# The Köthe Dual of an Abstract Banach Lattice

Authors: E. Jiménez Fernández; M. A. Juan; E. A. Sánchez-Pérez
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We analyze a suitable definition of Köthe dual for spaces of integrable functions with respect to vector measures defined onδ-rings. This family represents a broad class of Banach lattices, and nowadays it seems to be the biggest class of spaces supported by integral structures, that is, the largest class in which an integral representation of some elements of the dual makes sense. In order to check the appropriateness of our definition, we analyze how far the coincidence of the Köthe dual with the topological dual is preserved.


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## Body

## 1. Introduction

The aim of this paper is to extend the usual notion of Köthe dual of a Banach function space over aσ-finite measure to the general class of Banach lattices. Recall that the Köthe dual or associate space of a Banach function space X over a σ-finite measure μ is defined by the set of all the functionals belonging to the topological dual X* that can be represented as integrals; that is, if φ∈X* is such a functional, there is a measurable function gφ such that f⇝∫gφfdμ equals φ(f) for all f∈X.Integration with respect to Banach space valued vector measures onδ-rings is at the basis of a fruitful representation technique for abstract lattices that has been recently developed. Certainly, the spaces L1(ν) of integrable functions and Lw1(ν) of weakly integrable functions represent a large family of Banach lattices. Nowadays it is well known that each order continuous Banach lattice can be written (isometrically and in order) as an L1(ν)-space of a certain vector measure ν on a δ-ring, and an equivalent result holds for Banach lattices with the Fatou property and some additional requirements with the spaces Lw1(ν) (see [1, Theorems 4 and 8] and [2]; see also [3, pp. 22-23]). Similar results follow also under the assumption of p-convexity of the Banach lattices, using in this case spaces of p-integrable functions (see [4–6]).Far from being a formal requirement, the use of vector measures onδ-rings is essential for our purposes. The reason is that this is the way of extending the classical representation results—that hold for Banach function spaces based on σ-finite measure spaces—to the abstract case, that include for instance representations of spaces ℓp(Γ) for an uncountable set of indexes Γ based on an integration structure. On the other hand, it must be said that the representation for the σ-finite case is completely covered by the integration structure provided by vector measures on σ-algebras. Regarding spaces of integrable functions with respect to such vector measures, several descriptions of the dual space of an L1-space (an Lp-space) of a vector measure are nowadays known (see [7–10]). Notice that, since the spaces are order continuous, these results give directly a description of the corresponding dual spaces.In this paper we propose a concrete definition of Köthe dual space for a general class of Banach lattices. Our purpose includes two natural properties that one can expect for such a space; that is,(1)
its elements must be—at least locally—integrals, that is, functionals defined by integrable functions, and(2)
the set of such elements (the Köthe dual) coincides with the whole topological dual whenever the original lattice is order continuous.As the reader may notice, we will give an easy description of such a space which covers the hole existing in the mathematical literature regarding the notion of the space of the integrals as elements of the dual space.


## 2. Preliminaries

### 2.1. Banach Lattices

We recall in this section the notions concerning Banach lattices required for a suitable reading of the paper.LetE be a Banach lattice with order ≤ and norm ∥·∥. A linear subspace F of E is called an ideal of E if x∈F,  |y|≤|x| implies y∈F. An ideal F in E is said to be super order dense in E if for every 0≤x∈E there exists an increasing sequence (xn)n⊂F so that 0≤xn↑x in E. If this property holds by means of upwards directed systems, F is said to be order dense in E. An element 0≤e∈E such that x∧e=0 implies x=0 is said to be a weak unit of E.The Banach latticeE is said to have σ-order continuous norm, or briefly, to be σ-order continuous if for every decreasing sequence (xn)n↓0 it follows that ∥xn∥↓0. Similarly, E is said to have order continuous norm, or briefly, to be order continuous if this property holds for downwards directed systems. We denote by Ea the σ-order continuous part of E, that is, the largest σ-order continuous ideal in E. We will use Ean for the order continuous part of E which in this case is the largest order continuous ideal in E.Finally, the Banach latticeE is said to have the σ-Fatou property if for every increasing sequence 0≤xn↑ in E with supn∥xn∥<∞, it follows that there exists x:=supnxn in E and ∥x∥=supn≥1∥xn∥. Again, E is said to have the Fatou property if this is the case for upwards directed systems.An operatorT:E→F between Banach lattices is said to be an order isometry if it is a linear isometry which is also an order isomorphism (i.e., T is also one to one, onto, ∥Tx∥F=∥x∥E for all x∈E and T(x∧y)=Tx∧Ty for all x,y∈E). In this case E and F are said to be order isometric.For these other issues related to Banach lattices, see for instance [11–15].


### 2.2. Integration with respect to Vector Measures onδ-Rings

We recall here the integration theory with respect to vector measures introduced by Lewis in the context ofσ-algebras [16] and extended by Masani and Niemi and Stefansson in the context of δ-rings [17–19]. We refer also to [2, 20].Givenℛ a δ-ring of subsets of an abstract set Ω, that is, a ring of sets of Ω closed under countable intersections, we can consider ℛloc the associated σ-algebra to ℛ defined by ℛloc={B⊂Ω:B∩A∈ℛ,  for every A∈ℛ}. Take the space ℳ(ℛloc) of measurable real functions on (Ω,ℛloc) and denote by 𝒮(ℛloc) and 𝒮(ℛ) the space of simple functions with support in ℛloc and ℛ, respectively.Consider alsoν:ℛ→X a vector measure, that is, a set function from the δ-ring ℛ to a real Banach space X for which ν(∪nAn)=∑n‍ν(An) in the norm topology of X for all sequences (An) of pairwise disjoint members of ℛ such that ∪nAn∈ℛ. Recall the definition of the semivariation of ν which is given by ∥ν∥:ℛloc→[0,∞] with ∥ν∥(A)=sup{|x*ν|(A):x*∈BX*} for all A∈ℛloc, where X* denotes the topological dual of X, BX* denotes the unit ball in X*, and |x*ν| is the usual variation of scalar measure x*ν:ℛ→ℝ. The semivariation of ν is monotone increasing, countably subadditive, finite on ℛ, and satisfying for all A∈ℛloc,
(1)12∥ν∥(A)≤sup{∥ν(B)∥X:B∈ℛ∩2A}≤∥ν∥(A).
We will say that a set B∈ℛloc is ν-null if ∥ν∥(B)=0 and that a property holds ν-almost everywhere (ν-a.e.) if it holds except on a ν-null set. Also, a vector measure ν:ℛ→E with values in a Banach lattice E is positive if ν(A)≥0 for all A∈ℛ.Taking into account the previous definitions, the integrability with respect to a vector measure is defined as follows. First, consider the spaceLw1(ν) of functions in ℳ(ℛloc) such that
(2)∫Ωfd|x*ν|<+∞
for all x*∈X* (i.e., which are integrable with respect to |x*ν| for all x*∈X*), where functions which are equal ν-a.e. are identified. Endow this space with the norm
(3)∥f∥ν=supx*∈BX*∫Ω|f|d|x*ν|
and the ν-a.e. pointwise order. Then Lw1(ν) is a Banach lattice containing 𝒮(ℛ) in which convergence in norm of a sequence implies ν-a.e. convergence of some subsequence (see [18, Lemma 3.13]). Even more, the space Lw1(ν) is an ideal of measurable functions in the sense that, if |f|≤|g|  ν-a.e. with f∈ℳ(ℛloc) and g∈Lw1(ν), it follows that f∈Lw1(ν).Now, take the closed idealL1(ν) in Lw1(ν) of functions which are integrable with respect toν, that is, the functions in Lw1(ν) for which there exists a vector denoted by ∫Afdν∈X, such that
(4)x*(∫Afdν)=∫Afdx*ν∀x*∈X*,
for each A∈ℛloc. Then L1(ν) is a Banach lattice with the norm and the order inherited from Lw1(ν) in which 𝒮(ℛ) is dense. The space L1(ν) is also an ideal of measurable functions, and the space L1(ν) coincides with Lw1(ν) whenever X does not contain any linear subspace isomorphic to c0 [16, Theorem 5.1].Regarding the lattice structure of these spaces, recall thatLw1(ν) has the σ-Fatou property, L1(ν) is order continuous, Lw1(ν)a=Lw1(ν)an=L1(ν), and L1(ν) is always order dense in Lw1(ν).Finally, recall the integration operatorIν:L1(ν)→X given by Iν(f)=∫Ωfdν  which is linear and continuous with ∥Iν(f)∥X≤∥f∥ν.


## 2.1. Banach Lattices

We recall in this section the notions concerning Banach lattices required for a suitable reading of the paper.LetE be a Banach lattice with order ≤ and norm ∥·∥. A linear subspace F of E is called an ideal of E if x∈F,  |y|≤|x| implies y∈F. An ideal F in E is said to be super order dense in E if for every 0≤x∈E there exists an increasing sequence (xn)n⊂F so that 0≤xn↑x in E. If this property holds by means of upwards directed systems, F is said to be order dense in E. An element 0≤e∈E such that x∧e=0 implies x=0 is said to be a weak unit of E.The Banach latticeE is said to have σ-order continuous norm, or briefly, to be σ-order continuous if for every decreasing sequence (xn)n↓0 it follows that ∥xn∥↓0. Similarly, E is said to have order continuous norm, or briefly, to be order continuous if this property holds for downwards directed systems. We denote by Ea the σ-order continuous part of E, that is, the largest σ-order continuous ideal in E. We will use Ean for the order continuous part of E which in this case is the largest order continuous ideal in E.Finally, the Banach latticeE is said to have the σ-Fatou property if for every increasing sequence 0≤xn↑ in E with supn∥xn∥<∞, it follows that there exists x:=supnxn in E and ∥x∥=supn≥1∥xn∥. Again, E is said to have the Fatou property if this is the case for upwards directed systems.An operatorT:E→F between Banach lattices is said to be an order isometry if it is a linear isometry which is also an order isomorphism (i.e., T is also one to one, onto, ∥Tx∥F=∥x∥E for all x∈E and T(x∧y)=Tx∧Ty for all x,y∈E). In this case E and F are said to be order isometric.For these other issues related to Banach lattices, see for instance [11–15].


## 2.2. Integration with respect to Vector Measures onδ-Rings

We recall here the integration theory with respect to vector measures introduced by Lewis in the context ofσ-algebras [16] and extended by Masani and Niemi and Stefansson in the context of δ-rings [17–19]. We refer also to [2, 20].Givenℛ a δ-ring of subsets of an abstract set Ω, that is, a ring of sets of Ω closed under countable intersections, we can consider ℛloc the associated σ-algebra to ℛ defined by ℛloc={B⊂Ω:B∩A∈ℛ,  for every A∈ℛ}. Take the space ℳ(ℛloc) of measurable real functions on (Ω,ℛloc) and denote by 𝒮(ℛloc) and 𝒮(ℛ) the space of simple functions with support in ℛloc and ℛ, respectively.Consider alsoν:ℛ→X a vector measure, that is, a set function from the δ-ring ℛ to a real Banach space X for which ν(∪nAn)=∑n‍ν(An) in the norm topology of X for all sequences (An) of pairwise disjoint members of ℛ such that ∪nAn∈ℛ. Recall the definition of the semivariation of ν which is given by ∥ν∥:ℛloc→[0,∞] with ∥ν∥(A)=sup{|x*ν|(A):x*∈BX*} for all A∈ℛloc, where X* denotes the topological dual of X, BX* denotes the unit ball in X*, and |x*ν| is the usual variation of scalar measure x*ν:ℛ→ℝ. The semivariation of ν is monotone increasing, countably subadditive, finite on ℛ, and satisfying for all A∈ℛloc,
(1)12∥ν∥(A)≤sup{∥ν(B)∥X:B∈ℛ∩2A}≤∥ν∥(A).
We will say that a set B∈ℛloc is ν-null if ∥ν∥(B)=0 and that a property holds ν-almost everywhere (ν-a.e.) if it holds except on a ν-null set. Also, a vector measure ν:ℛ→E with values in a Banach lattice E is positive if ν(A)≥0 for all A∈ℛ.Taking into account the previous definitions, the integrability with respect to a vector measure is defined as follows. First, consider the spaceLw1(ν) of functions in ℳ(ℛloc) such that
(2)∫Ωfd|x*ν|<+∞
for all x*∈X* (i.e., which are integrable with respect to |x*ν| for all x*∈X*), where functions which are equal ν-a.e. are identified. Endow this space with the norm
(3)∥f∥ν=supx*∈BX*∫Ω|f|d|x*ν|
and the ν-a.e. pointwise order. Then Lw1(ν) is a Banach lattice containing 𝒮(ℛ) in which convergence in norm of a sequence implies ν-a.e. convergence of some subsequence (see [18, Lemma 3.13]). Even more, the space Lw1(ν) is an ideal of measurable functions in the sense that, if |f|≤|g|  ν-a.e. with f∈ℳ(ℛloc) and g∈Lw1(ν), it follows that f∈Lw1(ν).Now, take the closed idealL1(ν) in Lw1(ν) of functions which are integrable with respect toν, that is, the functions in Lw1(ν) for which there exists a vector denoted by ∫Afdν∈X, such that
(4)x*(∫Afdν)=∫Afdx*ν∀x*∈X*,
for each A∈ℛloc. Then L1(ν) is a Banach lattice with the norm and the order inherited from Lw1(ν) in which 𝒮(ℛ) is dense. The space L1(ν) is also an ideal of measurable functions, and the space L1(ν) coincides with Lw1(ν) whenever X does not contain any linear subspace isomorphic to c0 [16, Theorem 5.1].Regarding the lattice structure of these spaces, recall thatLw1(ν) has the σ-Fatou property, L1(ν) is order continuous, Lw1(ν)a=Lw1(ν)an=L1(ν), and L1(ν) is always order dense in Lw1(ν).Finally, recall the integration operatorIν:L1(ν)→X given by Iν(f)=∫Ωfdν  which is linear and continuous with ∥Iν(f)∥X≤∥f∥ν.


## 3.   Representation of Functionals on Spaces of Integrable Functions with respect to a Vector Measure on aδ-Ring

Letℛ be a δ-ring and ν:ℛ→X be a vector measure. Following [21, Section 2], we say that a countably additive measure λ:ℛ→[0,∞] is a local control measure for ν if it satisfies the following conditions:  (1)
limA⊂B,λ(A)→0∥ν(A)∥X=0, for every B∈ℛ,(2)
everyν-null set in ℛloc is λ-null. The first condition is equivalent to ν(A)=0 whenever A∈ℛ with λ(A)=0 (see [18, Proposition 3.6]), and this happens if and only if every λ-null set in ℛ is ν-null, due to the properties of the semivariation. Consequently, the countably additive measure λ is a local control measure for ν if and only if λ has the same null sets as ν. From [21, Theorem 3.2], there always exists such a measure, so Lw1(ν) and L1(ν) are Banach functions spaces related to λ. Here, by a Banach function space related to λ, we mean a Banach space X⊂L0(λ) satisfying that, if |f|≤|g| with f∈L0(λ) and g∈X then f∈X and ∥f∥≤∥g∥, where L0(λ) denotes the vector lattice of all measurable real functions on Ω (functions which are equal λ-a.e. are identified) endowed with the λ-a.e. pointwise order.In the proof of [21, Theorem 3.2], it is shown that there exists a maximal family {Aα}α∈Δ of nonnull sets members of ℛ satisfying that Aα∩Aβ is ν-null for α≠β. Considering that, for each α∈Δ the class of sets ℛ∩Aα:={A∈ℛ:A⊂Aα}, we have that ℛ∩Aα is a δ-ring as a collection of subsets of Ω and a σ-algebra as a collection of subsets of Aα. Taking να the restriction of ν to ℛ∩Aα, it is known that there always exists a finite local control measure λα (called a Rybakov control measure and of type |x*να|) for να. For this measure and for each B∈ℛ, it is proved that λα(Aα∩B)=0 for all α∈Δ except on a countable set. Now, the set function λ:ℛ→[0,∞] given by λ(A)=∑α∈Δ‍λα(A∩Aα) is a local control measure for ν.We start our study looking for a suitable definition for the Köthe dual space ofL1(ν). Not only the existence of a local control measure λ for ν but also its particular construction will be the key of our work. Firstly, we establish a useful representation of functionals on spaces L1(ν). So, let ℛ be a δ-ring and consider a (countably additive) vector measure ν:ℛ→X. Fix a decomposition {Aα}α∈Δ of the measure space as the one explained previously and take a countable family {Aαn}n∈ℕ of elements of {Aα}α∈Δ. In order to avoid misunderstandings, we advise that, throughout all the paper, the families {Aαn}n∈ℕ are supposed to be ν-a.e. disjoint; that is, all the indexes αn are supposed to be different. Let λ be a local control measure for ν with the properties of the ones constructed by Brooks and Dinculeanu. Since by the construction λ is finite in all A∈ℛ, we have that it is σ-finite in C:=⋃n∈ℕAαn∈ℛloc.Lemma 1.
Let{Aαn}n∈ℕ in {Aα}α∈Δ and C:=⋃n∈ℕAαn as explained above. Then there is a finite measure λC:ℛloc∩C→ℝ such that, for each φ∈(L1(ν))*, there is a measurable function gφ,C∈L0(λC) such that
(5)φ(f)=∫C‍fgφ,CdλC
for all f∈L1(ν) so that supp{f}⊆C.Proof.
Letλ be a fixed local control measure for ν constructed as explained previously. For each n∈ℕ, consider the measure λn:ℛ∩C→ℝ given by λn:=λ|Aαn. Note that all these measures are finite by the construction of λ. Consider the finite measure λC:ℛloc∩C→ℝ given by λC(B):=∑n=1∞(λn(Aαn∩B)/2nλn(Aαn)), B∈ℛloc∩C.
Define also the finite measureλφ,C:ℛloc∩C→ℝ by λφ,C(B):=∑n=1∞(φ(χAαn∩B)/2n∥ν∥(Aαn)), B∈ℛloc∩C. Note that this measure is well defined and absolutely continuous with respect to λC:ℛloc∩C→ℝ. By the Radon-Nikodym theorem, there is an integrable function gφ,C0∈L1(λC) such that
(6)λφ,C(B)=∫Bgφ,C0dλC,B∈ℛloc∩C.
Thus, for each f∈𝒮(ℛ), we have that
(7)∫‍fdλφ,C=∫Bfgφ,C0dλC.
Take the function hC:=∑n=1∞2n∥ν∥(Aαn)χAαn, and note that, for such an ℛ-simple function f, we have that
(8)∫CfhCdλφ,C=∫Cf(∑n=1∞‍2n∥ν∥(Aαn)χAαn)dλφ,C=φ(fχC).
Thus, for such a function f we obtain that
(9)φ(f)=∫CfhCdλφ,C=∫CfhCgφ,C0dλC.
Write gφ,C:=hCgφ,C0. A direct calculation using the order continuity of L1(ν) gives the result for this function gφ,C.(1) Let us consider now the following construction in order to define the Köthe dual ofL1(ν). Fix again a maximal ℛ-decomposition {Aα}α∈Δ and λ:ℛ→[0,∞] a local control measure for ν. For every countable family {Aαn}n∈ℕ in {Aα}α∈Δ, we can define the following: the set C=⋃nAαn, a δ-ring ℛC of subsets of C by ℛC:={A∈ℛ:A⊆C}, and a countably additive positive measure λ^C:(ℛC)loc→[0,∞] given by λ^C(A)=∑n‍λ(A∩Aαn), A∈(ℛC)loc.Notice that the measureλ^C  is well defined; although the measure λ is defined just for elements of ℛ, using classical arguments, it can be immediately seen that this formula extends it to the whole (ℛC)loc.Note also thatℛC=ℛ∩C and remark that (ℛC)loc is computed with respect to the target space C that is, (ℛC)loc:={A⊆C:A∩B∈ℛC,B∈ℛC}. In order to clarify the relation between the measurability with respect to the different σ-algebras involved, let us establish the following notation. If B is a subset of ℛloc, we write (ℛB)locB for the σ-algebra associated with ℛB using as target space the set B. No explicit reference is made when B=Ω. The relation between (ℛC)locC and ℛloc is now done by the following lemma. The proof is straightforward.Lemma 2.
LetC=⋃n=1∞Aαn. Then (ℛC)locC=ℛloc∩C. In particular, the measurable functions in ℳ(Ω,ℛloc) when restricted to C are measurable with respect to (C,(ℛC)locC), and L0(λ) when the functions are restricted to C coincides with L0(λ|C).For a given functionf∈L1(ν), we can find a countable set {αn}n∈ℕ of different elements of Δ such that the support of f outside Cf=⋃nAαn is ν-a.e. null (and so |λ|(Ω∖Cf)=0; see [20, Theorem 3.6]), and we can consider the σ-finite measure λ^f:=λ^Cf. Therefore, for a given function f∈L1(ν), we find a set Cf, a δ-ring ℛCf, and a σ-finite measure λ^f.We describe now the adequate family of functions-like objects necessary to define the Köthe dual of an abstract spaceL1(ν). For α∈Δ, define the sets Lα0(λ) of classes of λ-a.e. equal ℛ∩Aα-measurable functions in ℝAα. Consider the Cartesian product ∏α∈Δ‍Lα0(λ). For each countable set C of subsets Aαn, consider gC an element of ∏α∈Δ‍Lα0(λ) that is a measurable function in each Aαn and 0 out of C. Take the vector space ℱ0—with the obvious sum and product—of the families G:={gC} of elements gC∈∏α∈Δ‍Lα0(λ) that are compatible; that is, for each couple of countable families C and C′, if there is an α∈Δ such that Aα belongs both to C and C′, then the respective “α-coordinates” (gC)Aα and (gC′)Aα coincide λ|Aα-a.e. A natural norm-type function to consider for the elements of this space is given by
(10)∥G∥ℱ:=sup{∫CgCf|Cdλ^C:f∈BL1(ν),CcountableunionofAα′s∫C‍},G∈ℱ0.
It can be easily seen that this expression is a norm for the space ℱ of all the elements of ℱ0 for which ∥·∥ℱ is finite. This construction allows to write the extension of the definition of Köthe dual for the setting of the abstract L1(ν) spaces as (ℱ,∥·∥ℱ); that is, we define
(11)(L1(ν))′:={G∈ℱ:gCf·f|Cf∈L1(Cf,(ℛCf)locCf,λ^f),∀f∈L1(ν)(ℛCf)locCf},
with the dual norm ∥G∥(L1(ν))′:=∥G∥ℱ.(2) In what follows, we will consider a slightly different context in order to obtain a better representation of the Köthe dual space defined as in previous. Using the fixed decomposition given by{Aα}α∈Δ, define the δ-ring ℛ0 of the elements A=⋃n=1kAn, An∈ℛ such that there is an αn∈Δ with An⊆Aαn, k∈ℕ. Note that ℛ0⊆ℛ and so we have that ℛloc⊆(ℛ0)loc. We can then define the measure λ¯:ℛ0→ℝ given by λ¯(A):=∑n=1k‍λ(An), where A∈ℛ0 is decomposed as mentioned earlier. The variation |λ¯| of this measure is well defined and allows its extension to (ℛ0)loc. Note also that ℳ(ℛloc)⊆ℳ(ℛ0)loc.Proposition 3.
For a vector measureν, consider a maximal ℛ-decomposition {Aα}α∈Δ. Then
(12)(L1(ν))′:={G∈ℱ:gC·f∈L1(Ω,(ℛ0)loc,|λ¯|),∀f∈L1(ν)(ℛ0)loc},
and the norm for this space can be computed by
(13)∥G∥ℱ:=supf∈BL1(ν)∫CfgCffd|λ¯|=sup{∫CgCfd|λ¯|:f∈BL1(ν),C=⋃n=1∞Aαn}.Proof.
It is a consequence of the following equivalences. LetC be a countable union of elements Aαn. If G∈ℱ and f∈L1(ν) such that supp{f}⊆C, the function f·gCf can be supposed to be 0 outside the set Cf, and so gCf=gCff. Then f·gCf∈L1(λ^f) if and only if
(14)∫fgCfdλ^f=∑n=1∞‍∫AαnfgCfdλ^f<∞.
Write λn for the measure λαn(·)=λ(·∩Aαn), where Aαn∈Cf, and note that
(15)∫fgCfdλ^f=∑n=1∞‍∫AαnfgCfdλn=∑n=1∞‍∫AαnfgCfd|λ¯|=∫‍fgCfd|λ¯|=∫CfgCd|λ¯|.
Therefore, integrability of fgCf|Cf with respect to λ^f is equivalent to integrability of fgC with respect to |λ¯|, and the proof is complete.Theorem 4.
For a vector measureν, (L1(ν))*=(L1(ν))′ isometrically, where (L1(ν))′ is defined from any maximal ℛ-decomposition {Aα}α∈Δ.Proof.
It is a consequence of Lemma1, and Proposition 3. Let us show the proof in two steps.
Step1. Let φ∈(L1(ν))* and C=⋃n=1∞Aαn. Then using Lemma 1 we find a function gφ,C representing the Radon-Nikodym derivative of the set function φ(·|C)—that is a measure by the order continuity of L1(ν)—in a way that
(16)φ(f)=∫Cfgφ,CdλC
for all f∈L1(ν) such that supp{f}⊆C. Therefore, computing in each C the Radon-Nikodym derivative τC of the measure λC appearing in Lemma 1, with respect to λ^C, and writing
(17)φ(f)=∫Cfgφ,CdλC=∫Cfgφ,CτCdλ^C=∫Cfgφ,CτCd|λ¯|
whenever supp{f}⊆C, we obtain a family G of the space ℱ0. Moreover, note that for such G, we have that
(18)∥G∥ℱ=sup{|∫C‍gCfdλ^C|≤∥φ∥:f∈BL1(ν),C=⋃n=1∞Aαn},
and so G∈ℱ. In fact, it can be easily seen that the identification φ⇝G is linear, continuous, and injective.
Step2.Let us show now that this identification is also surjective. We use Proposition 3. Let G be a compatible family of elements of ℱ0 such that ∥G∥ℱ<∞. For each f∈L1(ν), define
(19)φ(f)=∫CffgCfd|λ¯|,
and notice that, for each countable union C of sets Aαn, αn∈Δ, such that Cf⊆C,
(20)∫CffgCfd|λ¯|=∫CfgCd|λ¯|,
as a consequence of the compatibility of the elements of ℱ0 that define the family G. This allows to prove that φ is well defined and linear; the finiteness of ∥G∥ℱ gives also that supf∈BL1(ν)|φ(f)|≤∥G∥ℱ<∞, and so φ is an element of the dual space (L1(ν))*.
Thus, we have proved that(ℱ,∥·∥ℱ)=((L1(ν))*,∥·∥(L1(ν))*) isometrically.Remark 5.
Notice that(L1(ν))′ is a Banach lattice with the order of (L1(ν))*. It can be easily seen that it coincides with the natural order in ℱ0.Remark 6.
Note that, for the case that theδ-ring ℛ is a countable union of σ-algebras (i.e., the σ-finite case), a better representation for the Köthe dual is possible. Instead of defining it as the space of families of countable supported elements of a Cartesian product, a single function can play the role; that is,
(21)(L1(ν))′:={g∈L0(λ):g·f|Cf∈L1(Cf,(ℛCf)locCf,λ^f),∀f∈L1(ν)(ℛCf)locCf}.
This is a trivial consequence of the definition of the families, since in the σ-finite case just a function defines a family. Taking into account the definition of the measures involved in the expression above, it is clear that this representation equals
(22)(L1(ν))′:={g∈L0(λ):g·f∈L1(λ),∀f∈L1(ν)},
that is the well-known definition of Köthe dual for the σ-finite case.We extend now the definition of the Köthe dual to abstract Banach lattices. LetE be an order continuous Banach lattice. Then it is possible to define its Köthe dual by means of an integral representation (using vector measures over δ-rings) which can be described as follows. First, by [12, Proposition 1.a.9], E can be decomposed into an unconditionally direct sum of a family of mutually disjoints ideals {Eα}α∈Δ, each Eα having a weak unit. Each Eα is now an order continuous Banach lattice with a weak unit and so, from [22, Theorem 8], there exist a σ-algebra Σα of parts of an abstract set Ωα and a positive vector measure να:Σα→Eα such that the integration operator Iνα:L1(να)→Eα is an order isometry. Consider the set Ω=⋃α∈Δ{α}×Ωα and the δ-ring ℛ of subsets of Ω given by the sets ⋃α∈Δ{α}×Aα satisfying that Aα∈Σα for all α∈Δ and there exists a finite set I⊂Δ such that Aα is να-null for all α∈Δ∖I. Then, take the vector measure ν:ℛ→E defined by
(23)ν(⋃α∈Δ{α}×Aα)=∑α∈Δ‍να(Aα)
and compute the integration operator Iν:L1(ν)→E. This operator is an order isometry (see [1, Theorem 4] and [3, pp. 22-23]). Let (Ω,(ℛ0)loc,|λ¯|) be the measure space associated with ν. The definition of the Köthe dual of E is done using this integral representation adequately:
(24)E′=(L1(ν))′={G∈ℱ:gC·Iν-1(e)∈L1(Ω,(ℛ0)loc,|λ¯|),∀e∈E}.Corollary 7.
LetE be an order continuous Banach lattice. Let ν be the vector measure which appears in the representation explained previously. Then E*=E′.Let us prove now a related result regarding the Köthe dual ofLw1(ν). The representation of the Köthe dual of L1(ν) given in Proposition 3 suggests the following definition of Köthe dual space for any Banach lattice L of |λ¯|-a.e. classes of (Ω,(ℛ0)loc)-measurable functions, in particular, for Lw1(ν). We define the Köthe dual of L as
(25)L′:={G∈ℱ:gC·f∈L1(Ω,(ℛ0)loc,|λ¯|),∀f∈L},
endowed with the norm
(26)∥G∥L′:=supf∈BL∫CgCfd|λ¯|.
Let us show that, as in the finite measure case—that is, in the case of a vector measure on a σ-algebra—, the Köthe duals of L1(ν) and Lw1(ν) coincide. For a Banach function space on a σ-finite measure space, this is the well-known fact that (Xa)*=(Xa)′=X′—where Xa is the order continuous part of X—, whenever Xa is super order dense in X.Corollary 8.
Letν be a vector measure on a δ-ring. Then one has that (Lw1(ν))′=(L1(ν))′=L1(ν)*.Proof.
The inclusion(Lw1(ν))′⊆(L1(ν))′ is obvious. If G∈L1(ν)′, for each countable family C, we have that gCf|C∈L1(|λ-|) for each f∈L1(ν).
Consider the restriction ofL1(ν) to functions with support in C. For every function 0≤h∈Lw1(ν) with support in C, we have that there is an increasing sequence 0≤sn in 𝒮(ℛ) converging a.e. to h (see the proof of [2, Theorem 4.8]). Notice also that, for all n,
(27)∥sn∥L1(ν)≤∥h∥Lw1(ν)<∞.
Ifh∈Lw1(ν), we have that the restriction |gCh||C can be approximated |λ-|-a.e. using an increasing sequence sn of simple functions with support in C as previously mentioned: |gC|sn increases a.e. to |gCh| where the sequence is clearly in L1(|λ-|). The monotone convergence theorem when applied to the sequence (|gC|sn)n of L1(|λ-|) gives that gCh|C∈L1(|λ-|). Theorem 4 completes the proof.Corollary 9.
Letν be a vector measure on a δ-ring and consider L an ideal of Lw1(ν) containing L1(ν). Then L′=(L1(ν))′.LetE be a Banach lattice with the Fatou property having its σ-order continuous part Ea as an order dense subset. As explained for the order continuous case, it is again possible to define its Köthe dual using a suitable representation for E by means of a space of integrable functions. Let us describe briefly the elements we need. Consider the vector measure ν associated with Ean=Ea as in Corollary 7, then Iν:L1(ν)→Ean is an order isometry. Extend now Iν to the space Lw1(ν) as follows. For every 0≤f∈Lw1(ν), choose (φn)⊂𝒮(ℛloc) such that 0≤φn↑f. For each n and I⊂Δ finite, take ξ(n,I)=φnχ⋃α∈I{α}×Ωα∈𝒮(ℛ) and consider e=sup(n,I)Iν(ξ(n,I)) in E. Finally, define Tν(f)=e and extend to the general case. The extension Tν defined in this way is again an order isometry between Lw1(ν) and E (see [1, Theorem 8]). Consequently, the definition of the Köthe dual of E can be done using this representation; that is,
(28)E′=(Lw1(ν))′={G∈ℱ:gC·Tν-1(e)∈L1(Ω,(ℛ0)loc,|λ¯|),∀e∈E},
where (Ω,(ℛ0)loc,|λ¯|) is the measure space associated with ν.Corollary 10.
LetE be a Banach lattice with the Fatou property having its σ-order continuous part Ea as an order dense subset. Let ν be the vector measure which appears in the representation explained previously. Then (Ea)*=(Ea)′=E′.


## 4. Some Examples and Applications

(1) Consider1≤p<∞ and the space ℓp(Γ) for a noncountable index set Γ. It is an order continuous Banach lattice. Consider the δ-ring ℛΓ defined by the finite subsets of Γ and the vector measure νp:ℛΓ→ℓp(Γ). Then the space L1(νp) can be directly computed and coincides with ℓp(Γ) isometrically and in order (see example 2.2 in [20] and example 4.1 and the comments before Theorem 5.1. in [2]). Fixing the maximal decomposition of Γ given by the singletons Aγ:={γ}, γ∈Γ and after Proposition 3, we get that its Köthe dual space is given by
(29)(L1(νp))′:={G∈ℱ:gC·(ωΓ)∈ℓ1(Γ),∀(ωΓ)∈ℓp(Γ)},
since L1(Γ,(ℛΓ)loc,η) is in this case ℓ1(Γ), taking into account that the variation of the local control measure λ in this case can be taken to be the counting measure and also that obviously ℛ0=ℛΓ. Since the maximal decomposition that we are considering is given by all the singletons, we can identify each element G∈ℱ with an element of ℝΓ, and so (L1(ν))′ can be written as
(30)(L1(νp))′:={(ηγ):(ηγ)·(ωΓ)∈ℓ1(Γ),∀(ωΓ)∈ℓp(Γ)}.
This characterization and Theorem 4 give that the dual space coincides with this Köthe dual space. This description has been frequently used in examples and computations of general Banach space theory. The reader can find these arguments for instance in [23] (see Remark 2.1), where the computation of the Köthe dual is based on the fact that the related measure space is discrete.The same construction can be done for the case when the vector measure defined previously takes its values inℓ∞(Γ). Then it is known that L1(ν∞)=c0(Γ) and Lw1(ν∞)=ℓ∞(Γ). Thus the (Banach) lattice ℓ0∞(Γ) of the elements of ℓ∞(Γ) with countable support satisfies c0(Γ)⊆ℓ0∞(Γ)⊆ℓ∞(Γ). The constructive arguments given previously for c0(Γ) instead of ℓp(Γ) together with Corollaries 8 and 9 give that all the spaces appearing in the inclusion c0(Γ)⊆ℓ0∞(Γ)⊆ℓ∞(Γ) satisfy that their Köthe duals coincide and are equal to ℓ1(Γ), providing in this way a description of the Köthe dual of the relevant space ℓ0∞(Γ).(2) Although the construction of (1) regards discrete vector measures, the results in this paper can also be applied for general Banach lattices of functions involving measures that are not discrete. Let us write the Köthe dual of the space appearing in Remark 22 of [4]. Consider a family of disjoint probability spaces (Ωγ,Σγ,μγ)γ∈Γ, where Γ is an uncountable set of indexes. The δ-ring ℛ is defined by finite unions B=⋃i=1nAγi, Aγi∈Σγi. We consider the vector measure κ:ℛ→c0(Γ) given by κ(B)=∑i=1n‍μγi(Aγi)χ{γi}. As in example 2.2 of [20], it can be easily shown that the space L1(κ) is the direct sum ⨁c0(Γ)L1(μγ). Consequently, the support of an element of this space is contained in a countable union of sets Ωγ, γ∈Γ. However, Lw1(κ)=⨁ℓ∞(Γ)L1(μγ) and the functions of this space can be even strictly positive in all points of ⋃γ∈Γ‍Ωγ. This is suggested by the symbols ⨁c0(Γ) and ⨁ℓ∞(Γ) that we use: in the first case, countability of the support of each element is assumed but not in the second. Thus, we can describe an element f∈L1(κ) as a sequence (fγi)i=1∞, where each fγi∈L1(μγi). Let us consider the variation |η| of the local control measure η for κ that is defined for each element B=⋃i=1nAγi∈ℛ as η(B)=∑i=1n‍μγi(Aγi). The Köthe dual of L1(κ) is then given by
(31)(L1(κ))′:={G∈ℱ:gC·f∈L1(Ω,(ℛ0)loc,|η|),∀f∈L1(κ)},
that can also be written as
(32)(L1(κ))′={⋃i=1nAγiG∈ℱ:gC·(fγi)i=1∞∈L1(Ω,(ℛ)loc,|η|),⋃i=1nAγi⊆C,f∈L1(κ)},
where each gC of an element G of ℱ can be identified with an element of the direct sum ⨁ℓ1(Γ)L∞(μγ) in the usual manner. Note that, in this case, ℛ and ℛ0 coincide. After Theorem 4, we know that this provides a description of the dual space (L1(κ))*. By Corollary 8, this space coincides also with (Lw1(κ))′.



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