ , E ) = { µ : Σ → E : µ ( ∅ ) = 0 and | µ | ( X ) exists in E } , is a normed vector space when endowed with the norm k µ k bv = k| µ | ( X ) k . Associated to every set function µ ∈ bv(Σ , E ) are two positive vector-valued setfunctions, the inner upper variation of µ, defined by µ + ( A ) = sup X nk =0 ( µ ( A k ) − µ ( A k − )) + for A ∈ Σand the inner lower variation of µ , defined by µ − ( A ) = sup X nk =0 ( µ ( A k ) − µ ( A k − )) − for A ∈ Σ;in both cases the supremum is taken over all finite chains A = ∅ ⊂ A ⊂ · · · ⊂ A n ⊂ A of sets in Σ . Notice that µ = µ + − µ − and | µ | = µ + + µ − . Lemma 13.
We assume that E is a Banach lattice with order continuous norm.The following conditions are equivalent for a set function µ : Σ → E :( a ) µ ∈ bv(Σ , E );( b ) µ + and µ − are vector capacities ;( c ) there exist two vector capacities µ and µ on Σ such that µ = µ − µ . Moreover, for any such decomposition we have µ + ≤ µ and µ − ≤ µ . The details are similar to those presented in [1], Ch. 1, § Theorem 4.
Let X be a compact Hausdorff space and E be a Banach lattice withorder continuous norm. Then for every comonotonic additive operator I : C ( X ) → E with bounded variation there exists a unique upper continuous set function µ :Σ + up ( X ) → E with bounded variation such that I ( f ) = Z + ∞ µ ( { x : f ( x ) ≥ t } ) d t (5.2) + Z −∞ [ µ ( { x : f ( x ) ≥ t } ) − µ ( A )] d t. for all f ∈ C ( X ) . Proof.
By Lemma 12, there exist two functionals I , I : C ( X ) → E that are mono-tone, translation invariant, positively homogeneous, upper continuous and such that I = I − I . Define µ : Σ + up ( X ) → E by µ = µ − µ , where µ , µ are associatedto the functionals I and I via Theorem 3. Then µ is upper continuous and byLemma 13 it also has bounded variation. We now prove that the representationformula (5.2) holds.Suppose that f ∈ C ( X ) + . The integral Z + ∞ µ ( { x : f ( x ) ≥ t } ) d t is well defined as a Bochner integral (see Lemma 6). Given ε > , one can choosean equidistant division 0 = t < · · · < t m = k f k of [0 , k f k ] such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z + ∞ µ ( { x : f ( x ) ≥ t } ) d t − m − X k =0 µ ( { x : f ( x ) ≥ t k } ) ( t k +1 − t k ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z k f k µ ( { x : f ( x ) ≥ t } ) d t − m − X k =0 µ ( { x : f ( x ) ≥ t k } ) ( t k +1 − t k ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε (5.3)and k f k /m < ε .Denote C k = { x : f ( x ) ≥ t k } for k = 0 , ..., m − . By the definition of µ and µ (see the proof of Theorem 3) one can choose functions f C k n ∈ C ( X ) + such that(5.4) (cid:13)(cid:13) µ ( C k ) − I ( f C k n ) (cid:13)(cid:13) < ε/ k f k and n − < k f k /m. Because the functions f C k n are defined by the formula (4.4) and n − < k f k /m , itfollows that the functions f C i n ( t i +1 − t i ) and P m − k = i +1 f C k n ( t k +1 − t k ) are comonotonic for all indices i, so that I ( m − X k =0 f C k n ( t k +1 − t k )) = I ( f C n )( t − t ) + I ( m − X k =1 f C k n ( t k +1 − t k ))= · · · = m − X k =0 I ( f C k n )( t k +1 − t k );the property of positive homogeneity of I is assured by Lemma 12.Notice that f ( x ) ≤ m − X k =0 f C k n ( x )( t k +1 − t k ) ≤ f ( x ) + 2 ε for all x ∈ X. Since the operators I , I are monotone and translation invariant, it follows that I j ( f ) ≤ I j ( m − X k =0 f C k n ( t k +1 − t k )) ≤ I j ( f ) + 2 εI j (1) for j ∈ { , } . whence(5.5) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) I j ( m − X k =0 f C k n ( t k +1 − t k )) − I j ( f ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ε k I j (1) k for j ∈ { , } . Therefore (cid:13)(cid:13)(cid:13)(cid:13)Z + ∞ µ ( { x : f ( x ) ≥ t } ) d t − I ( f ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z + ∞ µ ( { x : f ( x ) ≥ t } ) d t − m − X k =0 I ( f C k n )( t k +1 − t k ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m − X k =0 I ( f C k n )( t k +1 − t k ) − I ( f ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) see ( . ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z + ∞ µ ( { x : f ( x ) ≥ t } ) d t − m − X k =0 I ( f C k n )( t k +1 − t k ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + 2 ε k I (1) k + 2 ε k I (1) k≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z + ∞ µ ( { x : f ( x ) ≥ t } ) d t − m − X k =0 µ ( C k ) ( t k +1 − t k ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − X k =0 µ ( C k ) ( t k +1 − t k ) − n − X k =0 I ( f C k n )( t k +1 − t k ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + 2 ε k I (1) k + 2 ε k I (1) k see ( . )&( . ) ≤ ε (1 + k I (1) k + k I (1) k ) . Since ε > If f / ∈ C ( X ) + , then f + k f k ∈ C ( X ) + and by the preceding considerations wehave I ( f ) + k f k I (1) = I ( f + k f k ) = Z + ∞ µ ( { x : f ( x ) + k f k ≥ t } ) d t = Z + ∞ µ ( { x : f ( x ) ≥ t } ) d t + Z −k f k µ ( { x : f ( x ) ≥ t } ) d t = Z + ∞ µ ( { x : f ( x ) ≥ t } ) d t + Z −k f k µ ( { x : f ( x ) ≥ t } ) − µ ( X )d t + k f k µ ( X )= Z + ∞ µ ( { x : f ( x ) ≥ t } ) d t + Z −∞ µ ( { x : f ( x ) ≥ t } ) − µ ( X )d t + k f k I (1) . The proof of the representation formula (5.2) is now complete.As concerns the uniqueness of µ , suppose that ν is another upper continu-ous monotone set function with bounded variations for which the formula (5.2)holds. Given a set K = { x : f ( x ) ≥ t } ∈ Σ + up ( X ) , it is known that the functions f Kn : X → [0 ,
1] defined by the formula (4.4) decrease to χ K . This implies that( (cid:8) x : f Kn ( x ) ≥ t (cid:9) ) n is decreasing to K for each t ∈ [0 , . Consider the sequence offunctions ϕ n : [0 , → E, ϕ n ( t ) = ν ( (cid:8) x : f Kn ( x ) ≥ t (cid:9) ) . Notice that all these functions have bounded variation and their variation is boundedby the variation of | ν | ( X ) . By Lebesgue dominated convergence,lim n →∞ Z ϕ n ( t ) dt = ν ( K ) . On ther hand, by (5.2) and the definition of µ , µ ( K ) = lim n →∞ I ( f Kn ) = lim n →∞ Z ϕ n ( t ) dt = ν ( K ) , which ends the proof of the uniqueness. (cid:3) Open problems
We end our paper by mentioning few open problems that might be of interest toour readers.
Problem 1.
Is the order continuity of the norm of E a necessary condition for thevalidity of Theorems 3 and 4? As was noticed by Bartle, Dunford and Schwartz [2], much of the theory of weaklycompact linear operators defined on a space C ( X ) is dominated by the concept ofabsolute continuity. For more recent contributions see [20], [21] and [22].Suppose that A is a σ -algebra and E is a Banach latttice with order continuousnorm.A vector capacity µ : A → E + is called absolutely continuous with respect to acapacity λ : A → [0 , ∞ ) (denoted µ ≪ λ ) if for every ε > δ > A ∈ A , λ ( A ) < δ = ⇒ k µ ( A ) k < ε. The following lemma extends a result proved by Pettis in the context of σ -additivemeasures. Lemma 14. If µ is upper continuous and λ is upper continuous and supermodularthen the condition µ ≪ λ is equivalent to the following one: A ∈ A and λ ( A ) = 0 = ⇒ µ ( A ) = 0 . The proof is immediate, by reductio ad absurdum.
Problem 2.
Suppose µ : A → E + is an upper continuous vector measure. Doesthere exist a capacity λ : A → [0 , ∞ ) such that µ ≪ λ ? If Yes, is it possible to choose λ of the form λ = x ∗ ◦ µ for a suitable x ∗ ∈ E ∗ + ?It would be also interesting the following operator analogue of Problem 2: Problem 3.
Does there exist for each Choquet operator I : C ( X ) → E a functional x ∗ ∈ E ∗ + such that for every ε > there is δ > with the property k I ( f ) k ≤ ε k f k + δx ∗ ( | f | ) for all f ∈ C ( X )? Declaration of interests : none.
References [1] R. Aumann, L. Shapley, Values of Non-Atomic Games, Princeton University Press, Princeton,1974.[2] R. G. Bartle, N. Dunford, J. Schwartz, Weak compactness and vector measures, Canad. J.Math. 7 (1955) 289–305.[3] R. Bhatia, Notes on functional analysis. Texts and Readings in Mathematics, vol. 50, Hin-dustan Book Agency, New Delhi (2009)[4] S. Cerreia-Vioglio, F. Maccheroni, M. Marinacci and L. Montrucchio, Signed integral repre-sentations of comonotonic additive functionals, J. Math. Anal. Appl. 385 (2) (2012) 895–912.[5] S. Cerreia-Vioglio, F. Maccheroni, M. Marinacci and L. Montrucchio, Choquet integrationon Riesz spaces and dual comonotonicity, Trans. Amer. Math. Soc. 367 (2015) 8521-8542.[6] G. Choquet, Theory of capacities, Annales de l’Institut Fourier 5 (1954) 131–295.[7] A.D.R. Choudary, C.P. Niculescu,, Real Analysis on Intervals, Springer, 2014.[8] E. B. Davies, The structure and ideal theory of the pre-dual of a Banach lattice, Trans. Amer.Math. Soc. 131 (1968) 544–555.[9] Cl. Dellacherie, Quelques commentaires sur les prolongements de capacit´es, S´eminaire Prob-abilit´es V, Strasbourg, Lecture Notes in Math., vol. 191, Springer-Verlag, Berlin and NewYork, 1970.[10] D. Denneberg, Non-Additive Measure and Integral, Kluwer Academic Publisher, Dordrecht,1994.[11] J. Diestel, J.J. Uhl, Vector Measures, Math. Surveys No. 15, Amer. Math. Soc, Providence,R.I., 1977.[12] N. Dinculeanu, Vector Measures, Pergamon Press, New York, 1967.[13] L.G. Epstein, T. Wang, ”Beliefs about Beliefs” without Probabilities, Econometrica 64 (6)(1996) 1343–1373.[14] S.G. Gal and C.P. Niculescu, A Nonlinear extension of Korovkin’s theorem, Mediterr. J.Math. 17 (6) (2020). Preprint arXiv:2003.00002/February 27, 2020.[15] S.G. Gal and C.P. Niculescu, A note on the Choquet type operators. PreprintarXiv:2003.00002/February 27, 2020.[16] V. Klee, Boundedness and continuity of linear functionals. Duke Math. J., 22 (1955) 263–270.[17] M. Marinacci, L. Montrucchio, Introduction to the mathematics of ambiguity, in: I. Gilboa(Ed.), Uncertainty in Economic Theory, Routledge, New York, 2004.[18] P. Meyer-Nieberg, Banach Lattices, Springer-Verlag, Berlin, 1991.[19] I. Namioka, Partially ordered linear topological spaces, Mem. Amer. Math. Soc, No. , 1957.[20] C.P. Niculescu, Absolute continuity and weak compactness, Bull. Amer. Math. Soc. 81 (1975)1064–1066.[21] C.P. Niculescu, Absolute continuity in Banach space theory. Rev. Roum. Math. Pures Appl.24 (1979) 413–423. [22] C.P.Niculescu, An overview of absolute continuity and its applications. Internat. Ser. Numer.Math., , pp. 201–214, Birkh¨auser, Basel (2009).[23] C. P. Niculescu and O. Olteanu, A note on isotonic vector valued convex functions.http://arxiv.org/abs/2005.01088.[24] C.P. Niculescu, L.-E. Persson, Convex Functions and their Applications. A ContemporaryApproach, Second Edition, CMS Books in Mathematics vol. 23, Springer-Verlag, New York(2018).[25] G.L. O’brien, W. Vervaat, Capacities, large deviations and loglog laws. In vol. Stable Pro-cesses and Related Topics (S. Cambanis, G. Samorodnitsky, M. S. Taquu eds.), pp. 43–83,Birkh¨auser, Boston, 1991.[26] H.H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag, Berlin, 1974.[27] H.H. Schaefer and M.P. Wolff, Topological Vector Spaces, Graduate Texts in Mathematics ,Second Edition, Springer Science+Business Media New York, 1999.[28] D. Schmeidler, Integral representation without additivity, Proc. Amer. Math. Soc. 97 (1986)255–261.[29] D. Schmeidler, Subjective probability and expected utility without additivity, Econometrica57 (1989) 571–587.[30] Z. Wang, G.J. Klir, Generalized Measure Theory, Springer-Verlag, New York, 2009.[31] L. Zhou, Integral representation of continuous comonotonically additive functionals, Trans.Amer. Math. Soc. 350 (1998) 1811–1822. Department of Mathematics and Computer Science, University of Oradea, Univer-sity Street No. 1, Oradea, 410087, Romania
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