Abstract

Let m be an 2-valued (countably additive) vector measure and consider the space 𝐿2(m) of square integrable functions with respect to m. The integral with respect to m allows to define several notions of orthogonal sequence in these spaces. In this paper, we center our attention in the existence of strongly m-orthonormal sequences. Combining the use of the Kadec-Pelczyński dichotomy in the domain space and the Bessaga-Pelczyński principle in the range space, we construct a two-sided disjointification method that allows to prove several structure theorems for the spaces 𝐿1(m) and 𝐿2(m). Under certain requirements, our main result establishes that a normalized sequence in 𝐿2(m) with a weakly null sequence of integrals has a subsequence that is strongly m-orthonormal in 𝐿2(𝐦), where 𝐦 is another 2-valued vector measure that satisfies 𝐿2(m) = 𝐿2(𝐦). As an application of our technique, we give a complete characterization of when a space of integrable functions with respect to an 2-valued positive vector measure contains a lattice copy of 2.

1. Introduction

In recent years, vector measure integration has been shown to be a good framework for the analysis of the properties of Banach function spaces and the operators defined on them. In particular, it is a powerful tool for representing Banach function spaces providing an additional integration structure. For instance, every 2-convex order continuous Banach function space with weak unit can be written as a space 𝐿2(𝐦) of integrable functions with respect to a suitable vector measure 𝐦 ([1, Th. 2.4]; see also [2, Ch.3] for more information). As in the case of the Hilbert spaces of square integrable functions, sequences in 𝐿2(𝐦) that satisfy some orthogonality properties with respect to the vector valued integral become useful both for studying the geometry of the space [35] and for applications, mainly in the context of the function approximation [3, 6, 7].

In contrast to the scalar case, several notions of 𝐦-orthogonality are possible in the case of an 2-valued (countably additive) vector measure 𝐦. A sequence {𝑓𝑖}𝑖=1 in 𝐿2(𝐦) is said to be strongly 𝐦-orthonormal if the integral of the product of two different functions is 0 and the integral of each 𝑓2𝑖 is 𝑒𝑖, where {𝑒𝑖}𝑖=1 is an orthonormal sequence in 2. In this paper, we center our attention in this strong version of 𝐦-orthogonal sequence, giving a complete characterization of the spaces 𝐿2(𝐦) in which such sequences exist; actually, we will show that this fact is closely connected to the existence of lattice copies of 2 in the corresponding space 𝐿1(𝐦) that is preserved by the integration map. In order to do this, we develop a sort of two-sided Kadec-Pelczyński disjointification technique. Roughly speaking, this procedure allows to produce sequences of normalized functions in 𝐿2(𝐦)—or 𝐿1(𝐦)—which are almost disjoint and have integrals that are almost orthogonal in 2: after an isomorphic change of vector measure, we obtain our results both for the existence of strongly 𝐦-orthogonal sequences in 𝐿2(𝐦) and the existence of lattice copies of 2 in 𝐿1(𝐦).

The paper is organized as follows. After the preliminary Section 2, we analyze in Section 3 the existence of strongly orthonormal sequences in 𝐿2(𝐦), and we show that it is a genuine vector valued phenomenon, in the sense that they do not exist for scalar measures and in the case of their natural extensions, vector measures with compact integration maps. Actually, later on we prove that they do not exist for 2-valued measures with disjointly strictly singular integration maps. In the positive, we show in Theorem 3.7 that under reasonable requirements, given an 𝐦-orthonormal sequence {𝑓𝑖}𝑖=1 in 𝐿2(𝐦), it is possible to construct another vector measure 𝐦 such that(1){𝑓𝑖}𝑖=1 is a strongly 𝐦-orthonormal sequence in 𝐿2(𝐦),(2)𝐿2(𝐦)=𝐿2(𝐦).

Combining with the Kadec-Pelczyński dichotomy, the requirement on the sequence of being 𝐦-strongly orthogonal can be relaxed to being weakly null (Corollary 3.9), obtaining in this case a sequence of functions satisfying (1) and (2) that approximates a subsequence of the original one. Some examples and direct consequences of this result are also given. Finally, Section 4 is devoted to show some applications in the context of the structure theory of Banach function spaces, focusing our attention in Banach function lattices that are represented as spaces of square integrable functions with respect to an 2-valued vector measure and are not Hilbert spaces. For the case of positive measures, we show that the existence of strongly 𝐦-orthonormal sequences is equivalent of the existence of lattice copies of 4 in 𝐿2(𝐦) and lattice copies of 2 in 𝐿1(𝐦) (Proposition 4.3 and Theorem 4.5). The translation of these results for the space 𝐿1(𝐦) gives the following result on its structure that can also be written in terms of the integration map (Theorem 4.7): the space 𝐿1(𝐦) contains a normalized weakly null sequence if and only if it has a reflexive sublattice if and only if it contains a lattice copy of 2.

2. Preliminaries

In this section, we introduce several definitions and comments regarding the spaces 𝐿2(𝐦). We refer to [8] for definitions and basic results on vector measures. Let 𝑋 be a Banach space. We will denote by 𝐵𝑋 the unit ball of 𝑋, that is 𝐵𝑋={𝑥𝑋𝑥𝑋1}. 𝑋 will be the topological dual of 𝑋. Let Σ be a 𝜎-algebra on a nonempty set Ω. Throughout the paper 𝐦Σ𝑋 will be a countably additive vector measure. The semivariation of 𝐦 is the nonnegative function 𝐦 whose value on a set 𝐴Σ is given by 𝐦(𝐴)=sup{|𝐦,𝑥|(𝐴)𝑥𝐵𝑋}. The variation |𝐦| of 𝐦 on a measurable set 𝐴 is given by |𝐦|(𝐴)=sup𝐵Π𝐦(𝐵) for 𝐴Σ, where the supremum is computed over all finite measurable partitions Π of 𝐴. The variation |𝐦| is a monotone countably additive function on Σ—a positive scalar measure—, while the semivariation 𝐦 is a monotone subadditive function on Σ, and for each 𝐴Σ we have that 𝐦(𝐴)|𝐦|(𝐴).

For each element 𝑥𝑋, the formula 𝐦,𝑥(𝐴)=𝐦(𝐴),𝑥, 𝐴Σ, defines a (countably additive) scalar measure. As usual, we say that a sequence of Σ-measurable functions converges |𝐦,𝑥|-almost everywhere if it converges pointwise in a set 𝐴Σ such that |𝐦,𝑥|(Ω𝐴)=0. A sequence converges 𝐦-almost everywhere if it converges in a set 𝐴 that satisfies that the semivariation of 𝐦 in Ω𝐴 is 0.

Let 𝜇 be a positive scalar measure. The measure 𝐦 is absolutely continuous with respect to 𝜇 if lim𝜇(𝐴)0𝐦(𝐴)=0; in this case we write 𝐦𝜇 and we say that 𝜇 is a control measure for 𝐦. Countably additive vector measures always have control measures. It is known that there exists always an element 𝑥𝑋 such that 𝐦|𝐦,𝑥|. We call such a scalar measure a Rybakov measure for 𝐦 (see [8, Ch.IX,2] ). If |𝐦,𝑥| is a Rybakov measure for 𝐦, a sequence of Σ-measurable functions converges 𝐦-almost everywhere if and only if it converges |𝐦,𝑥|-almost everywhere.

A Σ-measurable function 𝑓 is integrable with respect to 𝐦 if it is integrable with respect to each scalar measure 𝐦,𝑥, and for every 𝐴Σ there is an element 𝐴𝑓𝑑𝐦𝑋 such that 𝐴𝑓𝑑𝐦,𝑥=𝐴𝑓𝑑𝐦,𝑥 for every 𝑥𝑋. The set of all the (classes of 𝐦—a.e. equal) 𝐦—integrable functions 𝐿1(𝐦) defines an order continuous Banach function space with weak unit 𝜒Ω—in the sense of [9, p.28]—over any Rybakov measure for 𝐦 that is endowed with the norm 𝑓𝐿1(𝐦)=sup𝑥𝐵𝑋Ω||𝑓||𝑑||𝐦,𝑥||,𝑓𝐿1(𝐦).(2.1) The reader can find the definitions and fundamental results concerning the space 𝐿1(𝐦) in [2, 1012].

The spaces 𝐿𝑝(𝐦) are defined extending the definition above in a natural manner [1, 2, 13]. They are 𝑝-convex order continuous Banach function spaces with weak unit 𝜒Ω over any Rybakov measure, with the norm 𝑓𝐿𝑝(𝐦)||𝑓||=𝑝𝐿1/𝑝1(𝐦),𝑓𝐿𝑝(𝐦).(2.2) It is also known that if 1/𝑝+1/𝑞=1, 𝑓1𝐿𝑝(𝐦) and 𝑓2𝐿𝑞(𝐦), then the pointwise product 𝑓1𝑓2 belongs to 𝐿1(𝐦) (see for instance [2, Ch.3]). We will consider the integration operator 𝐼𝐦𝐿1(𝐦)𝑋 associated to the vector measure 𝐦, that is defined by 𝐼𝐦(𝑓)=Ω𝑓𝑑𝐦, 𝑓𝐿1(𝐦). The properties of the integration map have been largely studied in several recent papers (see [2, 1417] and the references therein). If 𝑖,𝑗 are indexes of a set 𝐼, we write 𝛿𝑖,𝑗 for the Kronecker delta as usual. A sequence {𝑓𝑖}𝑖=1 in 𝐿2(𝐦) is called 𝐦-orthogonal if Ω𝑓𝑖𝑓𝑗𝑑𝐦=𝛿𝑖,𝑗𝑘𝑖 for positive constants 𝑘𝑖. If 𝑓𝑖𝐿2(𝐦)=1 for all 𝑖, it is called 𝐦-orthonormal. The properties of these sequences have been recently analyzed in a series of papers, and some applications have been already developed (see [37, 18]). In this paper, we deal with the following more restrictive version of orthogonality for 2-valued measures.

Definition 2.1. Let 𝐦Σ2 be a vector measure. We say that {𝑓𝑖}𝑖=1𝐿2(𝐦) is a strongly 𝐦-orthogonal sequence if Ω𝑓𝑖𝑓𝑗𝑑𝐦=𝛿𝑖,𝑗𝑒𝑖𝑘𝑖 for an orthonormal sequence {𝑒𝑖}𝑖=1 in 2 and for 𝑘𝑖>0. If 𝑘𝑖=1 for every 𝑖, we say that it is a strongly 𝐦-orthonormal sequence.

We need some elements on Banach-lattice-valued vector measures; in particular, on 2-valued measures when the order in 2 is considered. If 𝑋 is a Banach lattice, we say that a vector measure 𝐦Σ𝑋 is positive if 𝐦(𝐴)0 for all 𝐴Σ. Note that if 𝐦 is positive and 𝑥 is a positive element of the Banach lattice 𝑋, then the measure 𝐦,𝑥 coincides with its variation. We refer to [2, 9, 19] for general questions concerning Banach lattices and Banach function spaces. An operator between Banach lattices is called strictly singular if no restriction to an infinite dimensional subspace give an isomorphism, and 2-singular if this happens for subspaces isomorphic to 2. It is called disjointly strictly singular if no restriction to the closed linear span of a disjoint sequence is an isomorphism.

We use standard Banach spaces notation. A sequence {𝑥𝑛}𝑛=1 in a Banach space 𝑋 is called a Schauder basis of 𝑋 (or simply a basis) if for every 𝑥𝑋 there exists a unique sequence of scalars {𝛼𝑛}𝑛=1 such that 𝑥=lim𝑛𝑛𝑘=1𝛼𝑘𝑥𝑘. A sequence {𝑥𝑛}𝑛=1 which is a Schauder basis of its closed span is called a basic sequence. Let {𝑥𝑛}𝑛=1 and {𝑦𝑛}𝑛=1 be two basis for the Banach spaces 𝑋 and 𝑌, respectively. Then {𝑥𝑛}𝑛=1 and {𝑦𝑛}𝑛=1 are equivalent if and only if there is an isomorphism between 𝑋 and 𝑌 that carries each 𝑥𝑛 to 𝑦𝑛.

Let {𝑥𝑖}𝑖=1 be a basic sequence of a Banach space 𝑋 and take two sequences of positive integers {𝑝𝑖}𝑖=1 and {𝑞𝑖}𝑖=1 satisfying that 𝑝𝑖<𝑞𝑖<𝑝𝑖+1 for every 𝑖. A block basic sequence {𝑦𝑖}𝑖=1 associated to {𝑥𝑖}𝑖=1 is a sequence of vectors of 𝑋 defined as finite linear combinations as 𝑦𝑖=𝑞𝑖𝑘=𝑝𝑖𝛼𝑖,𝑘𝑥𝑘, where 𝛼𝑖,𝑘 are real numbers. We refer to [20, Ch.V] for the definition of block basic sequence and to [9, 20] for general questions concerning Schauder basis.

3. Strongly m-Orthogonal Sequences in 𝐿2(𝐦)

This section is devoted to show how to construct strongly 𝐦-orthonormal sequences in 𝐿2(𝐦). Let us start with an example of the kind of sequences that we are interested in.

Example 3.1 3.1. Let ([0,),Σ,𝜇) be Lebesgue measure space (Figure 1). Let 𝑟𝑘(𝑥)=sign{sin(2𝑘1𝑥)} be the Rademacher function of period 2𝜋 defined at the interval 𝐸𝑘=[2(𝑘1)𝜋,2𝑘𝜋], 𝑘. Consider the vector measure 𝐦Σ2 given by 𝐦(𝐴)=𝑘=1(1/2𝑘)(𝐴𝐸𝑘𝑟𝑘𝑑𝜇)𝑒𝑘2, 𝐴Σ.
Note that if 𝑓𝐿2(𝐦) then [0,)𝑓𝑑𝐦=((1/2𝑘)𝐸𝑘𝑓𝑟𝑘𝑑𝜇)𝑘2. Consider the sequence of functions 𝑓1(𝑥)=sin(𝑥)𝜒[𝜋,2𝜋]𝑓(𝑥)2𝜒(𝑥)=sin(2𝑥)[0,2𝜋](𝑥)+𝜒[(7/2)𝜋,4𝜋]𝑓(𝑥)3(𝜒𝑥)=sin(4𝑥)[0,4𝜋](𝑥)+𝜒[(23/4)𝜋,6𝜋](𝑓𝑥)𝑛2(𝑥)=sin𝑛1𝑥𝜒[0,2(𝑛1)𝜋](𝑥)+𝜒[(2𝑛2/2𝑛)𝜋,2𝑛𝜋](𝑥),𝑛2.(3.1)
This sequence can be used to define a strongly 𝐦-orthogonal sequence, since [0,)𝑓2𝑛𝑑𝐦,𝑒𝑛1=2𝑛𝐸𝑛𝑓2𝑛𝑟𝑛𝜋𝑑𝜇=22𝑛,[0,)𝑓2𝑛𝑑𝐦,𝑒𝑘1=2𝑘𝐸𝑘𝑓2𝑛𝑟𝑘𝑑𝜇=0,𝑘𝑛,[0,)𝑓𝑛𝑓𝑚𝑑𝐦,𝑒𝑘1=2𝑘𝐸𝑘𝑓𝑛𝑓𝑚𝑟𝑘𝑑𝜇=0,for𝑛𝑚and𝑘.(3.2) If we define the functions of the sequence {𝐹𝑛}𝑛=1 by 𝐹𝑛(𝑥)=(2𝑛/𝜋)𝑓𝑛(𝑥), we get [0,)𝐹2𝑛𝑑𝐦=𝑒𝑛,𝑛[0,)𝐹𝑛𝐹𝑘𝑑𝐦=0,𝑛,𝑘,𝑛𝑘.(3.3)

The starting point of our analysis is the Bessaga-Pelczyński selection principle. It establishes that if {𝑥𝑘}𝑘=1 is a basis of the Banach space 𝑋 and {𝑥𝑘}𝑘=1 is the sequence of coefficient functionals, if we take a normalized sequence {𝑦𝑛}𝑛=1 such that lim𝑛𝑦𝑛,𝑥𝑘=0, then {𝑦𝑛}𝑛=1 admits a basic subsequence that is equivalent to a block basic sequence of {𝑥𝑛}𝑛=1 (see for instance Theorem 3 in [20, 21], Ch.V). We adapt this result for sequences of square integrable functions {𝑓𝑛}𝑛=1 in order to identify when the sequence of integrals {Ω𝑓2𝑛𝑑𝐦}𝑛=12 is a basic sequence in 2. The following result is a direct consequence of the principle mentioned above. Notice that the first requirement in Proposition 3.2 is obviously satisfied in the case of 𝐦-orthonormal sequences. The second condition constitutes the key of the problem.

Proposition 3.2. Let 𝐦Σ2 be a vector measure, and consider the canonical basis {𝑒𝑘}𝑘=1 of 2. Let {𝑓𝑛}𝑛=1 be a sequence in 𝐿2(𝐦). If there is an 𝜀>0 such that the sequence {Ω𝑓2𝑛𝑑𝐦}𝑛=1 satisfies(1)inf𝑛Ω𝑓2𝑛𝑑𝐦2=𝜀>0, (2)lim𝑛Ω𝑓2𝑛𝑑𝐦,𝑒𝑘=0,𝑘, then {Ω𝑓2𝑛𝑑𝐦}𝑛=1 has a subsequence which is a basic sequence. Moreover, it is equivalent to a block basic sequence of {𝑒𝑘}𝑘=1.

Let us highlight with an example the geometrical meaning of the requirements above. This is, in a sense, the canonical situation involving disjointness.

Example 3.3 ([5, Ex.10]). Let (Ω,Σ,𝜇) be a probability measure space. Let us consider the following vector measure 𝐦Σ2, 𝐦(𝐴)=𝑖=1𝜇𝐴𝐴𝑖𝑒𝑖2,𝐴Σ,(3.4) where {𝐴𝑖}𝑖=1 is a disjoint measurable partition of Ω, with 𝜇(𝐴𝑖)0 for all 𝑖. Notice that Ω𝑓2𝑑𝐦=𝑖=1(𝐴𝑖𝑓2𝑑𝜇)𝑒𝑖2 for all 𝑓𝐿2(𝐦). Consider a sequence of norm one functions {𝑓𝑛}𝑛=1 in 𝐿2(𝐦) such that 𝑓𝑛=𝑓𝑛𝜒𝐴𝑛 for all 𝑛. For every 𝑘, the following equalities hold: lim𝑛Ω𝑓2𝑛𝑑𝐦,𝑒𝑘=lim𝑛𝑖=1𝐴𝑖𝑓2𝑛𝑒𝑑𝜇𝑖,𝑒𝑘=lim𝑛𝐴𝑘𝑓2𝑛𝑑𝜇=0,Ω𝑓2𝑛𝑑𝐦2=1𝑛.(3.5)
Therefore, condition (2.2) of Proposition 3.2 is fulfilled in this example: the role of disjointness is clear.

In what follows, we show that if the integration operator is compact then there are no strongly 𝐦-orthonormal sequences. In particular, this shows that the existence of such sequences is a pure vector measure phenomenon, since the integration map is obviously compact when the measure is scalar. Compactness of the integration map is nowadays well characterized (see [2, Ch.3] and the references therein); it is a strong property, in the sense that it implies that the space 𝐿1(𝐦) is lattice isomorphic to the 𝐿1 space of the variation of 𝐦, that is a scalar measure (see [2, Prop.3.48]). We need the next formal requirement for the elements of the sequence {𝑓𝑛}𝑛=1. We say that a function 𝑓𝐿2(𝐦) is normed by the integral if 𝑓𝐿2(𝐦)=Ω𝑓2𝑑𝐦1/2. This happens for instance when the vector measure 𝐦 is positive (see [22] or [2, Lemma 3.13]), since in this case the norm can be computed using the formula 𝑓𝐿1(𝐦)=Ω|𝑓|𝑑𝐦 for all 𝑓𝐿1(𝐦). We impose this requirement for the aim of clarity; some of the results could be adapted using a convenient renorming process in order to avoid it.

Remark 3.4. Let 𝐦Σ2 be a countably additive vector measure. If there exists a strongly 𝐦-orthonormal sequence in 𝐿2(𝐦) which elements are normed by the integrals, then the integration operator 𝐼𝐦𝐿1(𝐦)2 is not compact. To see this, let {𝑓𝑖}𝑖=1𝐿2(𝐦) be a strongly 𝐦-orthonormal sequence in 𝐿2(𝐦) and consider a orthonormal sequence {𝑒𝑖}𝑖=1. Then Ω𝑓𝑖𝑓𝑗𝑑𝐦=𝛿𝑖,𝑗𝑒𝑖, an thus Ω𝑓2𝑖𝑑𝐦=𝑒𝑖=𝐼𝐦𝑓2𝑖.(3.6) Therefore {𝑓2𝑖}𝑖=1𝐵𝐿1(𝐦), and so the sequence {𝐼𝐦(𝑓2𝑖)}𝑖=1 that satisfies that {𝐼𝐦(𝑓2𝑖)}𝑖=1𝐼𝐦(𝐵𝐿1(𝐦))𝐼𝐦(𝐵𝐿1(𝐦)) does not admit any convergent subsequence. It follows that 𝐼𝐦(𝐵𝐿1(𝐦)) is not compact and so, 𝐼𝐦(𝐵𝐿1(𝐦)) is not relatively compact. This allows to conclude that 𝐼𝐦 is not compact.

Theorem 3.7 below gives a necessary condition—and, in a sense, also a sufficient condition—for the existence of strongly orthonormal sequences in a space of functions 𝐿2(𝐦) starting from a given 𝐦-orthonormal sequence. The existence of such 𝐦-orthonormal sequences is always assured: just consider a sequence of normalized disjoint functions in 𝐿2(𝐦). The following result is an application of the Kadec-Pelczyński disjointification procedure for order continuous Banach function spaces—also called the Kadec-Pelczyński dichotomy, see Theorem 4.1 in [23, 24]—, in the following version, that can be found in [25] (see the comments after Proposition 1.1). Let 𝑋(𝜇) be an order continuous Banach function space over a finite measure 𝜇 with a weak unit (this implies 𝑋(𝜇)𝐿1(𝜇)). Consider a normalized sequence {𝑥𝑛}𝑛=1 in 𝑋(𝜇). Then (1)either {𝑥𝑛𝐿1(𝜇)}𝑛=1 is bounded away from zero,(2)or there exists a subsequence {𝑥𝑛𝑘}𝑘=1 and a disjoint sequence {𝑧𝑘}𝑘=1 in 𝑋(𝜇) such that 𝑧𝑘𝑥𝑛𝑘𝑘0.

Recall that the space 𝐿2(𝐦) is an order continuous Banach function space over any Rybakov (finite) measure 𝜇=|𝐦,𝑥0| for 𝐦.

Proposition 3.5. Let {𝑔𝑛}𝑛=1 be a normalized sequence in 𝐿2(𝐦). Suppose that there exists a Rybakov measure 𝜇=|𝐦,𝑥0| for 𝐦 such that {𝑔𝑛𝐿1(𝜇)}𝑛=1 is not bounded away from zero. Then there are a subsequence {𝑔𝑛𝑘}𝑘=1 of {𝑔𝑛}𝑛=1 and an 𝐦-orthonormal sequence {𝑓𝑘}𝑘=1 such that 𝑔𝑛𝑘𝑓𝑘𝐿2(𝐦)𝑘0.

Proof. By the criterion given above, there is a subsequence {𝑔𝑛𝑘}𝑘=1 of {𝑔𝑛}𝑛=1 and a disjoint sequence {𝑓𝑘}𝑘=1 such that 𝑔𝑛𝑘𝑓𝑘𝐿2(𝐦)𝑘0. Consider the sequence given by the functions 𝑓𝑘=𝑓𝑘/𝑓𝑘. Then 𝑔𝑛𝑘𝑓𝑘𝐿2(𝐦)𝑘0. Since Ω𝑓𝑘𝑓𝑗𝑑𝐦=0 for every 𝑘𝑗 due to the fact that they are disjoint, we obtain the result.

Although the existence of a strongly 𝐦-orthonormal subsequence of an 𝐦-orthogonal sequence cannot be assured in general, we show in what follows that under the adequate requirements it is possible to find a vector measure 𝐦 satisfying that 𝐿2(𝐦)=𝐿2(𝐦) and with respect to which there is a subsequence that is strongly 𝐦-orthonormal. We use the following lemma, which proof is elementary (see Lemma 3.27 in [2]).

Lemma 3.6. Let 𝐦Σ2 be a vector measure. Let 𝜑2𝐻 be an isomorphism, where 𝐻 is a separable Hilbert space, and consider the vector measure 𝐦=𝜑𝐦. Then the spaces 𝐿2(𝐦) and 𝐿2(𝐦) are isomorphic, and for every 𝑓𝐿2(𝐦), Ω𝑓2𝑑𝐦=𝜑(Ω𝑓2𝑑𝐦).

Theorem 3.7. Let us consider a vector measure 𝐦Σ2 and an 𝐦-orthonormal sequence {𝑓𝑛}𝑛=1 of functions in 𝐿2(𝐦) that are normed by the integrals. Let {𝑒𝑛}𝑛=1 be the canonical basis of 2. If lim𝑛Ω𝑓2𝑛𝑑𝐦,𝑒𝑘=0 for every 𝑘, then there exists a subsequence {𝑓𝑛𝑘}𝑘=1 of {𝑓𝑛}𝑛=1 and a vector measure 𝐦Σ2 such that {𝑓𝑛𝑘}𝑘=1 is strongly 𝐦-orthonormal.
Moreover, 𝐦 can be chosen to be as 𝐦=𝜙𝐦 for some Banach space isomorphism 𝜙 from 2 onto 2, and so 𝐿2(𝐦)=𝐿2(𝐦).

Proof. Consider an 𝐦-orthonormal sequence {𝑓𝑛}𝑛=1 in 𝐿2(𝐦) and the sequence of integrals {Ω𝑓2𝑛𝑑𝐦}𝑛=1. As an application of Proposition 3.2, we get a subsequence {Ω𝑓2𝑛𝑘𝑑𝐦}𝑘=1 that is equivalent to a block basic sequence {𝑒𝑛𝑘}𝑘=1 of the canonical basis of 2. Recall that according to the notation given in Section 2, 𝛼𝑖,𝑗 are the constants that appear in the definition of the block basic sequence. Associated to this sequence, there is an isomorphism 𝜑𝐴=𝑒span𝑛𝑘2𝜑𝐵=spanΩ𝑓2𝑛𝑘𝑑𝐦2(3.7) such that 𝜑(𝑒𝑛𝑘)=Ω𝑓2𝑛𝑘𝑑𝐦, 𝑘.
We can suppose without loss of generality that the elements of the sequence {𝑒𝑛𝑘}𝑘=1 have norm one. To see this, it is enough to consider the following inequalities. First note that there are positive constants 𝑄 and 𝐾 such that for every 𝑛, 𝑄=𝑄Ω𝑓2𝑛𝑘𝑑𝐦𝑒𝑛𝑘𝐾Ω𝑓2𝑛𝑘𝑑𝐦=𝐾 as a consequence of the existence of the isomorphism 𝜑. Let {𝜆𝑖}𝑖=1 be a sequence of real numbers. Then 𝑖=1𝜆𝑖𝑒𝑖𝑒𝑖22=𝑖=1||𝜆𝑖||2𝑞𝑖𝑗=𝑝𝑖||𝛼𝑖,𝑗||2𝑒𝑖2=𝑞𝑖=1𝑖𝑗=𝑝𝑖||𝜆𝑖||2||𝛼𝑖,𝑗||2𝑒𝑖2.(3.8) The existence of an upper and a lower bound for the real numbers 𝑒𝑖 given above provides the equivalence between this quantity and 𝑖=1𝜆𝑖𝑒𝑖22 for each sequence of real numbers {𝜆𝑖}𝑖=1.
Since each closed subspace of a Hilbert space is complemented, there is a subspace 𝐵𝑐 such that 2=𝐵2𝐵𝑐 isometrically, where this direct sum space is considered as a Hilbert space (with the adequate Hilbert space norm). We write 𝑃𝐵 and 𝑃𝐵𝑐 for the corresponding projections. Let us consider the linear map 𝜙=𝜑1Id𝐵2𝐵𝑐𝜙𝐴2𝐵𝑐, where Id𝐵𝑐𝐵𝑐 is the identity map.
Note that 𝐻=𝐴2𝐵𝑐 is a Hilbert space with the scalar product 𝑥+𝑦,𝑥+𝑦𝐻=𝑥,𝑥𝐻+𝑦,𝑦𝐻,𝑥+𝑦,𝑥+𝑦𝐴2𝐵𝑐,(3.9) that can be identified with 2. Obviously, 𝜙 is an isomorphism. Let us consider now the vector measure 𝐦=𝜙𝐦Σ𝐦2𝜙𝐴2𝐵𝑐. By Lemma 3.6, 𝐿2(𝐦)=𝐿2(𝜙𝐦)=𝐿2(𝐦). Let us show that {𝑓𝑛𝑘}𝑘=1 is a strongly 𝐦-orthonormal sequence. We consider the orthonormal sequence {(𝑒𝑛𝑘,0)}𝑘=1 in 𝐻. The first condition in the definition of strongly orthonormal sequence is fulfilled, since Ω𝑓2𝑛𝑘𝑑𝐦=Ω𝑓2𝑛𝑘𝑃𝑑(𝜙𝐦)=𝜙𝐵Ω𝑓2𝑛𝑘𝑑𝐦,𝑃𝐵𝑐Ω𝑓2𝑛𝑘=𝜑𝑑𝐦1Ω𝑓2𝑛𝑘=𝑒𝑑𝐦,0𝑛𝑘,,0(3.10) for every 𝑘. The second one is given by the following calculations. For 𝑘𝑙, Ω𝑓𝑛𝑘𝑓𝑛𝑙𝑑𝐦=Ω𝑓𝑛𝑘𝑓𝑛𝑙=𝜙𝑑(𝜙𝐦)Ω𝑓𝑛𝑘𝑓𝑛𝑙=𝑑𝐦𝜙(0)=0,(3.11) since 𝜙 is continuous and {𝑓𝑛𝑘}𝑘=1 is an 𝐦-orthonormal sequence. Thus we get Ω𝑓𝑛𝑘𝑓𝑛𝑙𝑑𝐦=0. This proves the theorem.

Remark 3.8. In a certain sense, the converse of Theorem 3.7 also holds. Take as the vector measure 𝐦 the measure 𝐦 itself with values in 2 and consider the canonical basis {𝑒𝑛}𝑛=1. Clearly, every strongly 𝐦-orthonormal sequence is 𝐦-orthonormal and satisfies the condition lim𝑛Ω𝑓2𝑛𝑑𝐦,𝑒𝑘=0, since Ω𝑓2𝑛𝑑𝐦,𝑒𝑘=𝑒𝑛,𝑒𝑘=0,𝑘𝑛.(3.12)

Corollary 3.9. Let 𝐦Σ2 be a countably additive vector measure. Let {𝑔𝑛}𝑛=1 be a normalized sequence of functions in 𝐿2(𝐦) that are normed by the integrals. Suppose that there exists a Rybakov measure 𝜇=|𝐦,𝑥0| for 𝐦 such that {𝑔𝑛𝐿1(𝜇)}𝑛=1 is not bounded away from zero. If lim𝑛Ω𝑔2𝑛𝑑𝐦,𝑒𝑘=0 for every 𝑘, then there is a (disjoint) sequence {𝑓𝑘}𝑘=1 such that (1)lim𝑘𝑔𝑛𝑘𝑓𝑘𝐿2(𝐦)=0 for a given subsequence {𝑔𝑛𝑘}𝑘=1 of {𝑔𝑛}𝑛=1, and(2)it is strongly 𝐦-orthonormal for a certain Hilbert space valued vector measure 𝐦 defined as in Theorem 3.7 that satisfies that 𝐿2(𝐦)=𝐿2(𝐦).

This is a direct consequence of Proposition 3.5 and Theorem 3.7. For the proof, just take into account the continuity of the integration map and the fact that the elements of the sequence {𝑔𝑛}𝑛=1 are normed by the integrals.

4. Applications: Copies of 2 in 𝐿1(𝐦) That Are Preserved by the Integration Map

One of the consequences of the results of the previous section is that the existence of strongly 𝐦-orthonormal sequences in 𝐿2(𝐦) is closely related to the existence of lattice copies of 2 in 𝐿1(𝐦). In this section, we show how to apply our arguments for finding some information on the structure of the spaces 𝐿1(𝐦) and the properties of the associated integration map.

Our motivation has its roots in the general problem of finding subspaces of Banach function spaces that are isomorphic to 2. It is well known that in general these copies are related to weakly null normalized sequences; the arguments that prove this relation go back to the Kadec-Pelczyński dichotomy and have been applied largely in the study of strictly singular embeddings between Banach function spaces [25, 26]. In some relevant classes of Banach function spaces—𝐿𝑝-spaces, Lorentz spaces, Orlicz spaces, and general rearrangement invariant (r.i.) spaces—these copies are related to subspaces generated by Rademacher-type sequences (see [2730] and the references therein). For instance, Corollary 2 in [27] states that for a r.i. Banach function space 𝐸 on [0,1], if the norms on 𝐸 and 𝐿1 are equivalent on some infinite dimensional subspace of 𝐸, then the Rademacher functions span a copy of 2 in 𝐸. However, our construction generates copies of 2 that are essentially different. Actually, they are defined by positive or even disjoint functions, and so the copies of 2 that our results produce allow to conclude that if there is a normalized sequence of positive functions with a weakly null sequence of integrals, the integration map is neither disjointly strictly singular nor 2-singular.

On the other hand, it is well known that strongly orthonormal sequences—that are called 𝜆-orthonormal systems in Definition 2 of [5]—define isometric copies of 4 in spaces 𝐿2(𝐦) of a positive vector measure 𝐦 (see Propositions 8 and 11 in [5]). In particular, this makes clear that the existence of these sequences imply that 𝐿2(𝐦) is not a Hilbert space, and so 𝐿1(𝐦) is not an 𝐿1-space. However, there is a big class of Banach function spaces that can be represented as 𝐿1(𝐦) of an 2-valued positive vector measure 𝐦 (see for instance Example 10 in [5] or Example 8 in [4]). The 𝐿1(𝐦)-spaces associated to such vector measures are sometimes called -sums of 𝐿1-spaces. In Section 4 of [31], a first attempt to study 2-convex subspaces—the natural extension of 2-copies in this setting—of -sums of 𝐿1-spaces was made. Also, a first analysis of the question of when 𝐿1(𝐦) is a Hilbert space—based on the behavior of specific sequences too—was made in [11, Section  4]. In what follows, we provide more information on the existence of copies of 2 in spaces 𝐿1(𝐦) of a positive vector measure, and the closely related problem of the existence of 4 in 𝐿2(𝐦). After that, some contributions to the analysis to the study of strictly singular integration maps are given. Recently, a new considerable effort has been made in order to find the links between the belonging of the integration map 𝐼𝐦 to a particular class of operators and the structure properties of the space 𝐿1(𝐦). For integration maps belonging to relevant operator ideals, this has been done in [16, 17, 32] (see also [2, Ch.5] and the references therein). For geometric and order properties of the integration map—mainly concavity and positive p-summing type properties—, we refer to [33, 34] and [2, Ch.6].

For the aim of clarity, in this section we deal with positive vector measures, that—as we said in the previous section—satisfy that all the elements of the spaces 𝐿2(𝐦) are normed by the integrals. In this case, it can be shown that there is an easy characterization of strongly 𝐦-orthonormal sequences, which simplifies the arguments.

Remark 4.1. Suppose that a vector measure satisfies that the set 𝑃={𝑥2𝐦,𝑥0} separates the points of 2 and assume that for a given sequence {𝑓𝑛}𝑛=1, Ω𝑓2𝑛𝑑𝐦,𝑥Ω𝑓2𝑘𝑑𝐦,𝑥=0 for every 𝑛,𝑘 such that 𝑛𝑘 and 𝑥𝑃. Then Ω𝑓𝑛𝑓𝑘𝑑𝐦=0 for every 𝑛𝑘. This is a direct consequence of Hölder's inequality and the integrability with respect to 𝐦 of all the functions involved. For the particular case of positive vector measures, the standard basis {𝑒𝑛}𝑛=1 of 2 plays the role of 𝑃; this means that the requirement Ω𝑓2𝑛𝑑𝐦=𝑒𝑛 for all 𝑛 automatically implies that {𝑓𝑛}𝑛=1 is a strongly m-orthonormal sequence.

Lemma 4.2. Let 𝐦Σ2 be a positive vector measure, and suppose that the bounded sequence {𝑔𝑛}𝑛=1 in 𝐿2(𝐦) satisfies that lim𝑛Ω𝑔2𝑛𝑑𝐦,𝑒𝑘=0 for all 𝑘. Then there is a Rybakov measure 𝜇 for 𝐦 such that lim𝑛𝑔𝑛𝐿1(𝜇)=0.

Proof. Take for instance the sequence 𝑥0={(1/2)𝑛/2}𝑛=12. Since 𝐦 is positive, the measure 𝜇=𝐦,𝑥0 is positive and defines a Rybakov measure for 𝐦. Since Ω𝑔2𝑛𝑑𝐦,𝑥0=𝑔𝑛2𝐿2(𝜇) for all 𝑛 and the requirement on {Ω𝑔2𝑛𝑑𝐦}𝑛=1 imply that it is weakly null, we obtain by Hölder's inequality that lim𝑛𝑔𝑛𝐿1(𝜇)lim𝑛𝑔𝑛𝐿2(𝜇)𝐦1/2=0.

Proposition 4.3. Let 𝐦Σ2 be a positive (countably additive) vector measure. Let {𝑔𝑛}𝑛=1 be a normalized sequence in 𝐿2(𝐦) such that for every 𝑘, lim𝑛Ω𝑔2𝑛𝑑𝐦,𝑒𝑘=0. Then 𝐿2(𝐦) contains a lattice copy of 4. In particular, there is a subsequence {𝑔𝑛𝑘}𝑘=1 of {𝑔𝑛}𝑛=1 that is equivalent to the unit vector basis of 4.

Proof. By Lemma 4.2, we can use Corollary 3.9 to produce a disjoint sequence {𝑓𝑘}𝑘=1 in 𝐿2(𝐦) that approximates a subsequence {𝑔𝑛𝑘}𝑘=1 of {𝑔𝑛}𝑛=1 and is strongly 𝐦-orthogonal. The same computations that can be found in the proof of Proposition 8 in [5] show that for finite sums 𝑁𝑘=1𝛼𝑘𝑓𝑘, the norm in 𝐿2(𝐦) is equivalent to the norm of {𝛼𝑘}𝑁𝑘=1 in 4. Consequently, the closure of these finite sums in 𝐿2(𝐦) provides a copy of 4. The disjointness of {𝑓𝑘}𝑘=1 implies that in fact it is a lattice copy. Note also that {𝑓𝑘}𝑘=1 is equivalent to {𝑔𝑛𝑘}𝑘=1 and so to the unit vector basis of 4.

As a direct consequence, we obtain that for a positive vector measure 𝐦, the existence of a normalized sequence of functions such that the sequence of square integrals is weakly null implies that 𝐿2(𝐦) cannot be a Hilbert space. On the other hand, if the integration map is compact, then 𝐿1(𝐦)=𝐿1(|𝐦|) isomorphically (see Proposition 3.48 in [2]), and thus 𝐿2(𝐦) is (isomorphic to) a Hilbert space. Notice first the following obvious consequence of this fact: 𝐿1(𝐦)is isomorphic to an 𝐿1(𝜇)-space of a finite measure 𝜇 if and only if there is a positive 2-valued vector vector measure 𝐦0  such that 𝐿1(𝐦)=𝐿1(𝐦0)  such that the integration map is compact; the converse statement is proved by considering the vector measure 𝑛(𝐴)=𝜇(𝐴)𝑒12, 𝐴Σ. However, as the next example shows, there are spaces 𝐿2(𝐦) for positive 2-valued vector measures with noncompact integration map that are Hilbert spaces. We will find in Corollary 4.6 that this conclusion—𝐿1(𝐦) not being an 𝐿1-space, and so 𝐿2(𝐦) not to be a Hilbert space—can be extended to the case of strictly singular integration maps.

Example 4.4. (1) An 2-valued measure such that 𝐿1(𝐦) is a Hilbert space and the integration map is not compact. Consider the Hilbert space 𝐿2[0,1] and a orthonormal basis 𝑆 for it. Consider the associated isomorphism 𝜙𝑆𝐿2[0,1]2 that carries each function to the 2-summable sequence of its fourier coefficients. Take the vector measure 𝑚𝑆Σ2 given by 𝑚𝑆(𝐴)=𝜙𝑆(𝜒𝐴) for each Lebesgue measurable set 𝐴Σ. Then 𝐿1(𝑚𝑆)=𝐿2[0,1], although 𝐼𝑚𝑆 is in fact an isomorphism.
(2) A positive 2-valued measure with noncompact integration map such that 𝐿1(𝐦) is a Hilbert space. Consider a vector measure 𝐦 as in Example 3.3 and define the positive measure 𝑛Σ2 by 𝑛(𝐴)=𝜇(𝐴)𝑒1+𝐦(𝐴), 𝐴Σ. A direct computation shows that the norm in 𝐿1(𝑛) is equivalent to the one in 𝐿1(𝜇). Then 𝐿2(𝑛)=𝐿2(𝜇) isomorphically, and 𝐼𝑛Σ2 is clearly noncompact.

Next result shows the consequences on the structure of 𝐿1(𝐦) of our arguments about the existence of strongly orthonormal sequences in 𝐿2(𝐦).

Theorem 4.5. Let 𝐦Σ2 be a positive (countably additive) vector measure. Let {𝑔𝑛}𝑛=1 be a normalized sequence in 𝐿2(𝐦) such that for every 𝑘, lim𝑛Ω𝑔2𝑛𝑑𝐦,𝑒𝑘=0 for all 𝑘. Then there is a subsequence {𝑔𝑛𝑘}𝑘=1 such that {𝑔2𝑛𝑘}𝑘=1 generates an isomorphic copy of 2 in 𝐿1(𝐦) that is preserved by the integration map. Moreover, there is a normalized disjoint sequence {𝑓𝑘}𝑘=1 that is equivalent to the previous one and {𝑓2𝑘}𝑘=1 gives a lattice copy of 2 in 𝐿1(𝐦) that is preserved by 𝐼𝐦.

Proof. By Corollary 3.9 and Lemma 4.2, there is a (normalized) disjoint sequence {𝑓𝑘}𝑘=1 in 𝐿2(𝐦) that is equivalent to a subsequence {𝑔𝑛𝑘}𝑘=1 of {𝑔𝑛}𝑛=1. Let us prove directly that {𝑔2𝑛𝑘}𝑘=1 generates an isomorphic copy of 2 in 𝐿1(𝐦). Let 𝐦=𝜙𝐦 the vector measure given in Corollary 3.9 and let 𝐾 be the norm of 𝜙1. Since lim𝑘𝑔𝑛𝑘𝑓𝑘𝐿2(𝐦)=0 for every 𝜀>0 there is a subsequence of the one above (that we denote as the previous one) that satisfies that 𝑛𝑘=1Ω|𝑔𝑛𝑘𝑓𝑘|2𝑑𝐦1/2<𝜀2.(4.1) Fix an 𝜀>0. We have that, by Hölder inequality, 𝑛𝑘=1Ω||𝑔2𝑛𝑘𝑓2𝑘||𝑑𝐦221/2=𝑛𝑘=1Ω||𝑔𝑛𝑘𝑓𝑘||||𝑔𝑛𝑘+𝑓𝑘||𝑑𝐦221/2𝑛𝑘=1Ω||𝑔𝑛𝑘𝑓𝑘||2𝑑𝐦Ω||𝑔𝑛𝑘+𝑓𝑘||2𝑑𝐦1/22𝑛𝑘=1Ω||𝑔𝑛𝑘𝑓𝑘||2𝑑𝐦1/2<𝜀.(4.2) This means that Ω|||||𝑛𝑘=1𝛼𝑘𝑔2𝑛𝑘|||||𝑑𝐦2Ω|||||𝑛𝑘=1𝛼𝑘𝑔2𝑛𝑘𝑓2𝑘|||||𝑑𝐦2+Ω𝑛𝑘=1||𝛼𝑘||𝑓2𝑘𝑑𝐦2𝑛𝑘=1𝛼2𝑘1/2𝜀+𝐾Ω𝑛𝑘=1||𝛼𝑘||𝑓2𝑘𝑑𝐦2(𝜀+𝐾)𝑛𝑘=1𝛼2𝑘1/2.(4.3) Similar computations give the converse inequality. The construction of 𝐦 and the disjointness of the functions of the sequence {𝑓2𝑘}𝑘=1 give the last statement.

Corollary 4.6. Let 𝐦Σ2 be a positive (countably additive) vector measure. The following assertions are equivalent. (1)There is a normalized sequence in 𝐿2(𝐦) satisfying that lim𝑛Ω𝑔2𝑛𝑑𝐦,𝑒𝑘=0 for all the elements of the canonical basis {𝑒𝑘}𝑘=1 of 2. (2)There is an 2-valued vector measure 𝐦=𝜙𝐦---𝜙 an isomorphism—such that 𝐿2(𝐦)=𝐿2(𝐦) and there is a disjoint sequence in 𝐿2(𝐦) that is strongly 𝐦-orthonormal.(3)There is a subspace 𝑆𝐿1(𝐦) that is fixed by the integration map 𝐼𝐦 which satisfies that there are positive functions 𝑛𝑆 such that {Ω𝑛𝑑𝐦}𝑛=1 is an orthonormal basis for 𝐼𝐦(𝑆).(4)There is an 2-valued vector measure 𝐦 defined as 𝐦=𝜙𝐦---𝜙 an isomorphism—such that 𝐿1(𝐦)=𝐿1(𝐦) and a subspace 𝑆 of 𝐿1(𝐦) such that the restriction of 𝐼𝐦 to 𝑆 is a lattice isomorphism in 2.

Proof. (1)(2) is a direct consequence of Theorem 4.5. For (2)(3), just notice that the strong 𝐦-orthogonality of a disjoint sequence {𝑔𝑛}𝑛=1 implies that {𝑔2𝑛}𝑛=1 gives a lattice copy of 2 preserved by the integration map 𝐼𝐦. Since 𝜙1𝐦=𝐦, we obtain that {𝜙1(Ω𝑔2𝑛𝑑𝐦)}𝑛=1={Ω𝑔2𝑛𝑑𝐦}𝑛=1 is a basis for 2.
(3)(1). There is a bounded sequence {𝑛}=1 in 𝐿1(𝐦) such that Ω𝑛𝑑𝐦=𝑎𝑛, where 𝑎𝑛 is a orthonormal basis of closure of the subspace 𝐴=span{Ω𝑛𝑑𝐦𝑛}, and an isomorphism 𝜙 from 𝐴 to 2 such that 𝜙(Ω𝑛𝑑𝐦)=𝑒𝑛. By composing with 𝜙 the integration map, the copy of 2 that is fixed by 𝐼𝐦 can be considered in such a way that 𝜙(Ω𝑔2𝑛𝑑𝐦)=𝑒𝑛. Consequently, lim𝑛𝜙(Ω𝑛𝑑𝐦),𝑒𝑘=0 for all 𝑘, and so lim𝑛𝑒𝑘,Ω𝑛𝑑𝐦=0 for all 𝑘. It is enough to take 𝑔𝑛=𝑛1/2.
(3)(4) is obvious.
(4)(1). Take the normalized sequence of positive functions {𝑛}𝑛=1 in 𝑆 such that {Ω𝑛𝑑𝜙𝐦}𝑛=1 is equivalent to the standard basis of 2, and define 𝑔𝑛=𝑛1/2. The weak to weak continuity of 𝜙 gives the result.

We have shown that the existence of lattice copies of 4 in 𝐿2(𝐦) is directly connected with the existence of lattice copies of 2 in 𝐿1(𝐦). Thus, and summarizing the results in this section, we finish the paper with a complete characterization of this property for 𝐿1(𝐦) of a positive 2-valued vector measure 𝐦.

Theorem 4.7. The following assertions for a positive vector measure 𝐦Σ2 are equivalent. (1)𝐿1(𝐦) contains a lattice copy of 2. (2)𝐿1(𝐦) has a reflexive infinite dimensional sublattice. (3)𝐿1(𝐦) has a relatively weakly compact normalized sequence of disjoint functions. (4)𝐿1(𝐦) contains a weakly null normalized sequence. (5)There is a vector measure 𝐦 defined by 𝐦=𝜙𝐦 such that integration map 𝐼𝐦 fixes a copy of 2. (6)There is a vector measure 𝐦 defined as 𝐦=𝜙𝐦 that is not disjointly strictly singular.

Proof. (1)(2)(3) are obvious. For (3)(4), just take into account that disjoint normalized sequences in weakly compact sets of Banach lattices are weakly null (see for instance the proof of Proposition 3.6.7 in [19]).
(4)(5). Take a weakly null normalized sequence {𝑔𝑗}𝑛𝑗=1 in 𝐿1(𝐦). By (the arguments used in) Lemma 4.2 we can find a Rybakov measure 𝜇 for 𝐦 such that 𝑔𝑗𝐿1(𝜇)0. Now we use the same arguments that lead to Theorem 3.7 and Corollary 3.9; by the Kadec-Pelczyński dichotomy, there exists a subsequence {𝑔𝑗𝑙}𝑙=1 of {𝑔𝑗}𝑗=1 and a disjoint sequence {𝑧𝑙}𝑙=1 in the unit sphere of 𝐿1(𝐦) such that lim𝑙𝑔𝑗𝑙𝑧𝑙=0. Notice that {𝑧𝑙}𝑙=1 also converges weakly to 0, so by taking a subsequence and after restricting the supports of the functions 𝑧𝑙 and renorming if necessary, we obtain a normalized weakly null positive disjoint sequence {𝑣𝑘}𝑘=1. This gives the copy of 2 that is fixed by the integration map associated to a vector measure 𝐦=𝜙𝐦 satisfying 𝐿1(𝐦)=𝐿1(𝐦) by 𝐼𝐦(𝑣𝑘)=𝑒𝑘 for the canonical basis {𝑒𝑘}𝑘=1 of 2. Finally, (5)(6) and (6)(1) are evident.

Acknowledgment

The support of the Ministerio de Economía y Competitividad, under Project no.MTM2009-14483-C02-02 (Spain) is gratefully acknowledged.