CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector J. Math. Anal. Appl. 354 (2009) 641–647 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa The space of scalarly integrable functions for a Fréchet-space-valued measure ∗ R. del Campo a, ,1,W.J.Rickerb a Departamento Matemática Aplicada I, EUITA, Ctra. de Utrera Km. 1, 41013-Sevilla, Spain b Math.-Geogr. Fakultät, Katholische Universität Eichstätt-Ingolstadt, D-85072 Eichstätt, Germany article info abstract Article history: 1 The space L w (ν) of all scalarly integrable functions with respect to a Fréchet-space-valued Received 18 November 2008 vector measure ν is shown to be a complete Fréchet lattice with the σ -Fatou property Available online 24 January 2009 which contains the (traditional) space L1(ν),ofallν-integrable functions. Indeed, L1(ν) is Submitted by B. Cascales 1 the σ -order continuous part of L w (ν). Every Fréchet lattice with the σ -Fatou property and containing a weak unit in its -order continuous part is Fréchet lattice isomorphic to Keywords: σ 1 Fréchet space (lattice) a space of the kind L w (ν). Vector measure © 2009 Elsevier Inc. All rights reserved. Fatou property Lebesgue topology Scalarly integrable function 1. Introduction Let X be a Fréchet space and ν : Σ → X be a vector measure (i.e. ν is σ -additive), where (Ω, Σ) is a measurable space. A measurable function f : Ω → R is said to be ν-integrable if ∗ ∗ (I-1) f is scalarly ν-integrable, that is, f is integrable with respect to the scalar measure ν, x :A → ν(A), x ,foreach ∗ ∗ x ∈ X (the continuous dual space of X), and ∈ ∈ (I-2) for each A Σ there exists an element A fdν X such that ∗ ∗ ∗ ∗ fdν, x = fdν, x , x ∈ X . A A For X a Banach space, the space L1(ν), consisting of all (equivalence classes of) ν-integrable functions equipped with the norm ∗ 1 f ν := sup | f | d ν, x , f ∈ L (ν), (1) ∗∈ ∗ x B X Ω ∗ ∗ ∗ ∗ ∗ is also a Banach space. Here B X∗ is the closed unit ball of X and |ν, x | is the variation measure of ν, x ,forx ∈ X .The space L1(ν) was introduced and investigated in [14], even for vector measures taking values in a locally convex Hausdorff * Corresponding author. E-mail addresses: [email protected] (R. del Campo), [email protected] (W.J. Ricker). 1 The author thanks the support of the Ministerio de Educación y Ciencia of Spain, under the program “José Castillejo”. 0022-247X/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2009.01.036 642 R. del Campo, W.J. Ricker / J. Math. Anal. Appl. 354 (2009) 641–647 space (briefly, lcHs). Since then, L1(ν) has been intensively studied in the Banach setting by many authors; see, e.g. [4,25] and the references therein. 1 G. Stefansson introduced the space L w (ν) of all (classes of) scalarly ν-integrable functions and showed that the same 1 functional (1) above is a norm in L w (ν) [28]. This is to be interpreted to mean that if f is scalarly ν-integrable, then ∞ 1 1 · f ν < and that (1) specifies a norm in L w (ν). Actually, Stefansson also showed that L w (ν) is complete for ν .Itis 1 1 then clear that L (ν) is a closed subspace of L w (ν). 1 1 In contrast to L (ν), the Banach space L w (ν) has received relatively little attention up to now; see, for example, [12], [25, 1 Chapter 3], [28]. Recently however, new features of L w (ν) have emerged, based on the theory of Banach function spaces, which indicate the importance of this space in its own right [3,5]. In particular, it is closely related to the “optimal domain property” for certain kernel operators. For a Fréchet space X and a vector measure ν : Σ → X, the corresponding (Fréchet) space L1(ν) is well understood; see, e.g. [8–11,14,21,23,24,26] and the references therein. Although the notion of an individual scalarly ν-integrable function already occurs in the Fréchet and even more general lcHs setting [14,16,17], there is no study made there or elsewhere (as 1 far as we aware) of what the corresponding space L w (ν) should be and what properties it might have. Our aim here is to 1 address this deficiency: we introduce and investigate various properties of L w (ν),forν a Fréchet space-valued measure, and identify its connection to L1(ν). 1 Let us indicate a sample of some of the results. In Section 2 we establish that L w (ν) is always complete, even if X is merely a metrizable lcHs. Even for normed spaces, this seems to have been overlooked. In contrast, L1(ν) is typically not complete if the metrizable lcHs X fails to be complete. In Section 3 it is shown that every Fréchet lattice F with the σ -Fatou 1 property and possessing a weak unit in its σ -order continuous part is Fréchet lattice isomorphic to L w (ν) for a suitable 1 vector measure ν. This shows how extensive the family of all spaces of the type L w (ν) is within the class of all Fréchet lattices. 1 2. The space L w (ν) (n) Let X be a metrizable lcHs generated by a fundamental sequence of increasing seminorms (· )n∈N.ThesetsBn := { ∈ (n) } ◦ := { ∗ ∈ ∗ | ∗| x X: x 1 form a fundamental sequence of zero neighbourhoods for X and their polars Bn x X : x, x 1, ∀ ∈ } ∈ N { ◦ ∈ N} x Bn ,forn , are absolutely convex [19, Theorem 23.5]. Moreover, Bn: n is a fundamental sequence of bounded ∗ ∗ ◦ ∈ N sets in the strong dual Xβ (i.e., each bounded set in Xβ is contained in a multiple of B n for some n ). In addition, each ◦ ∈ N ◦ = ◦ ◦ ∗ set Bn,forn ,isaBanach disc [19, Lemma 25.5], that is, the linear hull Lin(Bn) t>0 tBn of Bn (formed in X )isa Banach space when equipped with its Minkowski functional ∗ ∗ ◦ ∗ ◦ x ◦ := inf s > 0: x ∈ sB , x ∈ Lin B . Bn n n For each n ∈ N,thelocal Banach space Xn is the completion of the quotient X/Mn endowed with the quotient norm (n) (n) ∗ induced by · , where Mn := {x ∈ X: x = 0}.Letπn : X → Xn be the quotient map. Then its dual map π is an ∗ ◦ n isometric bijection between the Banach spaces Xn and Lin(Bn) [19, Remark 24.5(b)]. Given a vector measure ν : Σ → X, defined on a measurable space (Ω, Σ),asetA ∈ Σ is ν-null if ν(B) = 0forall B ∈ Σ with B ⊆ A.LetL0(ν) denote the σ -Dedekind complete Riesz space of all (classes, modulo ν-a.e., of) scalar-valued, Σ-measurable functions defined on Ω,withrespecttotheν-a.e. pointwise order [18, pp. 126–127]. For each n ∈ N,define (n) a [0, ∞]-valued seminorm · in L0(ν) by ν (n) ∗ 0 f ν := sup | f | d ν, x , f ∈ L (ν). (2) ∗∈ ◦ x Bn Ω The map ν : Σ → X given by n n νn(A) := πn ν(A) , A ∈ Σ, (3) is a Banach-space-valued vector measure. Observe that A ∈ Σ is ν-null if and only if it is νn-null for all n ∈ N. ∈ 0 ∈ 1 (n) ∞ Proposition 2.1. Let X be a metrizable lcHs, ν an X-valued measure and f L (ν).Thenf L w (ν) if and only if f ν < ,for n ∈ N. (n) ∗ ∗ ∗ (m) Proof. Let f satisfy f ν < ∞,foralln ∈ N.Givenx ∈ X ,thereexistm ∈ N and C > 0 such that |x, x | Cx for − ∗ ◦ all x ∈ X,i.e.C 1x ∈ B , and so m ∗ −1 ∗ (m) | f | d ν, x = C | f | d ν, C x C f ν < ∞. Ω Ω ∈ 1 ∈ N ∗ ∗ = ∗ ∗ ∈ ∗ Conversely, if f L w (ν),then f is scalarly νn-integrable, for n ,since ν, πn (ξ ) νn,ξ as measures, for ξ Xn . Furthermore, := | | ∗ = | | ∗ ∗ f νn sup f d νn,ξ sup f d ν, πn ξ ξ ∗∈B ∗ ξ ∗∈B ∗ Xn Ω Xn Ω R. del Campo, W.J. Ricker / J. Math. Anal. Appl. 354 (2009) 641–647 643 and hence, (n) := | | ∗ = f ν sup f d ν, x f νn (4) ∗∈ ◦ x Bn Ω ∗ ∗ ◦ ∗ = where, for the last equality, we use the fact that πn is an isometry and so πn (B X ) Bn.But,Xn is a Banach space and so ∞ 2 n f νn < . 1 0 Corollary 2.2. Let X be a metrizable lcHs and ν be an X-valued vector measure. Then, L w (ν) is an ideal in L (ν) and the restricted ·(n) : 1 →[ ∞ ∈ N 1 functionals ν L w (ν) 0, ) given by (2), for each n , are an increasing sequence of Riesz seminorms which turn L w (ν) into a metrizable, locally convex-solid Riesz space. ⊆ ·(n) 1 ∈ 0 ∈ 1 | | | | Proof. As Bn+1 Bn, the seminorms ( ν )n∈N are increasing in L w (ν).Also,ifg L (ν) and f L w (ν) with g f , ∈ 1 (n) (n) ∈ N 1 0 ·(n) then g L w (ν) and g ν f ν ,forn . This shows that L w (ν) is an ideal in L (ν) and each ν is a Riesz 1 seminorm in L w (ν). ∈ 1 (n) = ∈ N ∈ 1 = (n) = Suppose that f L w (ν) satisfies f ν 0foralln .