Reflexivity of function spaces associated to a σ-finite vector measure

Reflexivity of function spaces associated to a σ-finite vector measure ✩ Ricardo del Campo a, Antonio Fernández b,∗, Fernando Mayoral b, Francisco Naranjo b a Dpto. Matemática Aplicada I, Universidad de Sevilla, EUITA, Ctra. de Utrera Km. 1, 41013 Sevilla, Spain b Dpto. Matemática Aplicada II, Escuela Técnica Superior de Ingeniería, Camino de los Descubrimientos, s/n, 41092 Sevilla, Spain a b s t r a c t Keywords: Reflexivity Integrable function Vector measure δ-ring Locally strongly additive measure For a vector measure νdefined on a δ-ring with values in a Banach space and 1 < p <∞, we characterize the reflexivity of the different spaces Lp w(ν) (integrability in the weak sense), Lp(ν) (integrability in the strong sense), and Lp(ν) (integrability in the Choquet sense). 1. Introduction From the point of view of functional analysis the second most desired property of infinite spaces is reflexivity (the first one is completeness) and probably it is the most used in applications due to the weak compactness of its unit ball. Typical undergraduate examples of reflexive Banach spaces are Lebesgue Lp-spaces (1 <p <∞) of a positive σ-finite measure. The corresponding scalar function spaces associated to a vector measure νwith values into a Banach space have been long studied (see, for example [18] and most of the references in the present paper). In this new context the things are really different. There appear several Lp-spaces associated to the vector measure: in the weak sense Lp w(ν), in the strong sense Lp(ν), and finally, integrability in the Choquet sense Lp(ν), of course for 1 ≤p <∞. These kind of spaces are, in general, different from each other and nonreflexive, even for 1 <p <∞. When the vector measure νis ✩This research has been partially supported by La Junta de Andalucía. The authors acknowledge the support of the Ministerio de Economía y Competitividad of Spain and FEDER, under the project MTM2012-36740-C02-01. *Corresponding author. E-mail addresses: rcamp[email protected] (R. del Campo), [email protected] (A. Fernández), may[email protected] (F. Mayoral), [email protected] (F. Naranjo). . defined on a σ-algebra the reflexivity of Lp w(ν)and Lp(ν)has been studied in [12]. Roughly speaking, for 1 <p <∞, the space Lp w(ν), or equivalently Lp(ν), is reflexive if and only if they coincide. Also in the same context of a vector measure defined on a σ-algebra, the reflexivity of Lp(ν)is obtained as a byproduct of a general result about interpolation from [10], namely, Lp(ν)is always reflexive for all 1 <p <∞. In the present paper we study the reflexivity of these spaces when the measure is defined on a δ-ring, a more general (but natural) structure than a σ-algebra. In this new context we can say that a similar result characterizing reflexivity of Lp w(ν)and Lp(ν)holds (see Theorem 2.3). Nevertheless Lp(ν)is not always reflexive. We characterize those vector measures for which Lp(ν)is reflexive as the locally strongly additive vector measures (see Theorem 4.3). Much of this work deals with this kind of measures. 2. Reflexivity of Lpand Lp w The basic references for us about integration will be [7,13,16,17] and [18, Chapter 3]. Throughout this paper we will consider a vector measure ν:R →Xdefined on a δ-ring Rof subsets of some nonempty set Ωwith values in a real Banach space X, with dual X. We denote by Rloc the σ-algebra of subsets A ⊆Ω such that A ∩B∈Rfor each B∈R. Measurability of functions f:Ω −→ Rwill be considered with respect to the measurable space (Ω, Rloc). The semivariation of νis the set function ν :Rloc →[0, ∞] defined by ν(A) := sup {|ν, x| (A):xX≤1}, where |ν, x| is the variation of the scalar measure ν, x:A∈R−→  ν, x(A):=ν(A),x ∈R. Recall that for every subset A ∈Rloc, we have the following inequalities 1 2ν(A)≤sup{ν(B):B∈R,B ⊆A}≤ν(A). The semivariation is a subadditive set function that may be nonadditive. A set N∈Rloc is called ν-null if ν(N) =0, and a property holds ν-almost everywhere (ν-a.e.) if it holds except on a ν-null set. In what follows we will always consider vector measures ν:R →Xwhich are σ-finite, that is, there exist a pairwise disjoint sequence (Ωk)kin R, and a ν-null set N∈Rloc, such that Ω =(∪k≥1Ωk)∪N. Simple examples of σ-finite vector measures defined on δ-rings are given by the Lebesgue measure λdefined on the δ-ring R := {A∈M:λ(A)<∞}, where Mis the σ-algebra of all Lebesgue measurable subsets of the real line R, and the counting measure defined on the δ-ring Pf(N)of finite subsets of the natural numbers N. Other examples of σ-finite vector measure will be considered in Examples 3.2 and 4.5 below. Moreover, σ-finite vector measures have special scalar control measures as we see in the following result (see [7, Theorem 3.3]). Lemma 2.1. Let νbe a σ-finite vector measure. Then there exists x 0∈X, with x 0X≤1, such that |ν, x 0|(A) =0if and only if ν(A) =0, with A ∈Rloc. Proof. If νis σ-finite, then there exists 0 <f∈L1(ν). Consider the vector measure νf:Rloc →Xdefined by νf(A) := Afdν ∈X. Note that νfis defined on a σ-algebra, and νf(A) =fχ AL1(ν), for all A ∈Rloc (see [13, Theorem 3.2]). Let x 0∈X, with x 0X≤1, such that |νf, x 0| is a Rybakov control measure for νf(see [9, Theorem IX.1.2]). Then |ν, x 0|(A) =0if and only if ν(A) =0, with A ∈Rloc, because we know that |νf, x 0|(A) =Afd|ν, x 0|, for all A ∈Rloc.2 A measurable function f:Ω −→ Ris called weakly integrable (with respect to ν) if f∈L1(|ν, x|)for all x∈X. A weakly integrable function fis said to be integrable (with respect to ν) if, for each A ∈Rloc there exists an element (necessarily unique) Afdν ∈X, satisfying  A fdν,x= A fdν, x,x ∈X. If 1 ≤p <∞, a measurable function f:Ω −→ Ris called weakly p-integrable (with respect to ν) if |f|p is weakly integrable and p-integrable (with respect to ν) if |f|pis integrable. The space Lp w(ν)of all (ν-a.e. equivalence classes of) weakly p-integrable functions becomes a Banach lattice when endowed with the usual ν-a.e. pointwise order and the norm fLp w(ν):= sup ⎧ ⎪ ⎨ ⎪ ⎩⎛ ⎝ Ω |f|pd|ν, x|⎞ ⎠ 1 p :xX≤1⎫ ⎪ ⎬ ⎪ ⎭ . Moreover, the space Lp(ν)of all (ν-a.e. equivalence classes of) p-integrable functions is a closed order continuous ideal of Lp w(ν). In fact, it is the closure of S(R), the space of simple functions supported on R(see [13, Theorem 3.5]). Recall that order continuous means that f−fnLp(ν)→0for every 0 ≤fn↑f∈Lp(ν). For p ≥1, note that Lp w(ν)={f:Ω−→ R:|f|p∈L1 w(ν)},fLp w(ν)=|f|p 1 p L1 w(ν). These Banach lattices Lp(ν)and Lp w(ν)were initially studied in [12] and [19] for vector measures νdefined on a σ-algebra and its basic properties can be extended and remain true for vector measures defined on δ-rings (see [4]). Let us mention, in particular, that Lp w(ν)is p-convex, that is, there is a constant K>0 such that   (|f1|p+···+|fn|p)1 p  Lp w(ν)≤Kf1p Lp w(ν)+···+fnp Lp w(ν)1 p, for every election of vectors f1, ..., fnin Lp w(ν), as we can see directly from the definition of the norm  · Lp w(ν). The following result has been borrowed from [4, p. 75] (see also [2, Corollary 5.7]). We include here the proof for the sake of completeness. Proposition 2.2. Let 1 ≤p <∞and let 0 ≤fn↑in Lp w(ν)such that supnfnLp w(ν)<∞. Then, there exists supnfn∈Lp w(ν). Moreover supnfnLp w(ν)=supnfnLp w(ν). That is, Lp w(ν)has the sequential Fatou property. Proof. There exists a ν-null set N∈Rloc such that 0 ≤fn(w) ↑for all w∈Ω N. Consider the function g:Ω −→ [0, ∞] defined by g(w) := supnfn(w), if w∈Ω Nand g(w) =0, if w∈N. Then we have 0 ≤fp nχΩN↑gppointwise, and the Lebesgue monotone convergence theorem assures that  Ω gpd|ν, x| = lim n Ω fp nχΩNd|ν, x| ≤ xsup n fnp Lp w(ν)<∞, for all x∈X. In this way g∈Lp(|ν, x|)for all x∈X, and sup ⎧ ⎨ ⎩ Ω gpd|ν, x| :x≤1⎫ ⎬ ⎭ ≤sup n fnp Lp w(ν)<∞. In particular, by applying the above for the vector x 0of Lemma 2.1, we deduce that gis finite ν-a.e. and, in fact, it equals with supnfn. Thus g=sup nfn∈Lp w(ν), and moreover     sup n fn   Lp w(ν) =gLp w(ν)≤sup n fnLp w(ν)≤    sup n fn   Lp w(ν) .2 Recall that a Banach lattice is a KB-space whenever every norm bounded, positive, increasing sequence is norm convergent [1, Definition 14.10]. Thus every reflexive space is a KB-space (see the comments to the aforementioned definition), and it is clear that every KB-space has order continuous norm. Moreover every KB-space has the sequential Fatou property because every convergent (in norm) increasing sequence, necessarily converges to its supremum. The next result is the analogue to [12, Corollary 3.10] for vector measures defined on δ-rings. Its proof is a small modification of that, but we include it here for the sake of completeness. The equivalence of d)and h)has been proved independently by Avalos-Ramos and Galaz-Fontes in [2, Corollary 5.20]. Theorem 2.3. For every p >1, the following conditions are equivalent: a) Lp w(ν)has order continuous norm. b) Lp w(ν)is a KB-space. c) Lp w(ν)is reflexive. d) Lp(ν)is reflexive. e) Lp(ν)is a KB-space. f) Lp(ν)has the sequential Fatou property. g) Lp w(ν) =Lp(ν)as Banach lattices. h) L1 w(ν) =L1(ν)as Banach lattices. All eight assertions are true whenever the Banach space Xis weakly sequentially complete. Proof. a) =⇒b)Let (fn)nbe a norm bounded, positive, increasing sequence in Lp w(ν). By applying Proposition 2.2, there exists fin Lp w(ν)such that fn↑f. Then, from order continuity of the norm, we have that (fn)nconverges to fin Lp w(m). b) =⇒c)S ince Lp w(ν)is a p-convex (with p >1) Banach lattice, the space of summable sequences 1is not lattice embeddable in Lp w(ν)(see [14, p. 51]). Moreover, Lp w(ν)does not contain a lattice copy of the space of null sequences c0since it is a KB-space by hypothesis (see [1, Theorem 14.12]). The result then follows from Lozanovskii’s result (see [1, Theorem 14.23]). c) =⇒d)Lp(ν)is a closed subspace of Lp w(ν). d) =⇒e)It is well known that reflexive spaces are KB-spaces. e) =⇒f)E very KB-space has the sequential Fatou property. f) =⇒g)S ee [4, Proposition 5.4]. g)⇐⇒ h)It is enough to observe that f∈L1 w(ν)if and only if |f|1 p∈Lp w(ν). g) =⇒a)Note that Lp(ν)has always order continuous norm. See [19, Proposition 6] or [13, Theorem 3.3]. For the last claim in the statement of the theorem, recall that L1 w(ν) =L1(ν) whenever the Banach space Xis weakly sequentially complete. See [13, Theorem 5.1].2 3. Fatou property and order continuity of Lpof the semivariation Now we are going to consider, for 1 ≤p <∞, the spaces denoted by Lp(ν). These spaces appear in a natural way, as Lorentz spaces with respect to the semivariation ν, when we describe the interpolation spaces obtained by applying the real interpolation method to couples of Lp-spaces of a vector measure ν:R →X(see [6] and [10]). Let us introduce it briefly and describe some basic properties of them. Given a measurable function f:Ω −→ R, we shall consider its distribution function (with respect to the semivariation of the vector measure ν) νf:t ∈[0, ∞) −→  νf(t) ∈[0, ∞], defined by νf(t):=ν({w∈Ω:|f(w)|>t}),t≥0. This distribution function has similar properties as in the scalar case (see [10]). For instance, νfis non-increasing and right-continuous. Recall that L1(ν)is the space of (ν-a.e. equivalence classes of) measurable functions f:Ω −→ Rsuch that the integral ∞ 0νf(t)dt <∞. Then L1(ν), with the quasi-norm fL1(ν):= ∞ 0νf(t)dt and the usual ν-a.e. pointwise order, becomes a quasi-Banach lattice. For 1 <p <∞, we also consider the space Lp(ν):=f:Ω−→ R:|f|p∈L1(ν), with the quasi-norm fLp(ν):= |f|p 1 p L1(ν). We would need to mention that a consequence of [6, Remark 3.8.1] is that Lp(ν)is normable for every 1 <p <∞. This means that there is a lattice norm  · pequivalent to the quasi-norm  · Lp(ν). The case p =1is something special because we don’t know if L1(ν)is normable (see [11] for details). The following result is the analogue to Proposition 2.2. Proposition 3.1. Let 1 ≤p <∞and let 0 ≤fn↑in Lp(ν)such that supnfnLp(ν)<∞. Then, there exists supnfn∈Lp(ν). Moreover supnfnLp(ν)=supnfnLp(ν). That is, Lp(ν)has the sequential Fatou property. Proof. There exists a subset N∈Rloc, with ν(N) =0, such that 0 ≤fn(w) ↑for all w∈Ω N. Consider the function g:Ω −→ [0, ∞] defined by g(w) := supnfn(w), if w∈Ω Nand g(w) =0, if w∈N. Then we have 0 ≤fp nχΩN↑gppointwise, and νfp nχΩN(t)↑νgp(t)for all t ≥0. By applying the Lebesgue monotone convergence theorem we obtain ∞  0 νgp(t)dt = lim n ∞  0 νfp nχΩN(t)dt = lim n ∞  0 νfp n(t)dt =sup n fnp Lp(ν)<∞. Then νgp(t) <∞for all t >0and gis finite ν-a.e. We conclude that supnfn∈Lp(ν)and moreover supnfnLp(ν)=supnfnLp(ν).2 As it has been pointed out in [10], in general, the spaces Lp(ν), Lp(ν)and Lp w(ν)do not coincide, and the three spaces can be different. If the measure νis defined on a σ-algebra, we have the following inclusions L∞(ν) ⊆Lp(ν) ⊆Lp(ν) ⊆Lp w(ν), and all these inclusions are continuous for all 1 ≤p <∞ (see [10, Proposition 7]). Here L∞(ν) denotes the space of (classes ν-a.e. of) essentially bounded measurable functions f:Ω −→ Rwith the essential supremum norm. However, if the vector measure νis defined on a δ-ring instead of a σ-algebra, the inclusion Lp(ν) ⊆Lp(ν)is in general false as the following example points out. Example 3.2. (See [6, Example 2.1].) Consider the σ-finite vector measure ν:A∈Pf(N)→ν(A):=χA∈c0. For every 1 ≤p <∞, it is easy to check that Lp w(ν) =∞, the space of bounded sequences, and Lp(ν) =c0. In what follows it will be interesting to note that ν(A) =1, for every nonempty A ⊆N, and ν(∅) =0. This means, in particular, that νf=χ[0,∞)if fis an unbounded sequence, but νf=χ[0,f∞)if f∈∞. Consequently, L1(ν) =∞=L1 w(ν), and L1(ν) ⊆ L1(ν). Nevertheless, the inclusion L1(ν) ⊆L1 w(ν)remains and it is continuous for every vector measure νdefined on a δ-ring. And, moreover, the inclusion L1(ν) ⊆L1(ν)holds if and only if the measure νis locally strongly additive (see [6, Proposition 3.2]). In particular, if L1(ν) =L1 w(ν), then the measure νis locally strongly additive. Recall that a vector measure νis locally strongly additive if for every disjoint sequence (An)n⊆R, with ν (∪n≥1An)<∞, we have ν(An)X→0. See [5] and [6], where these measures were introduced in connection with real and complex interpolation methods and function spaces associated to a vector measure. Note that Example 3.2 tells us that S(R), the set of simple functions supported on subsets of the δ-ring R, is not always a dense subset of L1(ν). The things are different if the measure is locally strongly additive. The following technical results will be used to prove that S(R)is dense in L1(ν)when the vector measure is locally strongly additive. In what follows it will be convenient to consider the following notation. For a measurable function f:Ω −→ Rand a real number M, consider the measurable subset [f>M]: ={w∈Ω:f(w)>M}. Similar meaning have [f≤M]or [f=0]. Lemma 3.3. Let ν:R →Xbe a vector measure and let 0 ≤f∈L1(ν). Then ν ([f>M]) <∞for each M>0, and lim M→0 fχ[f≤M] L1(ν)=0. Proof. Note that f≥Mχ [f>M], for each M>0, and so fL1(ν)≥Mχ[f>M]L1(ν)=Mν([f>M]) . Thus, ν ([f>M]) ≤1 MfL1(ν)<∞. For the second assertion note that [fχ[f≤M]>t] =∅, if t ≥M>0, and so ν [fχ[f≤M]>t]=0for those t. On the other hand, if 0 ≤t <M, then [fχ[f≤M]> t]=[t <f≤M] and, in this case, ν [fχ[f≤M]>t]=ν ([t<f≤M]). Thus lim M→0 fχ[f≤M] L1(ν)= lim M→0 ∞  0 ν[fχ[f≤M]>t]dt = lim M→0 M  0 ν([t<f≤M]) dt ≤lim M→0 M  0 ν([f>t]) dt =0, since f∈L1(ν). 2 Lemma 3.4. Let ν:R →Xbe a vector measure. The following conditions are equivalent: 1) νis locally strongly additive. 2) ν(En) →0for each sequence (En)n⊆Rloc, such that En↓∅and ν(E1) <∞. In particular, if νis locally strongly additive, then for every A ∈Rloc, with ν(A) <∞, and every ε >0 there exists Bε∈R, with Bε⊆A, such that ν(A Bε) =χA−χBεL1(ν)<ε. Proof. 1) =⇒2) Suppose that (En)n⊆Rloc, with En↓∅and ν(E1) <∞. Then χEn∈L1 w(ν)for all n ≥1 because the sequence (En)nis decreasing and ν(E1) <∞. Now, locally strongly additivity of ν implies that χEn∈L1(ν)for all n ≥1(see [6, Lemma 3.1]), and moreover χEn↓0pointwise in L1(ν). The order continuity of the norm implies that ν(En) =χEnL1(ν)→0as we want to see. 2) =⇒1) Let (An)n⊆Rbe a disjoint sequence with ν (∪n≥1An)<∞. Put E1:= ∪n≥1Anand En:= E1(A1∪···∪An−1)for each n ≥2. Then it is clear that (En)n⊆Rloc, En↓∅and ν(E1) <∞. Moreover An⊆Enfor all n ≥1. Thus ν(An) ≤ν(En) →0and νis locally strongly additive. For the last assertion take A ∈Rloc, with ν(A) <∞, and recall that νis σ-finite. This allows us to choose a sequence (Ωn)n⊆R, with Ωn↑Ω. Then A A ∩Ωn↓∅and ν(A A ∩Ω1) ≤ν(A) <∞. Now the equivalence 2) assures that ν(A A ∩Ωn) →0, but ν(A A ∩Ωn) =χA−χA∩ΩnL1(ν).2 Here is the result about density of simple functions. Proposition 3.5. Let ν:R →Xbe a locally strongly additive vector measure. Then S(R)is dense in L1(ν). Proof. Decomposing functions into positive and negative parts, it is enough to consider only nonnegative functions. Note that f=fχ[f>M]+fχ[f≤M]for each 0 ≤f∈L1(ν)and M>0. Then Lemma 3.3 assures that the set L1 fs(ν):=g∈L1(ν):ν([g=0])<∞ is dense in L1(ν). Now we are going to prove that S(Rloc) ∩L1 fs(ν)is dense in L1 fs(ν). Take 0 ≤ g∈L1 fs(ν)and ε >0. Consider the sequence gn:= inf{g, n}for all n ≥1. Then 0 ≤gn↑gand [gn=0] ⊆[g=0]for all n ≥1. Then lim n→∞ g−gnL1(ν)= lim n→∞ ∞  0 ν([g−gn>t]) dt = lim n→∞ ∞  0 ν([g>n+t]) dt = lim n→∞ ∞  n ν([g>s]) ds =0. This means that the there exists m ≥1such that g−gmL1(ν)<ε 4. Since gmis bounded and [gm= 0] ⊆[g=0] there exists a simple function ϕ := N k=1 αkχAk, with Ak∈Rloc, Ak⊆[g=0], αk>0, for all 1 ≤k≤Nand 0 ≤ϕ ≤gmsuch that gm−ϕL∞(ν)<ε 4ν([g=0]) . Thus, having in mind that [gm−ϕ =0] ⊆[g=0], we obtain gm−ϕL1(ν)= ∞  0 ν([gm−ϕ>t])dt = ε 4ν([g=0])  0 ν([gm−ϕ>t])dt <ε 4ν([g=0])ν([g=0])=ε 4 and, consequently, g−ϕL1(ν)≤2g−gmL1(ν)+2gm−ϕL1(ν)<ε. Finally, note that Lemma 3.4 assures that S(R)is dense in S(Rloc) ∩L1 fs(ν). Indeed, given 0 ≤ϕ := n k=1 αkχAk∈S(Rloc) ∩L1 fs(ν)and ε >0 there exists Bk∈Rsuch that χAk−χBkL1(ν)<ε n2nn k=1 αk, for all k=1, ..., n. Now taking φ := n k=1 αkχBk∈S(R), we obtain that ϕ −φL1(ν)<ε, and the proof is over. 2 Proposition 3.6. Let ν:R →Xbe a vector measure. The following conditions are equivalent: 1) νis locally strongly additive. 2) fχEnL1(ν)→0for every f∈L1(ν)and every sequence (En)n⊆Rloc, with En↓∅. 3) f−fnL1(ν)→0for every sequence (fn)nand fof L1(ν)such that 0 ≤fn↑f. That is, L1(ν) is order continuous. 4) Lp(ν)is order continuous for every (some) 1 ≤p <∞. Proof. 1) =⇒2) Note that Lemma 3.4 assures that every simple function ϕ ∈S(R)satisfies the above condition 2). Given the function f∈L1(ν), the sequence (En)n⊆Rloc, with En↓∅and ε >0, from Proposition 3.5, we know that there exists ϕ ∈S(R)such that f−ϕL1(ν)<ε 4. Then we have fχEnL1(ν)≤2fχEn−ϕχEnL1(ν)+2ϕχEnL1(ν) ≤2f−ϕL1(ν)+2ϕχEnL1(ν)<ε 2+2ϕχEnL1(ν) and knowing that ϕχEnL1(ν)→0, it follows that fχEnL1(ν)→0. 2) =⇒3) Let 0 ≤fn↑f∈L1(ν)and let ε >0. The Lemma 3.3 assures that there exists B∈Rloc, with 0 <ν(B) <∞(we assume that fis not the null function), such that fχΩBL1(ν)<ε 24 . For every n ≥1 consider the measurable subsets En:= f−fn>ε 12ν(B)∈Rloc. Note that En↓∅. By the hypothesis fχEnL1(ν)<ε 24 for large enough n. Then for those nwe have that f−fnL1(ν)≤2(f−fn)χΩBL1(ν)+2(f−fn)χBL1(ν) ≤4fχΩBL1(ν)+4fnχΩBL1(ν) +4(f−fn)χEnL1(ν)+4(f−fn)χBEnL1(ν) ≤8fχΩBL1(ν)+8fχEnL1(ν) +4ε 12ν(B)ν(BEn)<8ε 24 +8ε 24 +4ε 12 =ε, and f−fnL1(ν)→0. 3) =⇒1) Let (An)n⊆Rbe a disjoint sequence with ν (∪n≥1An)<∞. Put Bn:= A1∪···∪Anfor every n ≥1. Then 0 ≤χBn↑χA, where A := ∪n≥1An, since the sequence (An)nis pairwise disjoint. Moreover χA∈L1(ν), as ν(A) <∞. By the hypothesis it follows that χA−χBnL1(ν)→0, but ν(An+1)X≤ν(An+1)≤ν(Bn+1)=χBn+1 L1(ν)≤χA−χBnL1(ν). 3) ⇐⇒ 4) This equivalence follows from the definition of the space Lp(ν)and the fact that it is normable as we have commented previously. 2 Remark 3.7. Now, knowing that Lp(ν)has order continuous norm if the measure νis locally strongly additive, it is not difficult to see that S(R)is dense in Lp(ν)for every 1 ≤p <∞. 4. Reflexivity of Lpof the semivariation Example 3.2 tells us that not always Lp(ν)is a reflexive space even for p >1. In this section we characterize those vector measures ν:R →Xsuch that Lp(ν)is reflexive. First we need the following technical results which are interesting in themselves. Proposition 4.1. For every p >1, the space Lp(ν)is a r-convex Banach lattice for every 1 ≤r<p. Proof. As commented above, we know that Ls(ν)is a Banach lattice for the equivalent lattice norm  ·s whenever s >1. In order to prove that Lp(ν)is r-convex it is enough to show that there exists K>0 such that   (|f1|r+···+|fn|r)1 r  Lp(ν)≤Kf1r Lp(ν)+···+fnr Lp(ν)1 r, for every election of vectors f1, ..., fnin Lp(ν). Take into account that s := p r>1, and so there exist two constants C1, C2>0such that C1hLs(ν)≤hs≤C2hLs(ν),h∈Ls(ν). Recall also that fLp(ν)=|f|r1 r Ls(ν)for all f∈Lp(ν)or, equivalently,   |h|1 r  Lp(ν)=h1 r Ls(ν) for all h ∈Ls(ν). Then, for every election of vectors f1, ..., fnin Lp(ν), we have      n  k=1 |fk|r1 r     Lp(ν) =     n  k=1 |fk|r     1 r Ls(ν) ≤1 C1     n  k=1 |fk|r     1 r s ≤1 C1n  k=1 |fk|rs1 r ≤C 1 r 2 C1n  k=1 |fk|rLs(ν)1 r ≤C 1 r 2 C1n  k=1 fkr Lp(ν)1 r as we want to prove. 2 Proposition 4.2. Let ν:R →Xbe a vector measure. For every 1 <p <∞, the inclusions L1 w(ν) ∩L∞(ν) ⊆ Lp(ν) ⊆L1 w(ν) +L∞(ν)hold. Proof. For the second inclusion note that Lp(ν) ⊆Lp w(ν). Now, if f∈Lp w(ν) decompose it as f= fχ[|f|>1] +fχ[|f|≤1]. It is clear that fχ[|f|≤1] ∈L∞(ν). On the other hand, for p ≥1, we have |f|χ[|f|>1] ≤|f|pχ[|f|>1] ≤|f|p∈L1 w(ν), and |f|χ[|f|>1] ∈L1 w(ν). Consequently f∈L1 w(ν) +L∞(ν).