Choquet type L1-spaces of a vector capacity
Abstract
Given a set function Λ with values in a Banach space X, we construct an integration theory for scalar functions with respect to Λ by using duality on X and Choquet scalar integrals. Our construction extends the classical Bartle–Dunford–Schwartz integration for vector measures. Since just the minimal necessary conditions on Λ are required, several -spaces of integrable functions associated to Λ appear in such a way that the integration map can be defined in them. We study the properties of these spaces and how they are related. We show that the behavior of the -spaces and the integration map can be improved in the case when X is an order continuous Banach lattice, providing new tools for (non-linear) operator theory and information sciences.
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Choquet type L1-spaces of a vector capacity O. Delgado a, E.A. Sánchez Pérez b,∗ a Departamento de Matemática Aplicada I, E. T. S. de Ingeniería de Edificación, Universidad de Sevilla, Avenida de Reina Mercedes, 4 A, Sevilla 41012, Spain b Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain Abstract Given a set function with values in a Banach space X, we construct an integration theory for scalar functions with respect to by using duality on Xand Choquet scalar integrals. Our construction extends the classical Bartle–Dunford–Schwartz integration for vector measures. Since just the minimal necessary conditions on are required, several L1-spaces of integrable functions associated to appear in such a way that the integration map can be defined in them. We study the properties of these spaces and how they are related. We show that the behavior of the L1-spaces and the integration map can be improved in the case when Xis an order continuous Banach lattice, providing new tools for (non-linear) operator theory and information sciences. Keywords: Non-additive measures; Measures of information; Choquet integral; Fuzzy measure 1. Introduction Lebesgue type integration of scalar functions with respect to a vector measure was originally developed by Bartle, Dunford and Schwartz [3] in order to extend the classical Riesz representation theorem for vector valued operators. Later, an equivalent integration theory was constructed by Lewis [23] using the duality of the vector measure’s codomain. Nowadays this theory is well understood and has found many important applications in functional analysis and operator theory, among others the extension of linear operators to larger domains, see [29] and the references therein for an outlook of this topic. Similar and other applications would be desirable in the case of non-finitely additive vector valued set functions, as for instance the study of non-linear operators or the construction of some measuring tools in information science (e.g. [4,18]). The aim of this paper is to create an integration theory for vector-valued capacities—sometimes also called fuzzy capacities—and the corresponding spaces of integrable functions, which fits with the integration with respect to vector measures and allows to address this kind of applications in next works. As a first step in this process, Choquet type integrals will provide the appropriated framework and will play a fundamental role for our goal. *Corresponding author. Fax. +34 963877669. E-mail addresses: [email protected] (O. Delgado), [email protected].es (E.A. Sánchez Pérez).
Let us explain our motivation regarding possible applications of our work. Our interest in this topic is twofold. On the one hand, the relevance of our paper from the point of view of the mathematical analysis may be found in the aim of finding extensions of recent vector-measure-based developments in operator theory and harmonic analysis. Indeed, Bartle–Dunford–Schwartz integration has found an upturn of interest due to its use for computation of maximal domains of operators, with applications in several applied problems (see for example [10,12,11,29]). In this direction, similar techniques would be applied for non-linear operators using the tools developed here, where linearity—vector measures—is substituted by non-linearity—fuzzy vector capacities—. Concretely, we are thinking about fundamental non-linear operators as the Hardy–Littlewood maximal operator. On the other hand, as a continuation of our research in collaboration with information scientists, we are also interested in the theoretical development of mathematical instruments for the bibliometric analysis of the impact of the scientific research and its multiple applications—altmetrics, research assessment, big data analysis, just to mention some of them (see [6,16])—. Current research on the topic shows that two lines of research make sense. The first one stresses the fact that non-additive set functions and integration are more and more needed for constructing such measuring tools (see [4,18,21,26]). The second one highlights the fact that multiple scalar indexes—that is, vector-valued indexes—are sometimes needed for a suitable mathematical treatment of the information (see [6,17] and the references therein). Integration with respect to general set functions has a long tradition, and current contributions on this subject are hot research topics in several branches of both pure and applied mathematics. In the Introduction of [22], an excellent historical review on this integration can be found; see also [24] for an overview of non-additive monotonic measures and their properties. There are a lot of mathematical developments related to non-additive integration; the references in both papers just mentioned provide a nice selections of works regarding this subject and its broad class of applications. We must also refer to the so called pseudo analysis, that is based in the study of the properties of the pseudo measures. Different terms are used for related types of set functions, as for example fuzzy measures, capacities, non-monotonic measures, pseudo-additive measures and null-additive set functions. Starting from the early work of Choquet ([9]), Sugeno and others which may be mentioned here (see for example [27,28] and the references therein), solid integration theories have been constructed for related classes of non-additive measures (see the books [13,30,31]; also [33,35]). In particular, the Choquet integral has been deeply studied in the case of scalar functions and scalar positive capacities (e.g. [13]) and even there are studies about the L1and other function spaces associated to this integral (e.g. [7]). Regarding general (topological) spaces of scalar functions that are integrable, a lot of work has also been made in recent years, also from different—theoretical and applied—points of view. Although the literature on the subject is really broad, let us mention here the works that are more related to our developments, that are the papers [7,8] for the classical Choquet integral for scalar functions, and [32] for general pseudo measures. Regarding the vector valued case, also vector capacities with values in a Riesz space has been studied (e.g. [20]). In this case, thanks to the order structure, a Choquet type vector integral can be defined by using vector Riemann integration. Our approach will be different. Starting with a Banach space valued capacity (without any order) we will use Choquet scalar integrals and duality in a similar way as in Lewis integration to define a vector-valued integral. Several L1-spaces associated to the vector capacity appear along the paper as soon as some specific conditions are required. We study the relation among them and the role played by the Dunford and Bochner integration when the distribution functions with respect to the vector capacity are considered; we must mention here the paper [15] for a similar study in the case of vector measures, that in some sense inspired our work. Finally, we will see how much the behavior and the properties of the L1-spaces and the integration map improve when the vector capacity take values in an order continuous Banach lattice. We must remark that the novelty of our results mainly concerns the vector nature of the proposed Choquet integral, and the fact that we are directly interested in the structure of the associated spaces of integrable functions, besides the properties of the integrals. This is so because spaces of integrable functions are central in both motivations that led us to start this study. From the pure-analytic point of view, domains of (linear and non-linear) classical operators are normally function spaces, and so the maximal domain must be expected to be such an space too. From the point of view of the applications, all the functions that can be used for representing a measuring tool with some fixed requirements in information science are also elements of such an space. The contents of this paper are structured as follows. In the preliminaries we collect the basic concepts and facts on scalar capacities and function spaces that are needed. Section 3is devoted to the space L1(λ) of a scalar positive capacity λ, that is, the space of (λ-a.e. classes of) measurable functions fsuch that Iλ|f|dm <∞where mis the Lebesgue measure on the interval I=[0, ∞)and λ|f|is the distribution function of |f|with respect to λ. Although
this space has already been studied in [7], as it will be our main tool we have preferred to include here a detailed outline in which minimal conditions on λare required taking care that the identification of functions which are equal λ-a.e. is correct. In Section 4we consider a family F=(λα)of scalar positive capacities and construct two quasi-Banach function spaces associated to F, namely, the space L1(F)of the capacity F =supαλαand the space L1(F)of (F-a.e. classes of) measurable functions fsuch that supαI(λα)|f|dm <∞. A Banach space valued capacity : →X come into play in Section 5. As particular cases of the results of Section 4, we obtain the quasi-Banach function spaces L1(), w-L1 v(), L1(||||||), w-L1 qv() associated to the families (|x∗|)x∗∈BX∗and (qx∗)x∗∈BX∗, where |x∗| and qx∗are the variation and quasi-variation respectively of the scalar capacity x∗given by the composition of with the element x∗in the closed unit ball BX∗of the topological dual X∗of X. Under the appropriate conditions, in Section 6we associate to an integration map I: w-L1 qv() →X∗∗, where I(f ), x∗ =Ix∗fdm for all positive f∈w-L1 qv() and x∗∈X∗. For non-positive fwe use its positive and negative parts. Then two new sets appear in the case when Xis non-reflexive: the set L1 qv() of functions f∈ w-L1 qv() such that I(f χA) ∈j(X)for all A ∈, where jis the canonical embedding of Xinto X∗∗, and L1 v() = L1 qv() ∩w-L1 v(). We find the following containment relations L1()⊂L1 v() ⊂w-L1 v() ∩∩∩ L1(||||||)⊂L1 qv() ⊂w-L1 qv(). In the case when is a vector measure the vertical inclusions are equalities, w-L1 v() and L1 v() coincide with the spaces of weakly integrable and integrable functions (in the sense of Lewis) with respect to respectively, and Iis the integration operator with respect to . In general, since Iis not additive, we cannot know even if L1 qv() and L1 v() are vector spaces. Section 7gives conditions on Xand under which these two sets are Banach function spaces with the norms of w-L1 qv() and w-L1 v() respectively. The key is that under these conditions Xis a Banach lattice satisfying that j(X) is an ideal of X∗∗ and the map Iis increasing and subadditive on positive functions. Moreover, Iturns out to be continuous, the space w-L1 qv() coincides with the space of (-a.e. classes of) measurable functions such that |f|is Dunford integrable with respect to mand L1(||||||)with the space of functions such that |f|is Bochner integrable. We end with Section 8by showing an example of a vector capacity which satisfies all the conditions required along the paper and giving easier descriptions of its associated L1-spaces. 2. Preliminaries Throughout this paper (, ) will denote a measurable space. Let λ: →[0, ∞] be a set function satisfying that λ(∅) =0. Such a set function λwill be called a capacity. A set Z∈is λ-null if λ(A) =0for all A ∈such that A ⊂Z. Note that every measurable subset of a λ-null set is λ-null. A property holds λ-a.e. if it holds except on a λ-null set. If (An) ⊂is an increasing sequence with A =∪Anwe will write An↑A. If the sequence is decreasing with A =∩Anwe will write An↓A. The following properties of a capacity will be used in the sequel: (P1) λis increasing if λ(A) ≤λ(B)for every A, B∈such that A ⊂B. (P2) λis null-additive if λ(A ∪Z) =λ(A) for all A, Z∈with Zbeing λ-null. (P3) λis quasi-subadditive if there exists a constant K≥1 such that λ(A ∪B) ≤K(λ(A) +λ(B)) for every disjoint sets A, B∈. If K=1it is called subadditive. (P4) λis superadditive if λ(A) +λ(B) ≤λ(A ∪B) for every disjoint sets A, B∈. (P5) λis submodular if λ(A ∪B) +λ(A ∩B) ≤λ(A) +λ(B) for all A, B∈. (P6) λis continuous from below if λ(An) →λ(A) whenever An, A ∈with An↑A. (P7) λis continuous from above at ∅if λ(An) →0 whenever An∈with An↓∅and λ(A1) <∞.
Denote by L0() the space of all measurable functions f: →Rand by L0()+the positive cone of L0(), that is the set of functions in L0() which take values in [0, ∞). Write Nλfor the set of functions f∈L0() such that f=0λ-a.e. In what follows we require that ∪Znis λ-null whenever (Zn)is a sequence of λ-null sets. This fact is obtained for instance if λis continuous from below and has any of the properties (P2,3,5). Under this requirement Nλis a vector space and for fn−gn, f−g∈Nλwith fn→fλ-a.e. it follows that gn→gλ-a.e. The support of a function f∈L0() will be denoted by supp(f ). Note that f∈Nλif and only if supp(f ) is λ-null. Denote by L0(λ) the quotient space L0()/Nλ. That is, L0(λ) is the space of all real measurable functions fdefined on , where functions which are equal λ-a.e. are identified. For f∈L0() we will denote by f+and f−the positive and negative parts of frespectively, that is, f+=fχ Pfand f−=(−f)χ Nfwhere Pf={ω∈ :f(ω) >0}and Nf={ω∈ : f(ω) <0}. We will write Sfor the space of simple functions on and Sλ={ϕ∈S:λ(supp(ϕ)) <∞}. By a λ-quasi-Banach function space (briefly, λ-quasi-B.f.s.) we mean a quasi-Banach space E⊂L0(λ) with quasinorm · E, satisfying that if f∈L0(λ), g∈Eand |f| ≤|g|λ-a.e. then f∈Ewith fE≤gE. If Eis a Banach space we will refer it as a λ-Banach function space (briefly, λ-B.f.s.). In particular, a λ-quasi-B.f.s. is a quasi-Banach lattice for the λ-a.e. pointwise order. Note that all inclusions between λ-quasi-B.f.s. are continuous, see the argument given in [25, p. 2]. A λ-quasi-B.f.s. Eis σ-order continuous if for every sequence (fn) ⊂Ewith fn↓0λ-a.e. it follows that fnE↓0. In this case S∩Eis dense in E. It is said that Ehas the σ-Fatou property if for every (fn) ⊂E such that 0 ≤fn↑fλ-a.e. and supfnE<∞we have that f∈Eand fnE↑fE. Let ρbe a λ-quasi-norm function, that is a map ρ: L0() →[0, ∞] satisfying the following conditions: (C1) ρ(f) ≤ρ(g) whenever f, g∈L0() with |f| ≤|g|λ-a.e. (C2) ρ(f) =0i f and only if f=0λ-a.e. (C3) ρ(af) =|a|ρ(f)1for all a∈Rand f∈L0(). (C4) There exists K≥1such that ρ(f +g) ≤K(ρ(f) +ρ(g)) for all f, g∈L0(). Condition (C1) guarantees that ρ: L0(λ) →[0, ∞] is well defined. Then Xρ={f∈L0(λ) :ρ(f) < ∞} is a vector space and ρis a quasi-norm on it. Moreover, if f∈L0(λ), g∈Xρand |f| ≤|g|λ-a.e. then f∈Xρwith ρ(f) ≤ρ(g). The λ-quasi-norm function ρis said to have the σ-Fatou property if ρ(fn) ↑ρ(f) whenever fn, f∈L0() with 0 ≤fn↑fλ-a.e. In this case, it is known that Xρis complete. For the sake of completeness we include a proof of this fact; it can be obtained by adapting the proof of Theorem 1.6 in [5], taking into account that in this book the definition of function norm includes the σ-Fatou property (see also [7] and Sections 2 and 3 in [8]). Proposition 1. Let ρbe a λ-quasi-norm function with the σ-Fatou property. Then Xρis a λ-quasi-B.f.s. with the σ-Fatou property and ρis a quasi-norm on it. Proof. Let r>0be such that 2K=21 rwhere Kis the constant of condition (C4). Then ρn j=1 fj≤41 rn j=1 ρ(fj)r1 r(1) for all finite subset (fj)n j=1⊂Xρ, see [19, Lemma 1.1]. Consider a Cauchy sequence (fn) ⊂Xρand take (nk) strictly increasing such that ρ(fnk+1−fnk) ≤1 2k. Denote gk=fnk+1−fnk, A ={ω∈ :k≥1|gk(ω)| <∞} and g=k≥1gkχA. Since m k=1|gk|χA↑k≥1|gk|χApointwise, it follows that ρ(g) ≤ρ k≥1 |gk|χA=lim m→∞ ρm k=1 |gk|χA≤41 r k≥1 ρ(gk)r1 r<∞ 1We use the convention 0 ·∞ =∞ ·0 =0.
and so g∈Xρ. Similarly, ρ(k≥mgkχA) ≤41 r(k≥mρ(gk)r)1 rfor each m ≥1. Consider the sets AN m={ω∈ : m k=1|gk(ω)| >N}for m, N≥1 and note that χAN m≤1 Nm k=1|gk|pointwise and χAN m↑χ∪m≥1AN mas m →∞. Then, ρ(χ∪m≥1AN m)=lim m→∞ ρ(χAN m)≤1 Nlim m→∞ ρm k=1 |gk|≤41 r N k≥1 ρ(gk)r1 r. Since \A =∩ N≥1∪m≥1AN mand so ρ(χ\A) ≤ρ(χ∪m≥1AN m)for all N, taking N→∞we have that ρ(χ\A) =0. This implies that \Ais λ-null. Noting that g+fn1−fnm=k≥mgkχAλ-a.e., given ε>0it follows that ρ(g +fn1−fn)≤Kρ k≥m gkχA+ρ(fnm−fn)<ε for large enough nand m. Hence, fn→g+fn1in Xρand so Xρis complete. The σ-Fatou property of Xρfollows clearly from the σ-Fatou property of ρ.2 Let ξ: →Rbe a set function satisfying that ξ(∅) =0. Such a set function ξwill be called a real capacity. A set Z∈is ξ-null if ξ(A) =0for all A ∈such that A ⊂Z. The variation of ξis the set function |ξ|: →[0, ∞] defined by |ξ|(A) =supn i=1 |ξ(Ai)|: (Ai)n i=1⊂is a partition of A for A ∈. The quasi-variation of ξis the set function qξ: →[0, ∞] defined by qξ(A) =sup{|ξ(B)|: B∈with B⊂A} for A ∈. Note that both |ξ|and qξare increasing capacities. We also consider the capacity |ξ(·)|: →[0, ∞)given by |ξ(·)|(A) =|ξ(A)|for A ∈. The following lemma collects several properties involving the capacities |ξ|, qξ and |ξ(·)|which can be routinely checked. The reader can find more information on the variation of non-additive set functions in [30,36,37]. Lemma 2. Let ξbe a real capacity on . The following statements hold: (a) |ξ(·)| ≤qξ≤|ξ|. (b) ξ-null, |ξ|-null, qξ-null and |ξ(·)|-null sets coincide. (c) |ξ|is superadditive. (d) |ξ(·)| =qξ⇔|ξ(·)|is increasing, and qξ=|ξ|⇔qξis superadditive. (e) If (ξ) denotes any one of |ξ|, qξ, |ξ(·)|, then (e1) (a ξ) =|a| (ξ) for all a∈R, (e2) (ξ +η) ≤(ξ) +(η) for ηbeing another real capacity on , and (e3) if (ξn)is a sequence of real capacities on such that ξn(A) →ξ(A) for all A ∈then (ξ) ≤ liminf(ξn). (f) If |ξ(·)|has any of the properties (P2,3,6) then qξhas the same property. (g) If qξhas any of the properties (P2,3,6) then |ξ|has the same property. The quasi-additivity constant is preserved in (f) and (g). Note that in the case when ξtakes values in [0, ∞), from Lemma 2.(d), in general ξdoes not coincide with any of qξor |ξ|. 3. L1-space of a capacity Let λ: →[0, ∞] be an increasing capacity and denote by mthe Lebesgue measure on the interval I=[0, ∞). For f∈L0()+, the distribution function of fwith respect to λis the map λf: I→[0, ∞] defined by
λf(t) =λ({ω∈:f(ω)>t}) for t∈I. Since λis increasing we have that λfis decreasing and so measurable. Then we can consider the Lebesgue integral Iλ(f ) = I λfdm ∈[0,∞]. Remark 3. If f∈L0()+is such that λ(supp(f )) <∞then λfis bounded and so Riemann integrable in every interval [0, a]with 0 <a<∞. Then Iλ(f ) = ∞ 0 λf(t) dt is the Choquet integral of fwith respect to λcreated in [9]. A positive function ϕ∈Salways can be written in its standard representation, that is ϕ=n j=1αjχAjwhere (Aj)n j=1⊂is a finite collection of pairwise disjoint sets and 0 <α 1<α 2<···<α n. Setting α0=0, we have that λϕ= n k=1 λn j=k Ajχ[αk−1,αk) and so Iλ(ϕ) = n k=1 λn j=k Aj(αk−αk−1). (2) In particular, Iλ(χA) =λ(A) for all A ∈. Let us show some properties of the integration map Iλ: L0()+→[0, ∞] which will be needed later on. Similar result in a slightly different context can be found in recent papers on the Choquet integral. (See [22, Proposition 3.4], and in general Section 3in this paper; Section 4gives similar results for other non-additive integrals. See also the references in this paper for more information on these matters. The reader can find more information on continuous monotone set functions in [24] and the papers quoted in it.) Lemma 4. The following statements hold: (a) Iλ(af ) =aIλ(f ) for all f∈L0()+and 0 ≤a∈R. (b) Iλ(f ) ≤Iλ(g) for every f, g∈L0()+such that f≤gpointwise. (c) Iλ(f ) =Iλ(g) whenever f, g∈L0()+with f=gλ-a.e. if and only if λis null-additive. (d) Iλ(fn) ↑Iλ(f ) whenever fn, f∈L0()+with fn↑fpointwise if and only if λis continuous from below. (e) Iλ(fn) ↓0whenever fn∈L0()+with fn↓0pointwise and Iλ(f1) <∞if and only if λis continuous from above at ∅. Proof. An appropriate change of variables gives (a). Part (b) is also clear as λis increasing. (c) Suppose that Iλ(f ) =Iλ(g) for every f, g∈L0()+such that f=gλ-a.e. For A, Z∈with Zbeing λ-null, taking f=χA∪Zand g=χAwe have that f=gλ-a.e. and so λ(A ∪Z) =Iλ(f ) =Iλ(g) =λ(A). Conversely, suppose that λis null-additive and consider f, g∈L0()+with f=gexcept on a λ-null set Z. For every t∈I, denote At={ω∈ :f(ω) >t}and Bt={ω∈ :g(ω) >t}. Noting that At∩Z, Bt∩Zare λ-null and At∩\Z= Bt∩\Z, we have that λ(At)=λ(At∩\Z) =λ(Bt∩\Z) =λ(Bt). Then λf=λgpointwise and so Iλ(f ) =Iλ(g). (d) Suppose that Iλ(fn) ↑Iλ(f ) for every fn, f∈L0()+such that fn↑fpointwise. For An, A ∈with An↑A, taking fn=χAnand f=χAwe have that fn↑fpointwise and so λ(An) =Iλ(fn) ↑Iλ(f ) =λ(A). Conversely,
suppose that λis continuous from below and consider fn, f∈L0()+with fn↑fpointwise. Noting that {ω∈ : fn(ω) >t} ↑{ω∈ :f(ω) >t}for all t∈I, we have that λfn↑λfpointwise. Then, by applying the monotone convergence theorem for the Lebesgue integral with respect to m, it follows that Iλ(fn) ↑Iλ(f ). (e) Suppose that Iλ(fn) ↓0for every fn∈L0()+such that fn↓0 pointwise and Iλ(f1) <∞. For An∈ with An↓∅and λ(A1) <∞, taking fn=χAnwe have that fn↓0 pointwise and Iλ(f1) =λ(A1) <∞. So λ(An) = Iλ(fn) ↓0. Conversely, suppose that λis continuous from above at ∅and consider fn∈L0()+with fn↓0 pointwise and Iλ(f1) <∞. Note that λf1(t) <∞for all t>0as λf1is decreasing. Since {ω∈ :fn(ω) >t} ↓∅for all t>0, we have that λfn↓0m-a.e. Then, by applying the dominated convergence theorem for the Lebesgue integral with respect to m, it follows that Iλ(fn) ↓0. 2 Related results on the additivity properties of the integral under further requirements on the capacity are known; see for example Theorem 4.1 in [7], and Section 4in this work. See also [30] for a systematic approach to non-additive set functions. Consider the map · λ: L0() →[0, ∞] defined by fλ=Iλ(|f|)for f∈L0(). The following proposition gives conditions under which · λis a λ-quasi-norm function with the σ-Fatou property. Proposition 5. Suppose that λis null-additive, quasi-subadditive and continuous from below. Then, the following statements hold: (a) fλ≤gλwhenever f, g∈L0() with |f| ≤|g|λ-a.e. (b) fλ=0if and only if f=0λ-a.e. (c) af λ=|a| fλfor all a∈Rand f∈L0(). (d) f+gλ≤2K(fλ+gλ)for all f, g∈L0(), with Kbeing the constant of the quasi-subadditivity of λ. (e) fnλ↑fλwhenever fn, f∈L0() with 0 ≤fn↑fλ-a.e. Proof. (a) Let f, g∈L0() be such that |f|χA≤|g|χApointwise for some A ∈with \Abeing λ-null. Since |f|χA=|f|and |g|χA=|g|λ-a.e., from Lemma 4.(b) and (c) it follows that fλ=Iλ(|f|)=Iλ(|f|χA)≤Iλ(|g|χA)=Iλ(|g|)=gλ. (b) If f=0 except on a λ-null set Z, then {ω∈ :|f(ω)| >t} ⊂Zfor all t∈Iand so λ|f|=0 pointwise. Hence fλ=0. Conversely, suppose that fλ=0 and denote An={ω∈ :|f(ω)| >1 n}. Since 1 nχAn≤|f|pointwise we have that 0=fλ≥ 1 nχAn λ=Iλ1 nχAn=1 nλ(An) and so λ(An) =0. Note that An↑supp(f ) from which λ(supp(f )) =0as λis continuous from below. Then f=0 λ-a.e. (c) Clear from Lemma 4.(a). (d) For every f, g∈L0(), noting that {ω∈:|(f +g)(ω)|>t}⊂{ω∈:|2f(ω)|>t}∪{ω∈:|2g(ω)|>t} for all t∈Iand since λis increasing and quasi-subadditive with constant K, it follows that λ|f+g|≤K(λ2|f|+λ2|g|) pointwise. Then, f+gλ≤K(2fλ+2gλ)=2K(fλ+gλ). (e) Let fn, f∈L0() be such that 0 ≤fnχA↑fχ Apointwise for some A ∈with \Abeing λ-null. Since fnχA=|fn|and fχ A=|f|λ-a.e., from Lemma 4.(c) and (d) it follows that fnλ=Iλ(|fn|)=Iλ(fnχA)↑Iλ(f χA)=Iλ(|f|)=fλ.2 In the remainder of this section we will assume that the capacity λsatisfies the properties (P1,2,3,6). These properties guarantee the good behavior of the L1-space of λdefined as L1(λ) ={f∈L0(λ) :fλ<∞}.
Note that a simple function ϕ∈L1(λ) if and only if ϕ∈Sλ, see (2). Theorem 6. The space L1(λ) is a λ-quasi-B.f.s. with the σ-Fatou property and · λis a quasi-norm on it. Moreover, L1(λ) is σ-order continuous if and only if λis continuous from above at ∅. In this last case, Sλis dense in L1(λ). Proof. The first part follows from Propositions 1 and 5. Suppose that L1(λ) is σ-order continuous and let fn∈L0()+with fn↓0 pointwise and Iλ(f1) <∞. Since fn∈L1(λ) as fnλ≤f1λ=Iλ(f1), we have that Iλ(fn) =fnλ↓0. Then λis continuous from above at ∅by Lemma 4.(e). Conversely, suppose that λis continuous from above at ∅and let fn∈L1(λ) be such that fn↓0λ-a.e. Taking A ∈such that \Ais λ-null and fnχA↓0 pointwise and noting that Iλ(f1χA) =f1χAλ=f1λ<∞, from Lemma 4.(e) we have that fnλ=fnχAλ=Iλ(fnχA) ↓0. So L1(λ) is σ-order continuous. Noting that Sλ=S∩L1(λ) we conclude the proof. 2 Remark 7. Let μ: →[0, ∞] be a measure. Obviously μis a capacity satisfying all the properties (P1–7). For every positive ϕ∈S, from (2) it follows that Iμ(ϕ) coincides with the Lebesgue integral ϕdμ. For f∈L0()+, taking a sequence (ϕn) ⊂Ssuch that 0 ≤ϕn↑fpointwise, from Lemma 4.(d) and applying the monotone convergence theorem for the Lebesgue integral with respect to μ, it follows that Iμ(f ) =limIμ(ϕn)=lim ϕndμ = fdμ. Then our space L1(μ) is just the classical space of Lebesgue integrable functions with respect to μ. Now we want to extend the integration map Iλto non-positive functions of L1(λ). In order to obtain the Lebesgue integral in the case when λis a measure, for a function f∈L1(λ) we define Iλ(f ) =Iλ(f +) −Iλ(f −). Note that the definition is meaningful as Iλ(f +), Iλ(f −) <∞and if g∈L1(λ) is such that f=gλ-a.e., since f+=g+and f−=g−λ-a.e., from Lemma 4.(c) it follows that Iλ(f ) =Iλ(g). It is important to emphasize that the integration map Iλ: L0()+→[0, ∞] is not additive in general. In fact this is only the case when λis a measure, since finite additivity and continuity from below imply countable additivity. So the definition of Iλfor a non-positive function depend on its positive and negative parts. From Lemma 4.(a) it follows that Iλ: L1(λ) →Ris homogeneous, that is, Iλ(af ) =aIλ(f ) for all f∈L1(λ) and a∈R. Indeed, we only have to note that (af )+=af +, (af )−=af −if a≥0 and (af )+=−af −, (af )−=−af + if a<0. It is well known that Iλis subadditive on the set of positive simple functions if and only if λis submodular, see for instance [13, Ch. 6] and the references therein or [1] for a nice proof. In this case, since λis continuous from below, it follows that Iλis subadditive on all L0()+and so · λis a norm, that is L1(λ) is a λ-B.f.s. Proposition 8. If λis submodular then Iλ: L1(λ) →Ris continuous. Proof. Suppose that λis submodular and so Iλis subadditive on L0()+. Given f, g∈L0()+, applying Lemma 4.(b) we have that Iλ(f ) ≤Iλ(|f−g|+g) ≤Iλ(|f−g|)+Iλ(g), from which it follows that |Iλ(f ) −Iλ(g)| ≤Iλ(|f−g|). For f, g∈L1(λ), since |f+−g+| ≤|f−g|and |f−−g−| ≤ |f−g|, we obtain that |Iλ(f ) −Iλ(g)|≤|Iλ(f +)−Iλ(g+)|+|Iλ(f −)−Iλ(g−)| ≤Iλ(|f+−g+|)+Iλ(|f−−g−|) ≤2Iλ(|f−g|)=2f−gλ.2 We end this section by constructing a class of capacities which satisfy all the necessary properties to obtain a good L1-space.
Example 9. Let μ: →[0, ∞] be a measure and : I→Ian increasing function vanishing only at zero, with limx→0+(x) =0 and being derivable in (0, ∞)with a decreasing derivative . For instance (x) =xpwith 0 <p<1, (x) =1 −e−xor (x) =ln(1 +x). Note that (a +b−c) +(c) ≤(a) +(b) (3) for all 0 ≤c≤a, b<∞. Consider the capacity λ: →[0, ∞] given by λ(A) =(μ(A)) for all A ∈. As usual, (∞) =limx→∞ (x). Note that the λ-null and μ-null sets coincide and so L0(λ) =L0(μ). It is direct to check that λis increasing, null-additive and continuous from below. Let us see that λis submodular and so subadditive. For every A, B∈with μ(A), μ(B) <∞, since μis a measure we have that μ(A ∪B) =μ(A) +μ(B) −μ(A ∩B). Applying (3) for a=μ(A), b=μ(B) and c=μ(A ∩B), we obtain that λ(A ∪B)+λ(A ∩B) =(μ(A) +μ(B) −μ(A ∩B))+(μ(A ∩B)) ≤(μ(A)) +(μ(B)) =λ(A) +λ(B). If any of Aor Bhas infinite μmeasure then the submodular inequality is clear. Therefore L1(λ) =f∈L0(μ) :fλ= I (μ|f|)dm<∞ is a μ-B.f.s. with the σ-Fatou property and · λis a norm on it. Moreover, the integration map Iλ: L1(λ) →Ris continuous. The space L1(λ) turns out to be an intermediate space between L∞(μ) and L1(μ), that is, L∞(μ) ∩L1(μ) ⊂L1(λ) ⊂L∞(μ) +L1(μ). Let us show this fact. For every positive ϕ∈Swith standard representation ϕ=n j=1αjχAjsuch that αn≤1, since is concave (as decreases) and n k=1(αk−αk−1) =αn≤1, it follows that ϕλ= n k=1 μn j=k Aj(αk−αk−1) ≤n k=1 μn j=k Aj(αk−αk−1)= ϕdμ . Note that the concave inequality holds even if some ∪n j=kAjhas infinite μmeasure. Then, if f∈L0(μ) is such that |f| ≤1μ-a.e., taking a sequence (ϕn) ⊂Ssuch that 0 ≤ϕn↑|f|μ-a.e., we have that fλ=limϕnλ≤lim ϕndμ= |f|dμ. Hence every non-null function f∈L∞(μ) ∩L1(μ) satisfies that fλ≤f∞f1 f∞ and so L∞(μ) ∩L1(μ) ⊂L1(λ). On the other hand, since is decreasing we have that (x0)x ≤(x) (4) for all 0 ≤x<x 0<∞. Let f∈L1(λ) and denote An={ω∈ :|f(ω)| >n}. Note that there exists nsuch that μ(An) <∞as in other case (μ|f|) =(∞)which contradicts fλ<∞. Then limμ(An) =0as An↓{ω∈ :|f(ω)| =∞}. Take x0>0 such that (x0) >0 and n0such that μ(An0) <x 0. Then, since μ|f|χAn0≤μ(An0) pointwise, by (4) it follows that
(At)=(At∩\Z) =(Bt∩\Z) =(Bt). Then f=gpointwise. 2 Denote D() =f∈L0() : I |x∗|f||dm < ∞for all x∗∈X∗, that is, D() is the set of functions f∈L0() such that |f|is Dunford integrable with respect to m, see [14, Ch. II, §3]. For every f∈D() it follows that fD() =sup x∗∈BX∗ I |x∗|f||dm < ∞, see the proof of (c) ⇒(a) in Theorem 13. Remark 19. Note that in general D() is not even a vector space. If we can find M⊂BX∗satisfying that for every x∗∈BX∗there exists y∗∈Msuch that |x∗(·)| ≤|y∗(·)|and (|x∗(·)|)x∗∈Mis a uniformly quasi-subadditive family of capacities with the properties (P1,2,6), from Theorems 12 and 13 it follows that D() is a -quasi-B.f.s. with quasi-norm · D(). But in this case, qx∗=|x∗(·)|for all x∗∈Mas |x∗(·)|is increasing and Msatisfies the properties (qvN1,2,3), so w-L1 qv() =D() with equal norms. This is the reason why we do not consider this space in Section 5. For a positive function f∈D() we define the integral of fwith respect to as the element I(f ) ∈X∗∗ given by I(f ), x∗= I x∗fdm for all x∗∈X∗. That is, I(f ) is the Dunford integral of fwith respect to m. Note that |I(f ), x∗|≤x∗X∗fD(). For a general function f∈D() we cannot use its positive and negative parts for defining I(f ) as they could not be in D(), so we will consider functions in w-L1 qv(). Note that w-L1 qv() ⊂D() with fD() ≤fqv for f∈w-L1 qv(), as |x∗(·)| ≤qx∗for all x∗∈X∗. For a general function f∈w-L1 qv() we define I(f ) = I(f +) −I(f −). By Lemma 18.(b) we have that I(f ) =I(g) whenever f, g∈w-L1 qv() with f=g-a.e. So the integration map I: w-L1 qv() →X∗∗ is well defined. However, the definition of Ifor a non positive function depends on its positive and negative parts as Iis not additive in general. In fact it can be proved that the additivity is obtained only in the case when is a vector measure on the δ-ring of sets A ∈with ||||||(A) <∞. The following properties of Iwill be used later. Lemma 20. The following statements hold: (a) Iis homogeneous, that is, I(af ) =aI(f ) for all f∈w-L1 qv() and a∈R. (b) If fn, f∈w-L1 qv() are such that 0 ≤fn↑f-a.e. then I(fn) →I(f ) in the weak∗topology of X∗∗. Proof. (a) For a positive function f∈w-L1 qv() and a≥0, by making an appropriate change of variables it follows that Ix∗af dm =aIx∗fdm for all x∗∈X∗and so I(af ) =aI(f ). For general fand aonly note that (af )+=af +, (af )−=af −if a≥0 and (af )+=−af −, (af )−=−af +if a<0. (b) Let fn, f∈w-L1 qv() be such that 0 ≤fnχA↑fχ Apointwise with A ∈being such that \Ais -null. Since {ω∈:(fnχA)(ω) > t}↑{ω∈:(f χA)(ω) > t}
for all t∈I, from (1) it follows that x∗fnχA→x∗fχ Apointwise for every x∗∈X∗. On the other hand, since |x∗fnχA| ≤(qx∗)|f|pointwise, by applying the dominated convergence theorem for the Lebesgue integral with respect to mwe have that I(fnχA), x∗= I x∗fnχAdm → I x∗fχ Adm =I(f χA), x∗. Since I(fn) =I(fnχA)and I(f ) =I(f χA)the conclusion follows. 2 If ξ: →Ris a real capacity continuous from below, null-additive and qξsatisfies (P3), in which case qξalso satisfies (P2, 6), the integration map Iξ: L1(qξ) →Ris given by Iξ(f ) =Iξf+dm −Iξf−dm for all f∈L1(qξ). Remark 21. In the case when x∗∈X∗is such that x∗is null-additive and qx∗satisfies (P3), since x∗is continuous from below by (1), it follows that I(f ), x∗ =Ix∗(f ) for all f∈w-L1 qv() ⊂L1(qx∗). We have defined Iby using Dunford integration, a natural question now is what is the role of the Bochner integration in this play. For Bochner integration theory we refer to [2, Ch. 11, §8]. Denote by jthe canonical embedding of Xinto X∗∗ and B() ={f∈L0() :|f|is Bochner integrable}. Note that B() ⊂f∈L0() : I |f|Xdm < ∞⊂D(). (6) Every positive ϕ∈Swith standard representation ϕ=n j=1αjχAjsatisfies that ϕ= n k=1 (∪n j=kAj)χ[αk−1,αk)(7) is an X-step function and so Bochner integrable with respect to mwith Bochner integral Iϕdm = n k=1(∪n j=kAj)(αk−αk−1) ∈X. Hence, ϕ∈B() with I(ϕ) =j( Iϕdm). From this it follows that S⊂B(). In the case when the convergence in the property (1) is in X, i.e. is continuous from below, we find another functions in B(). Proposition 22. If is continuous from below then L1(||||||) ⊂B() and I(f ) =j I f+dm−j I f−dm for every f∈L1(||||||). Proof. Let f∈L0()+and take a sequence (ϕn) ⊂Ssuch that 0 ≤ϕn↑fpointwise. Since is continuous from below we have that f(t) −ϕn(t)X→0 for all t∈Iand so fis strongly m-measurable. If moreover f∈L1(||||||), since f(t) −ϕn(t)X≤f(t)X+ϕn(t)X≤2||||||f(t) for all t∈I, by applying the dominated convergence theorem for the Lebesgue integral with respect to mwe have that If−ϕn dm →0. This means that fis Bochner integrable with respect to mwith Bochner integral Ifdm such that Iϕndm →Ifdm in X. So, f∈B() and I(ϕn) →j( Ifdm) in X∗∗. On the other hand, from Lemma 20.(b) it follows that I(ϕn) →I(f ) weakly∗in X∗∗. So, I(f ) =j( Ifdm). For a general f∈L1(||||||), noting that |f|, f+, f−∈L1(||||||)we have the conclusion. 2
Remark 23. In the case when is continuous from below, if there exists a constant C≥1 such that |||||| ≤ C(·)X, from (6) and Proposition 22 it follows that L1(||||||)=B() =f∈L0() : I |f|Xdm < ∞. Now, two natural sets arise when we think about Pettis integration, see for instance [2, Ch. 11, § 10]. Namely, L1 qv() ={f∈w-L1 qv() :I(f χA)∈j(X)for all A∈} and L1 v() =L1 qv() ∩w-L1 v(). Since Iis not additive in general we cannot know even if L1 qv() and L1 v() are vector spaces. Our goal in the next section is to give conditions under which these spaces are closed subspaces of w-L1 qv() and w-L1 v() respectively. Of course, if Xis reflexive there is no problem as in this case L1 qv() = w-L1 qv() and L1 v() =w-L1 v(). Note that S|||||| ⊂L1 qv() and S⊂L1 v(). Even more, in the case when is continuous from below we have that L1(||||||) ⊂L1 qv() and L1() ⊂L1 v(). All these containments follows from (5), Proposition 22 and the preceding comments. Remark 24. If is a vector measure, in which case it is continuous from below and null-additive, as we have already pointed out in Remark 17, it follows that w-L1 v() =w-L1 qv() is the space of weakly integrable functions with respect to . Moreover, every f∈w-L1 qv() satisfies that I(f ) =w-fdis the weak integral of fwith respect to , see [34]. Indeed, for a positive ϕ∈S, by (7) and since x∗is a real measure for each x∗∈X∗, it follows that Ix∗ϕdm =ϕdx∗. For a positive function f∈w-L1 qv(), taking a sequence (ϕn) ⊂Ssuch that 0 ≤ϕn↑f-a.e., by Lemma 20.(b) and applying the dominated convergence theorem for the Lebesgue integral with respect to x∗we have that I(f ), x∗=limI(ϕn), x∗=lim ϕndx∗= fdx ∗=w- fd,x ∗. For a general f∈w-L1 v(), since the weak integration map with respect to is a linear operator the conclusion follows. Therefore, L1 v() =L1 qv() is the space of integrable functions with respect to and for every f∈L1 qv() we have that I(f ) =fdis the integral of fwith respect to , see [34]. 7. L1-spaces associated to a vector capacity Let Xbe an order continuous Banach lattice, that is, every order bounded increasing sequence in Xconverges in the norm of X, see [25, Proposition 1.a.8]. In this case, from [25, Theorem 1.b.16] we have that j(X) is an ideal of X∗∗, that is, if x∗∗ ∈X∗∗ and x∈Xwith |x∗∗| ≤|j(x)|then x∗∗ ∈j(X). Recall that X∗is also a Banach lattice with the order x∗≥0if and only if x∗, x ≥0for all 0 ≤x∈X. Consider a vector capacity : →Xwith the following properties: (o1) is increasing, that is, (A) ≤(B) for all A, B∈such that A ⊂B. (o2) is submodular, that is, (A ∪B) +(A ∩B) ≤(A) +(B) for all A, B∈. (o3) is continuous from below, that is, (An) →(A) in Xwhenever An, A ∈with An↑A. Note that by (o1) we have that is positive, that is, (A) ≥0for all A ∈. Then, from (o2) it follows that is subadditive, that is, (A ∪B) ≤(A) +(B) for all A, B∈. Lemma 25. The vector capacity has the properties (1,2) and the set N=x∗∈BX∗:x∗≥0satisfies (qvN1,2,3). Proof. The property (1) is obvious from (o3). For every A, Z∈with Z-null, by (o1) and since is subadditive, it follows that
(A) ≤(A ∪Z) ≤(A) +(Z) =(A) and so (2) holds. For every x∗∈BX∗it follows that |x∗| ∈Nand qx∗≤q|x∗|, as |x∗, x| ≤|x∗|, |x| for all x∈X. So, (qvN1) holds. For each 0 ≤x∗∈X∗, from (o1) we have that the real capacity x∗is increasing and takes values in [0, ∞). Then, qx∗=x∗, see Lemma 2.(d). Since by (o2) we have that x∗is submodular and so subadditive, (qvN3) holds for K=1. Moreover, by (o3) and since x∗is null-additive (as it is increasing and subadditive), it follows that (qvN2) holds. 2 From the previous lemma, all what we have seen in Section 6holds for . In particular, S⊂L1()⊂L1 v() ⊂w-L1 v() ⊂[i]L1(|x∗|) ∩∩ ∩ ∩ ∩ S|||||| ⊂L1(||||||)⊂L1 qv() ⊂w-L1 qv() ⊂[i]L1(x∗) (8) with 0 ≤x∗∈X∗. Moreover, we can add the following results. Proposition 26. The following statements hold: (a) w-L1 qv() =D() ={f∈L0() :Ix∗|f|dm <∞for all 0 ≤x∗∈X∗}and fqv =I(|f|)X∗∗ for all f∈w-L1 qv(). (b) L1(||||||) =B() ={f∈L0() :I|f|Xdm <∞} and f|||||| =I|f|Xdm for all f∈L1(||||||). (c) S|||||| =S. Proof. (a) For the first part only note that qx∗≤q|x∗|=|x∗|for all x∗∈X∗. In the second one we use that y∗Y∗=sup0≤y∈BYy∗, yfor any Banach lattice Yand 0 ≤y∗∈Y∗. Namely, for every f∈w-L1 qv() it follows that fqv =sup x∗∈N I (qx∗)|f|dm=sup x∗∈N I x∗|f|dm =sup x∗∈N I(|f|), x∗=I(|f|)X∗∗ . (b) Note that yY=sup0≤y∗∈BY∗y∗, yfor any Banach lattice Yand 0 ≤y∈Y. Then, for every A ∈we have that ||||||(A) =sup x∗∈N qx∗(A) =sup x∗∈N x∗(A) =(A)X. From this fact that the conclusion follows, see Remark 23. (c) Clear as |||||| =(·)X.2 For each 0 ≤x∗∈X∗, the capacity x∗satisfies the properties (P1,2,3,5,6) and I(f ), x∗ =Ix∗(f ) for all f∈w-L1 qv() ⊂L1(x∗), see Remark 21. This gives the following properties for I. Proposition 27. The following statements hold: (a) 0 ≤I(f ) ≤I(g) for all f, g∈w-L1 qv() such that 0 ≤f≤g-a.e. (b) I(f +g) ≤I(f ) +I(g) for all positive functions f, g∈w-L1 qv(). (c) I: w-L1 qv() →X∗∗ is continuous. Proof. (a) Let f, g∈w-L1 qv() be such that 0 ≤f≤g-a.e. For each 0 ≤x∗∈X∗, since every -null set is x∗-null, by Lemma 4.(b) and (c) we have that 0≤I(f ), x∗=Ix∗(f ) ≤Ix∗(g) =I(g), x∗. Since the canonical embedding of Xinto X∗∗ is order preserving (see [25, Proposition 1.a.2]) and so x∈Xis such that x≥0if and only if x∗, x ≥0for all 0 ≤x∗∈X∗, it follows that 0 ≤I(f ) ≤I(g).
(b) Let f, g∈w-L1 qv() be positive functions. For each 0 ≤x∗∈X∗, since x∗is submodular and so Ix∗is subadditive on L0()+(see the comments preceding Proposition 8), we have that I(f +g),x∗=Ix∗(f +g) ≤Ix∗(f ) +Ix∗(g) =I(f ), x∗+I(g), x∗=I(f ) +I(g), x∗. Then, I(f +g) ≤I(f ) +I(g). (c) For every f, g∈w-L1 qv() and 0 ≤x∗∈X∗it follows that |Ix∗(f ) −Ix∗(g)|≤2Ix∗(|f−g|), see the proof of Proposition 8. Then, using that y∗Y∗≤2 sup0≤y∈BY|y∗, y| for any Banach lattice Yand y∗∈Y∗, we have that I(f ) −I(g)X∗∗ ≤2sup x∗∈N |I(f ) −I(g), x∗| =2sup x∗∈N |Ix∗(f ) −Ix∗(g)| ≤4sup x∗∈N Ix∗(|f−g|)=4f−gqv.2 Note that the spaces w-L1 qv() and w-L1 v() are actually -B.f.s.’. Indeed, for 0 ≤x∗∈X∗, since x∗is increasing and submodular it can be proved that |x∗|is submodular, and so Ix∗and I|x∗|are subadditive on L0()+. Recalling that fqv =sup x∗∈N Ix∗(|f|)and fv=sup x∗∈N I|x∗|(|f|) for every f∈L0(), it follows that · qv and · vare norms. The properties of Ishown in Proposition 27 allow us to get the main result of this section. Theorem 28. The sets L1 qv() and L1 v() are -B.f.s.’ with norms · qv and · vrespectively. Proof. From Lemma 20.(a) it is clear that af ∈L1 qv() for all f∈L1 qv() and a∈R. If f, g∈L1 qv() are positive functions, since j(X) is an ideal of X∗∗, from Proposition 27.(b) it follows that f+g∈L1 qv(). For general f, g∈ L1 qv() and A ∈, noting that (f +g)+≤f++g+and (f +g)−≤f−+g−, by Proposition 27.(a) and (b) we have that |I((f +g)χA)|=|I((f χA+gχA)+)−I((f χA+gχA)−)| ≤I((f χA+gχA)+)+I((f χA+gχA)−) ≤I((f χA)++(gχA)+)+I((f χA)−+(gχA)−) ≤I((f χA)+)+I((gχA)+)+I((f χA)−)+I((gχA)−). Since h+=hχPhand h−=(−h)χNhfor any h ∈L0() (see the Preliminaries), it follows that the last element in the above inequality belongs to j(X) and so I((f +g)χA) ∈j(X). Hence, f+g∈L1 qv(). Therefore, L1 qv() is a linear subspace of w-L1 qv(), from which it is clear that L1 v() is a linear subspace of w-L1 v(). Since I: w-L1 qv() →X∗∗ is continuous and j(X)is closed in X∗∗, it follows that L1 qv() is closed in w-L1 qv(). From this, since w-L1 v() ⊂w-L1 qv() and all the containments between -quasi-B.f.s.’ are continuous, it follows that L1 v() is closed in w-L1 v(). Then, L1 qv() and L1 v() are Banach spaces with norms · qv and · vrespectively. On the other hand, if f∈L0() and g∈L1 qv() are such that |f| ≤|g|-a.e. then f∈w-L1 qv() with fqv ≤ gqv. Moreover, since (f χA)+, (f χA)−≤|g|-a.e. for every A ∈, by Proposition 27.(a) and (b) we have that I((f χA)+), I((f χA)−)≤I(|g|)≤I(g+)+I(g−).
Then I((f χA)+), I((f χA)−) ∈j(X)and so f∈Lqv(). Hence, L1 qv() and L1 v() are -B.f.s.’. 2 Finally we conclude this section by characterizing when the space L1 qv() is σ-order continuous. Theorem 29. The -B.f.s. L1 qv() is σ-order continuous if and only if is continuous from above at ∅, i.e. (An) → 0in Xwhenever An∈with An↓∅. Moreover, in this case Sis dense in L1 qv(). Proof. First note that S⊂L1 qv(), see (8) and Proposition 26.(c). Moreover, from Proposition 22 it follows that I(χA) =j((A)) for all A ∈. Suppose that L1 qv() is σ-order continuous. Given An∈with An↓∅, since χAn↓0 pointwise and (χAn) ⊂ L1 qv(), by using Proposition 26.(a) we have that (An)X=I(χAn)X∗∗ =χAnqv →0. Conversely suppose that is continuous from above at ∅and let (fn) ⊂L1 qv() be such that fn↓0-a.e. For each 0 ≤x∗∈X∗, since x∗is continuous from above at ∅, by Theorem 6 we have that L1(x∗) is σ-order continuous. Then, since fn↓0x∗-a.e. it follows that I(fn), x∗=Ix∗(fn)=fnx∗↓0. Hence, I(fn) ↓0in the order of X∗∗. Since j(X) is order continuous as Xis so, we have that fnqv = I(fn)X∗∗ ↓0. 2 8. Example Consider the measure space (N, P(N), c) with cbeing the counting measure. Note that L0(c) is just the space 0 of all real sequences and the c-a.e. pointwise order coincides with the coordinate order. Let Xbe a σ-order continuous saturated c-B.f.s. Recall that Xbeing saturated means that there exists a sequence x=(xn) ∈Xsuch that xn>0for all n. In this case, Xis a Köthe function space in the sense of [25, Definition 1.b.17]. In what follows we collect some properties of Xwhich will be used later in our example. Denoting the scalar product of two sequences x=(xn), y=(yn) ∈0by (x, y) =xnynprovided the sum exists, the Köthe dual of Xis given by X=y=(yn)∈0:(|x|,|y|)<∞for all x=(xn)∈X. The space Xendowed with the norm yX=supx∈BX|(x, y)|for y∈X, is a saturated c-B.f.s. The linear isometry η: X→X∗given by η(y), x =(x, y) for all y∈Xand x∈Xis surjective as Xis σ-order continuous, see [25, pg. 29]. If x∗∈X∗and y∈Xare such that x∗=η(y), then x∗≥0if and only if y≥0. Note that X⊂X. From [25, Proposition 1.b.18] it follows that xX=xX for all x∈X. The equality X=X holds with equal norms if and only if Xhas the Fatou property, see [25, pg. 30]. Also we will use the linear isometry π: X →X∗∗ defined by π(z), x∗ =z, η−1(x∗)for all z∈X and x∗∈X∗. Note that if z∈Xthen π(z) =j(z). Let us construct now a vector capacity with values in X. Consider a σ-finite measure μ: →[0, ∞] and let : I→Ibe a function as in Example 9. Then, the capacity λ: →[0, ∞] given by λ(A) =(μ(A)) for A ∈, satisfies the properties (P1,2,3,5,6) and the λ-null and μ-null sets coincide. Take a sequence of pairwise disjoint sets (n) ⊂such that =∪nand μ(n) <∞for all nand assume that e=(λ(n))n∈X. We define the vector capacity : →Xby (A) =(λ(A ∩n))n for A ∈. Note that is well defined as (A) ≤e. Remark 30. The condition e∈Xholds in many cases. For instance if μ(n) →0 then e∈c0. This happens whenever μis finite. An example of a non-finite μcould be the measure mon Iwith density h(x) =1 1+x, that is μ(A) = Ah dm. In this case, by taking n=[n −1, n) we have that μ(n) =ln(1+n n) <1 n. So e∈c0and for instance if (x) =xpwith 0 <p<1 then e∈qfor every 1 p<q<∞.
Let us see that satisfies the requirements of Section 7. Lemma 31. The vector capacity has the properties (o1,2,3) and is continuous from above at ∅. Moreover the -null and μ-null sets coincide. Proof. Note that is increasing and submodular as λis so. Given Ak, A ∈such that Ak↑A, since λis increasing and continuous from below, for each fixed nwe have that λ(Ak∩n) ↑λ(A ∩n). Then (Ak) ↑(A) pointwise and so in X, as Xis σ-order continuous. Hence, is continuous from below. Given Ak∈with Ak↓∅, since μis a measure, for each fixed nit follows that μ(Ak∩n) ↓0 and so λ(Ak∩ n) ↓0. That is, (Ak) ↓0 pointwise and so in X. Hence, is continuous from above at ∅. It is direct to check that a set Zis -null if and only if Z∩nis λ-null (equivalently, μ-null) for all n. This happens if and only if Z=∪Z∩nis μ-null. 2 Therefore, all what we have seen in Sections 5, 6and 7hold for . Moreover we can give nicer descriptions for the L1-spaces associated to in terms of λ. Proposition 32. The following statements hold: (a) L1(||||||) ={f∈L0(μ) :I(λ|f|χn)nXdm <∞} and for f∈L1(||||||)we have that f|||||| = I(λ|f|χn)nXdm. (b) w-L1 qv() ={f∈L0(μ) :(Iλ(|f|χn))n∈X}and for f∈w-L1 qv() we have that fqv = (Iλ(|f|χn))nX . Moreover, (Iλ(f χn))n∈X and I(f ) =π((Iλ(f χn))n). (c) L1 qv() ={f∈L0(μ) :(Iλ(|f|χn))n∈X}and for f∈L1 qv() we have that fqv =(Iλ(|f|χn))nX. Moreover, (Iλ(f χn))n∈Xand I(f ) =j ((Iλ(f χn))n). Proof. First note that L0() =L0(μ) as the -null and μ-null sets coincide. (a) For every f∈L0()+, since {ω∈:f(ω)>t}∩n={ω∈:(f χn)(ω) > t} for all t∈I, it follows that f=(λfχ n)n. Then, the conclusion follows from Proposition 26.(b). (b) Let 0 ≤x∗∈X∗and 0 ≤y=(yn) ∈Xbe such that x∗=η(y). For f∈L0()+we have that x∗f(t) =x∗, f(t)=ynλfχ n(t) for all t∈I. By applying the monotone convergence theorem we obtain that I x∗fdm =yn I λfχ ndm =ynIλ(f χn). From this it follows that f∈L0(μ) is such that Ix∗|f|dm <∞for all 0 ≤x∗∈X∗if and only if Iλ(|f|χn)n∈ X. Then, by Proposition 26.(a), the description of w-L1 qv() holds. Moreover, for a positive function f∈w-L1 qv() we have that I(f ), x∗= I x∗fdm =π((Iλ(f χn))n), x∗ for all 0 ≤x∗∈X∗. The same equality holds for a general x∗∈X∗by taking positive and negative parts of x∗. Hence, I(f ) =π((Iλ(f χn))n). Then, for a general f∈w-L1 qv() it follows that fqv =I(|f|)X∗∗ = Iλ(|f|χn)n X . For every nit is clear that fχ n∈L1(λ) as (Iλ(|f|χn))n∈X. Noting that (f χA)+=f+χAand (f χA)−=f−χA for all A ∈, we have that (Iλ(f χn))n=(Iλ(f +χn))n−(Iλ(f −χn))n∈X
as f+, f−∈w-L1 qv(). Moreover, I(f ) =I(f +)−I(f −) =π((Iλ(f +χn))n)−π((Iλ(f −χn))n) =π((Iλ(f χn))n). (c) For every f∈L1 qv() ⊂w-L1 qv() we have that (Iλ(f χn))n∈X and π((Iλ(f χn))n)=I(f ) ∈j(X). Since πcoincides with jon Xand πis injective it follows that (Iλ(f χn))n∈Xand I(f ) =j ((Iλ(f χn))n). Noting that |f| ∈L1 qv(), we have that (Iλ(|f|χn))n∈Xand fqv =(Iλ(|f|χn))nX =(Iλ(|f|χn))nX. On the other hand, if f∈L0(μ) is such that (Iλ(|f|χn))n∈X⊂X we have that f∈w-L1 qv(). Then, for every A ∈it follows that fχ A∈w-L1 qv() and |I(f χA)|=|I(f +χA)−I(f −χA)|≤I(f +χA)+I(f −χA) ≤2I(|f|)=2π((Iλ(|f|χn))n)=2j ((Iλ(|f|χn))n), and so I(f χA) ∈j(X). Hence, f∈L1 qv().2 Note that L1 qv() is σ-order continuous as is continuous from above at ∅, see Theorem 29. In order to give a description in terms of λfor the spaces L1(), w-L1 v() and L1 v(), we consider the variation |λ|of λdefined as in the real capacity case. Since |λ|is superadditive and λis subadditive it follows that |λ|is finitely additive. Moreover, since |λ|is continuous from below as λis so, we have that |λ|is a measure. We will need the following result. Lemma 33. Let x∗∈X∗and y=(yn) ∈Xbe such that x∗=η(y). For every A ∈it follows that |x∗|(A) =|yn||λ|(A ∩n). Proof. For every partition (Ai)m i=1⊂of A ∈we have that m i=1 |x∗(Ai)|= m i=1 n≥1 ynλ(Ai∩n)≤ m i=1 n≥1 |yn|λ(Ai∩n) = n≥1 |yn| m i=1 λ(Ai∩n)≤ n≥1 |yn||λ|(A ∩n). Hence, |x∗|(A) ≤|yn| |λ|(A ∩n). On the other hand, for a fixed ktake a partition (Bk i)mk i=1⊂of A ∩k. Since Bk i∩k=Bk iand Bk i∩n=∅whenever n = k, we have that |yk| mk i=1 λ(Bk i)= mk i=1 n≥1 ynλ(Bk i∩n)= mk i=1 |x∗(Bk i)|≤|x∗|(A ∩k). Then, |yk| |λ|(A ∩k) ≤|x∗|(A ∩k). Since |x∗|is superadditive and increasing it follows that n k=1 |yk||λ|(A ∩k)≤ n k=1 |x∗|(A ∩k)≤|x∗|(A) for all nand so |yn| |λ|(A ∩n) ≤|x∗|(A).2 Now from the previous lemma we can obtain the following conclusions. Proposition 34. The following statements hold:
(a) L1()is the space of functions f∈L0(μ) such that (|λ||f|χn(t))n∈X for all t>0and I(|λ||f|χn)nX dm <∞, and for f∈L1()we have that f=I(|λ||f|χn)nX dm. (b) w-L1 v() ={f∈L0(μ) :(I|λ|(|f|χn))n∈X}and for f∈w-L1 v() we have that fv=(I|λ|(|f|χn))nX . (c) L1 v() ={f∈L0(μ) :(Iλ(|f|χn))n∈Xand (I|λ|(|f|χn))n∈X}. Note that I|λ|(|f|χn) =n|f| d|λ|for all nas |λ|is a measure. Proof. (a) By Lemma 33 every A ∈satisfies that (A) =sup x∗∈BX∗ |x∗|(A) =sup y=(yn)∈BX|yn||λ|(A ∩n). Then (A) <∞if and only if (|λ|(A ∩n))n∈X, and in this case (A) =(|λ|(A ∩n))nX . If f∈L1(), since I|f|dm <∞and |f|is decreasing, we have that |f|(t) <∞for all t>0. So (|λ||f|χn(t))n∈X with (|λ||f|χn(t))nX =|f|(t) for all t>0. The converse inclusion follows as if f∈L0(μ) is such that (|λ|fχ n(t))n∈X for all t>0 then |f|(t) =(|λ|fχ n(t))nX for all t>0. (b) Let f∈L0(μ). For every x∗∈X∗and y=(yn) ∈Xwith x∗=η(y), from Lemma 33 and applying the monotone convergence theorem it follows that I |x∗||f|dm =|yn| I |λ||f|χndm =|yn|I|λ|(|f|χn). Then I|x∗||f|dm <∞for all x∗∈X∗if and only if (I|λ|(|f|χn))n∈X. From this the description of w-L1 v() holds. Moreover, for f∈w-L1 v() we have that fv=sup y=(yn)∈BX|yn|I|λ|(|f|χn)=(I|λ|(|f|χn))nX . (c) Clear from part (b) of this proposition and Proposition 32.(c). 2 Finally note that in the case when Xhas the Fatou property (i.e. X=X), we have that L1 qv() =w-L1 qv() and L1 v() =w-L1 v() ={f∈L0(μ) :(I|λ|(|f|χn))n∈X}. The first equality above follows as λ ≤|λ|and so Iλ(|f|χn) ≤I|λ|(|f|χn)for all n. Moreover, fv= (I|λ|(|f|χn))nXfor all f∈L1 v(). Also in this case we have that L1 v() is σ-order continuous. Indeed, given (fk) ⊂L1 v() such that fk↓0μ-a.e. (equivalently |λ|-a.e.), since L1(|λ|)is σ-order continuous as |λ|is a measure, for each fixed nit follows that I|λ|(fkχn) ↓0. Then (I|λ|(|fk|χn))n↓0 pointwise and so in X. Acknowledgements The first and second authors gratefully acknowledge the support of the Ministerio de Economía y Competitividad under projects MTM2015-65888-C4-1-P and MTM2016-77054-C2-1-P, respectively. The first author also acknowledges the support of the Junta de Andalucía (project FQM-7276), Spain. References [1] A. Alfonsi, A simple proof for the convexity of the Choquet integral, Stat. Probab. Lett. 104 (2015) 22–25. [2] C. Aliprantis, K. Border, Infinite Dimensional Analysis: A Hitchhiker’s Guide, Springer-Verlag, New York, 1999. [3] R.G. Bartle, N. Dunford, J. Schwartz, Weak compactness and vector measures, Can. J. Math. 7 (1955) 289–305. [4] G. Beliakov, S. James, Citation-based journal ranks: the use of fuzzy measures, Fuzzy Sets Syst. 167 (2011) 101–119. [5] C. Bennett, R.C. Sharpley, Interpolation of Operators, vol. 129, Academic Press, Boston, 1988. [6] J.M. Calabuig, A. Ferrer-Sapena, E.A. Sánchez-Pérez, Vector-valued impact measures and generation of specific indexes for research assessment, Scientometrics 108 (3) (2016) 1425–1443.
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