Kantorovich–Wright integration and representation of vector lattices

J. Math. Anal. Appl. 455 (2017) 554–568

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Kantorovich–Wright integration and representation of vector lattices A.G. Kusraev a,b,∗ , B.B. Tasoev a,∗∗ a

Southern Mathematical Institute, Vladikavkaz Science Center of the RAS, 22 Markus street, Vladikavkaz, 362027, Russia b K.L. Khetagurov North Ossetian State University, 44–46 Vatutin Street, Vladikavkaz, 362025, Russia

a r t i c l e

i n f o

Article history: Received 17 February 2017 Available online 31 May 2017 Submitted by J.D.M. Wright Keywords: Vector lattice Vector measure Kantorovich–Wright integration Smallest extension Space of integrable functions Direct sum of measures

a b s t r a c t The aim of this work is to introduce a purely order-based Kantorovich–Wright type integration of scalar functions with respect to a vector measure defined on a δ-ring and taking values in a Dedekind σ-complete vector lattice and use it for obtaining general representation theorems for Dedekind complete vector lattices. © 2017 Elsevier Inc. All rights reserved.

1. Introduction Integration with respect to a measure taking values in a vector lattice has its roots in spectral theory, in representation of linear operators by means of integration with respect to spectral measures. An order based integration theory of real-valued measurable functions with respect to countably additive vector measures with values in a Dedekind complete vector lattice was developed by Kantorovich [12,13]. A decisive contribution was made by Wright [29,30]. The existing literature is rather sparse; some aspects of the theory are reflected in the book by Kusraev [14, Ch. 6]. Bartle, Dunford, and Schwartz [2] introduced the theory of integration with respect to a σ-additive vector measure defined on a σ-algebra of subsets of some set and with values in a Banach space. Later, Lewis [15] gave an alternative duality based approach. The theory was extended to vector measures defined on δ-rings in Lewis [16] and Masani and Niemi [18,19]. The integration of scalar measurable functions with respect to * Corresponding author at: Southern Mathematical Institute, Vladikavkaz Science Center of the RAS, 22 Markus street, Vladikavkaz, 362027, Russia. ** Corresponding author. E-mail addresses: [email protected] (A.G. Kusraev), [email protected] (B.B. Tasoev). http://dx.doi.org/10.1016/j.jmaa.2017.05.059 0022-247X/© 2017 Elsevier Inc. All rights reserved.

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a vector measure with values in an F -space was developed in Rolewicz [22], [23, §III.6], Turpin [26], [27, Chap. 7], and Thomas [25]. For over recent 25 years the spaces of integrable functions with respect to a measure taking values in a Banach (quasi-Banach) lattice has been a field of increased interest. The spaces of integrable and weakly integrable functions with respect to a vector measure possess interesting order and metric properties and have been studied intensively by many authors. They find applications in important problems such as the representation of abstract quasi-Banach lattices as spaces of integrable functions, the study of the optimal domain of linear operators, domination and factorization of operators, spectral integration etc., see the survey paper by Curbera and Ricker [4] and the book by Okada, Ricker and Sánches Pérez [21] as well as the recent papers by Calabuig, Delgado, Juan, Sánchez Pérez, Tradacete [3,5–8,11,24] and the references therein. The aim of this work is to introduce a purely order-based Kantorovich–Wright type integration and apply it to general representation theorems for Dedekind complete vector lattices. Kantorovich–Wright type integration with respect to a vector measures defined on a δ-ring with values in a Dedekind σ-complete vector lattice is introduced in Section 2. In the context of Banach lattices a crucial role is played by the spaces L1 (μ) and L1w (μ) of integrable and weakly integrable functions with respect to a vector measure. Dealing with the functional representation of vector lattices or quasi-Banach lattices a duality based definition of L1w (μ) no longer works but a natural candidate for a space of weakly integrable functions is the concept of the smallest extension of the integration operator as presented in Section 3. Section 4 deals with a basic construction of the direct sums of a family of measure spaces. In section 5 we demonstrate that, given an arbitrary σ-Dedekind complete vector lattice, there exist an order dense ideal which is lattice isomorphic to the vector lattice L1o (μ) of equivalence classes of o-integrable functions with respect to a vector measure μ, while the whole vector lattice is lattice isomorphic to the domain of the smallest extension Iˆμo of Iμo . The definitions of countable additivity and integration are understood in the sense of order convergence. Therefore, it is sometimes convenient to speak of an order measure (o-measure) and the order integration (o-integration) operator Iμo . It is important to know under which condition a quasi-Banach lattice is order isometric to some quasiBanach function space. Applications of the Kantorovich–Wright integration to this problem as well as à comparison with the Bartle–Dunford–Schwartz integration will be presented in a forthcoming article. We use the standard notation and terminology of Aliprantis and Burkinshaw [1] for the theory of vector lattices (see also Luxemburg and Zaanen [17], Meyer-Nieberg [20]). Throughout the text we assume that all vector spaces are defined over the field of reals and all vector lattices are Archimedean. We let := denote the assignment by definition, while N and R symbolize the naturals and the reals. 2. Kantorovich–Wright integration In this section X is a Dedekind σ-complete vector lattice and Ω is a nonempty set. Let P(Ω) stand for the powerset of Ω. A ring (of subsets of Ω) is a subset R ⊂ P(Ω) such that A \ B ∈ R and A ∪ B ∈ R for all A, B ∈ R. A δ-ring is a ring, which is closed under the countable intersections. Let Rloc stand for the collection of sets A ⊂ Ω such that A ∩ B ∈ R for all B ∈ R; in symbols, Rloc := {A ∈ P(Ω) : A ∩ B ∈ R for all B ∈ R}.

(1)

If R is a δ-ring then the collection Rloc is a σ-algebra containing R. Indeed given A ∈ Rloc and a sequence ∞ ∞ loc (An )∞ , we have (Ω \ A) ∩ B = B \ (B ∩ A) ∈ R and ( n=1 An ) ∩ B = n=1 (An ∩ B) ∈ R for all n=1 in R ∞ B ∈ R. Thus, Ω \ A ∈ Rloc and n=1 An ∈ Rloc . Moreover R ⊂ Rloc trivially.

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Definition 2.1. A function μ : R → X+ is said to be a measure, if μ(∅) = 0 and for every sequence (An )∞ of ∞ n=1 ∞ ∞ pairwise disjoint sets An ∈ R with n=1 An ∈ R the series n=1 μ(An ) is order convergent to μ( n=1 An ); in symbols,  ∞

μ

n=1

 An

 ∞ ∞  n  = oμ(Ak ) := μ(Ak ) . n=1

n=1

k=1

A triple (Ω, R, μ) is said to be a vector measure space if Ω is a nonempty set, R is a δ-ring of subsets of Ω, and μ : R → X+ is a measure. Everywhere below An ↑ means that An ⊂ An+1 for all n ∈ N, while An ↑ A means that An ↑ and ∞ n=1 An = A. The meanings of An ↓ and An ↓ A are similar. Lemma 2.2. Let A, B ∈ R and (An )∞ n=1 be a sequence in R. Then for a measure μ : R → X+ the following hold: (1) (2) (3) (4)

If If If If

A ⊂ B then μ(B \ A) = μ(B) − μ(A) and μ(A)  μ(B). An ↑ A then μ(An ) ↑ and μ(A) = o- limn μ(An ). An ↓ ∅ then μ(An ) ↓ and o- limn μ(An ) = 0. ∞ ∞ ∞ o- n=1 μ(An ) exists in X and A = n=1 An , then μ(A)  o- n=1 μ(An ).

Proof. This can be proved by standard arguments from measure theory. 2 Definition 2.3. A set A ∈ Rloc is called negligible (or, more precisely, μ-negligible) if μ(B ∩ A) = 0 for all B ∈ R. Say that a property P (·) is true for almost all ω ∈ Ω or almost everywhere (μ-a.e., for short) on Ω whenever {ω ∈ Ω : P (ω) is false} is μ-negligible. Define a family R∗ of subsets of Ω and a function μ∗ : R∗ → X+ by putting, as R∗ := {B ∪ A : B ∈ R and A is negligible}, μ∗ (B ∪ A) := μ(B) (B ∈ R, A ∈ N ). It follows from Lemma 2.2 that the collection N (μ) of all μ-negligible sets is a σ-ideal in the sense that (1) ∅ ∈ N (μ), (2) if A ∈ Rloc , B ∈ N (μ), and A ⊂ B, then A ∈ N (μ), (3) countable union of negligible sets is negligible. Lemma 2.4. The family R∗ is a δ-ring containing R and the function μ∗ from R∗ to X+ is a measure extending μ from R to the δ-ring R∗ . Proof. Clearly, R∗ is closed under finite unions and intersections. Show that R∗ contains also differences. Assume first that B ∈ R and A is negligible. Since B∩A ∈ R and R is a ring, we have B\A = B\(B∩A) ∈ R. Now, for B1 ∪ A1 , B2 ∪ A2 ∈ R∗ with arbitrary B1 , B2 ∈ R and negligible A1 , A2 we deduce (B1 ∪ A1 ) \ (B2 ∪ A2 ) = (B1 ∪ A1 ) ∩ (B2c ∩ Ac2 ) = (B1 ∩ B2c ∩ Ac2 ) ∪ (A1 ∩ B2c ∩ Ac2 ) = ((B1 \ B2 ) \ A2 ) ∪ C, where C = A1 ∩ B2c ∩ Ac2 is negligible, since N (μ) is an ideal of sets, while (B1 \ B2 ) \ A2 ∈ R by virtue of the above observation. Consequently, (B1 ∪ A1 ) \ (B2 ∪ A2 ) ∈ R∗ . It remains to ensure that the family R∗ is closed under countable intersections. For a sequence ∞ ∞ ∗ (Bn ∪ An )∞ n=1 in R with Bn ∈ R and negligible An put B := n=1 Bn ∈ R and A := n=1 An . Then

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∞ B ⊂ n=1 (Bn ∪ An ) ⊂ B ∪ A. Taking into account the fact that N (μ) is a σ-ideal we conclude that A is ∞ negligible and the desired relation n=1 (Bn ∪ An ) ∈ R∗ holds. 2 By virtue of Lemma 2.4 we can assume without loss of generality that the δ-ring R contains all negligible sets and μ(A) = 0 for each negligible set A ∈ Rloc . Now we present briefly the construction of Kantorovich– Wright integration for positive vector measure taking values in a Dedekind σ-complete vector lattice coming back to Kantorovich [13,12] and Wright [29,30]. Definition 2.5. A function f : Ω → R is called R-simple if it is a finite linear combination of characteristic n functions of sets in R, that is, f admits a representation f = k=1 ak χAk with A1 , . . . , An ∈ R and a1 , . . . , an ∈ R. In this representation neither a1 , . . . , an are distinct nor A1 , . . . , An are nonempty. (By definition χ∅ = 0.) But A1 , . . . , An always may be chosen pairwise disjoint. Denote by S(R) the set of all

n R-simple functions. Given a R-simple function f = k=1 ak χAk , the integral f dμ is defined by  Iμo (f ) :=

f dμ :=

n 

ai μ(Ai ).

i=1

n Integral of a simple function is well-defined, i.e., if a simple function f is representable as f = i=1 ai χAi m m n and f = j=1 bj χBj , then i=1 ai μ(Ai ) = j=1 bj μ(Bj ). We omit the verification of this fact, as well as the proofs of the following two lemmas, as routine exercises. Lemma 2.6. The set S(R) with the point-wise operations and ordering is a vector lattice and the mapping Iμo : f → Iμo (f ) from S(R) to X is a positive linear operator. Moreover, Iμo (|f |) = 0 if and only if f = 0 μ-a.e. for all f ∈ S(R). n m loc Lemma 2.7. If 0  g = -simple i=1 bi χBi is a R-simple function and 0  f = j=1 aj χAj is a R function, then the function g ∧ f is R-simple. Lemma 2.8. Let (fn ) be a monotone decreasing sequence of positive functions in S(R) such that limn fn = 0 ∞ μ-almost everywhere. Then n=1 Iμo (fn ) = 0. Proof. The proof is similar to that of [29, Proposition 3.1].

2

Definition 2.9. Say that a Rloc -measurable real-valued function f defined on a conegligible subset of Ω is integrable, if there exists a sequence (fn )∞ n=1 of R-simple functions such that 0  fn ↑ f μ-a.e. and

∞

dμ exists in X . In this occasion we put f n + n=1  Iμo (f ) :=

f dμ :=

∞ 

fn dμ.

n=1

(A subset A ⊂ Ω is conegligible if Ω \ A is negligible.) An arbitrary Rloc -measurable function f is integrable, whenever so are f + and f − . The integral of f is defined as Iμo (f ) = Iμo (f + ) − Iμo (f − ). The following lemma says that the integral Iμo is well defined. Lemma 2.10. Let f : Ω → R+ ∪ {∞} be an Rloc -measurable function and let (fn ) and (gn ) be increasing sequences of positive R-simple functions. If limn fn = f = limn gn μ-almost everywhere then

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x :=

∞

∞

Iμo (fn ) =

n=1

Iμo (gn ) =: y,

n=1

provided that one of the two upper bounds exists in X. Proof. Observe that the sequence (fn ∧ gm )n∈N increases and converges μ-a.e. gm for each fixed m ∈ N. Moreover, (fn ∧ gm )n∈N lies in S(R) by Lemma 2.7. Applying Lemma 2.8 yields Iμo (gm ) =

∞

Iμo (fn ∧ gm ) 

n=1

∞

Iμo (fn ) = x.

n=1

It follows that y ≤ x and similarly x ≤ y. 2 Denote by L0 (μ) := L0 (Ω, Rloc , μ) the set of all real-valued Rloc -measurable functions defined on conegligible subsets of Ω. Say that two functions f, g ∈ L0 (μ) are equivalent and write f ∼ g if f (ω) = g(ω) for μ-a.e. ω ∈ Ω. Let L0 (μ) := L0 (Ω, Rloc , μ) stand for the set of equivalence classes in L0 (μ) under ∼. For f ∈ L0 (μ), write f˜ for its equivalence class in L0 (μ). The linear structure and the ordering of L0 (μ) are conventionally defined using pointwise operations and order relation, see Fremlin [9, §241]. Let L1o (μ) := L1o (Ω, Rloc , μ) be the set of real-valued μ-integrable functions defined on conegligible subsets of Ω. Let L1o (μ) := L1o (Ω, Rloc , μ) be the set of equivalence classes of members of L1o (μ). Define an operator Iμo : L1o (μ) → X by writing Iμo (f˜) := Iμo (f ) for every f ∈ L1o (μ). Lemma 2.11. The following assertions hold: (1) L0 (μ) is a Dedekind σ-complete vector lattice. (2) L1o (μ) is an order dense ideal in L0 (μ). (3) L0 (μ) is a Dedekind complete if and only if μ is localizable. Proof. The proof of (1) is standard and (3) is similar to [9, Theorem 241G]. The fact that L1o (μ) is an order ideal in L0 (μ) follows from Lemma 2.7. Show that L1o (μ) is order dense in L0 (μ). Take 0 < f˜ ∈ L0 (μ) and put {f  n−1 } := {ω ∈ Ω : f (ω)  n−1 } ∈ Rloc with f ∈ L0 (μ) and n ∈ N. If μ(B ∩ {f  n−1 }) = 0 for all B ∈ R and n ∈ N then {f  n−1 } is μ-negligible for all n ∈ N. Moreover, {f  n−1 } ∈ R and μ({f  n−1 }) = 0 for all n ∈ N by the remark after the proof of Lemma 2.4. It follows that f = 0 μ-a.e. which contradicts the choice of f˜ > 0. Thus, μ(B0 ∩ {f  n−1 0 }) > 0 for some n0 ∈ N and −1 1 B0 ∈ R. Evidently, n−1 χ ∈ L (μ) and 0 < n χ  f with C := B ∩ {f  n−1 C C 0 o 0 0 0 } ∈ R. Thus, we have −1 0 1 proved that for each 0 < f˜ ∈ L (μ) there exists g := n0 χC ∈ Lo (μ) such that 0 < g  f μ-a.e. It follows that g˜ ∈ L10 (μ) and 0 < g˜  f˜, that is, L1o (μ) is order dense in L0 (μ) by definition. 2 The variants of the convergence theorems of Lebesgue integration theory are true for Kantorovich–Wright integral. The proofs of following three results can be given along the lines of [29, Propositions 3.3–3.5] and [14, Theorems 6.1.4 and 6.1.5]. Theorem 2.12 (Monotone convergence). Let (fn ) be a sequence in L1o (μ) such that fn ≤ fn+1 μ-a.e. for each n ∈ N and {Iμo (fn ) : n ∈ N} is order bounded in X. Then there exists f ∈ L1o (μ) such that (fn ) converges to f μ-a.e. and Iμo (f ) = o- lim Iμo (fn ). n

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Theorem 2.13 (Fatou’s Lemma). Let (fn )∈N be a sequence in L1o (μ)+ such that 0 ≤ fn μ-a.e. for each n ∈ N

∞ and lim inf n Iμo (fn ) := n=1 kn Iμo (fk ) exists in X+ . Then the function lim inf n fn ∈ L1o (Ω, Rloc , μ)+ and Iμo (lim inf fn )  lim inf Iμo (fn ). n

n

Theorem 2.14 (Dominated convergence). Let (fn ) be a sequence of functions in L1o (μ) converging to f ∈ L0 (μ) μ-a.e. If there exists g ∈ L1o (μ) such that |fn | ≤ g μ-a.e. for each n ∈ N, then f is integrable, (Iμo (fn )) is order convergent, and Iμo (f ) = o- lim Iμo (fn ). n

Theorem 2.15. Let X be a Dedekind σ-complete vector lattice and μ : R → X+ a measure. Then L1o (Ω, Rloc , μ) is a Dedekind σ-complete vector lattice and the mapping Iμo : L1o (Ω, Rloc , μ) → X is a strictly positive order σ-continuous linear operator. Proof. This is immediate from Lemmas 2.6 and 2.11 and Theorem 2.14. 2 3. The smallest extension of the Kantorovich–Wright integration In the context of Banach lattices a crucial role is played by the space L1w (μ) of weakly integrable functions with respect to a vector measure. Dealing with vector lattices or quasi-Banach lattices a duality based definition of L1w (μ) no longer works but a natural candidate for a space of weakly integrable function is the concept of the smallest extension of the integration operator presented in this section. Definition 3.1. Let E and F be vector lattices with F Dedekind complete and let G be an order ideal of ˆ the collection of all x ∈ X such that the set E. Consider a positive operator S : G → F and denote by G ˆ {S(g) : g ∈ G, 0 ≤ g ≤ |x|} is order bounded in F . Then G is an order ideal in E and we may define ˆ := sup{Sg : g ∈ G, 0 ≤ g ≤ x} := sup{S(g ∧ x) : g ∈ G} (x ∈ G ˆ + ). Sx ˆ + → F is additive and positively homogeneous, so it can be extended to G ˆ by differences. The operator Sˆ : G The resulting operator, which we denote by Sˆ again, extends S and is less or equal to every other positive ˆ The operator Sˆ is naturally called the smallest extension 1 of S with respect to E, extension of S to all of G. see [1, Theorem 1.30] and [14, 3.1.3]. ˆ Specify some properties of the smallest extension. Below, in Lemmas 3.2, 3.3, 3.4 and 3.6, E, F , G, G, ˆ S, and S are the same as in Definition 3.1. Lemma 3.2. The minimal extension Sˆ is order continuous or order σ-continuous if and only if so is S. Proof. See [1, Theorem 1.64]. 2 Lemma 3.3. The following assertions hold: (1) Sˆ is a lattice homomorphism if and only if so is S. (2) If G is order dense in E then Sˆ is strictly positive if and only if so is S. 1 ˆ → F of S to all of E, see [1, p. 27]. At the same time G ˆ ≤ T for any other positive extension T : G ˆ is the largest Evidently, S order ideal of E to which S extends positively, [14, 3.6.1 (4)].

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Proof. In both cases the necessity is trivial. ˆ 2 ) = 0 for all x1 , x2 ∈ G ˆ with ˆ 1 ) ∧ S(x (1) Assume that S is a lattice homomorphism and ensure that S(x x1 ∧ x2 = 0. Indeed, making use of Definition 3.1 we deduce ˆ 1 ) ∧ S(x ˆ 2) = S(x



 g1 ∈G+

g1 ,g2 ∈G+

S(g1 ∧ x1 ) ∧



 g2 ∈G+

S(g2 ∧ x2 )

S(g1 ∧ x1 ) ∧ S(g2 ∧ x2 ) = 0

which implies that Sˆ is a lattice homomorphism by [1, Theorem 2.14]. ˆ we can pick g ∈ G with 0 < g ≤ x, since G is order dense in E and hence in G. ˆ (2) Given 0 < x ∈ G, ˆ ˆ Now, if S is strictly positive then S(x) ≥ S(g) = S(g) > 0. 2 ˆ + and there exists x = supα xα in E. If Sˆ is Lemma 3.4. Assume that (xα )α∈A is an increasing net in G ˆ ˆ ˆ α ). A similar statement holds ˆ order continuous and supα S(xα ) exists in F , then x ∈ G and S(x) = supα S(x ˆ+. also for an order σ-continuous Sˆ and an increasing sequence (xn ) in G ˆ By Lemma 3.2 S is also order continuous, so that it Proof. We restrict ourselves to an order continuous S. ˆ suffices to show x ∈ G. Take an arbitrary g ∈ G with 0  g  x and note that (xα ∧ g) is an increasing net ˆ α ). It in G+ and g = supα xα ∧ g. Order continuity of S yields then 0  S(g) = supα S(xα ∧ g)  supα S(x ˆ follows that S([0, x] ∩ G) is order bounded in F and hence x ∈ G. 2 Lemma 3.5. Let X be a universally complete vector lattice and (xα )α∈A an increasing net in X+ . Then there exists a band projection π on X such that supα πxα exists in X, while for the complementary band projection π  := IX − π we have N π  e = supα π  (xα ∧ N e) for all N ∈ N and e ∈ X+ . Proof. There is no loss of generality in assuming that X = C∞ (Q) with extremely disconnected compact space Q. (Recall that the symbol C∞ (Q) denotes the universally complete vector lattice of all continuous functions f : Q → [−∞, ∞] for which the open set {q ∈ Q : −∞ < f (q) < ∞} is dense in Q.) Let (xα ) be an increasing net in C∞ (Q) and define two functions x ¯, x : Q → [0, ∞] by x ¯(q) = sup{xα (q) : α ∈ A} (q ∈ Q), x(q) :=

inf

sup x ¯(q  )

U ∈N (q) q  ∈U

(q ∈ Q),

where N (q) is a basis of neighborhoods of q. Then x ¯ is lower semicontinuous and x is continuous, see [28, Lemma V.1.4 and Theorem V.1.1]. Consider an open set Q0 := {q ∈ Q : x(q) < ∞} and observe that ¯ 0 is clopen. Now, let π stands for the band projection of C∞ (Q) corresponding to Q ¯ 0 and πx its closure Q ¯ ¯ stands for the function coinciding with x on Q0 and vanishing on Q1 := Q \ Q0 . Evidently, πx ∈ C∞ (Q)

¯(q) = ∞ and πx = supα πxα , see [28, Theorem V.2.2]. At the same time x(q) = ∞ for all q ∈ Q1 , so that x for all q ∈ Q1 \ A where A is a meager subset of Q1 . The latter implies that N e(q) = supα xα (q) ∧ N e(q) for all q ∈ Q1 \ A, whence the desired equation N π  e = supα π  (xα ∧ N e) follows. 2

ˆ is order dense ideal in E, and Sˆ is strictly positive Lemma 3.6. Assume that E is universally complete, G ˆ + such that y := supα S(x ˆ α ) exists in and order continuous. Assume further that (xα ) is increasing net in G ˆ ˆ F . Then there exists x ∈ G with x = supα xα and y = S(x). Proof. By Lemma 3.5 there exists a band projection π in E and an element x ∈ E+ such that x = supα πxα , ˆ + and while for the complementary projection π  := IE − π we have kπ  e = supα π  (xα ∧ ke) for all e ∈ G  k ∈ N. If π = 0 then x = supα πxα and the claim follows from Lemma 3.4. Using order continuity of Sˆ

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ˆ  e) = supα S(π ˆ  (xα ∧ ke)) ≤ supα S(x ˆ α ) = y. Since k ∈ N is arbitrary, S(π ˆ  e) ≤ y/k implies yields 0 ≤ kS(π   ˆ ˆ S(π e) = 0 and hence π e = 0 for all e ∈ G+ because S is strictly positive. It follows π  = 0 and the proof is complete. 2 Given a vector measure μ : R → X+ , apply Definition 3.1 to E := L0 (μ), G := L1o (μ) and S := Iμo . Denote ˆ of Iˆo . Then L1 (μ) ⊂ L1 (μ) ⊂ L0 (μ). by Iˆμo the smallest extension of Iμo and by L1ow (μ) the domain G μ o ow Definition 3.7. The vector lattice L1ow (μ) ⊂ L0 (μ) is called the space of weakly integrable function with respect to μ, while a measurable function f ∈ L0 (μ) is called weakly integrable with respect to μ if f˜ ∈ L1ow (μ). In view of Lemma 3.6 it is important to know when the vector lattice L0 (μ) is Dedekind complete. As in the case of scalar measures the answer is given in terms of localizability. Definition 3.8. A measure μ : R → X is said to be localizable if for every collection A ⊂ Rloc , there exists B ∈ Rloc such that (i) A \ B is μ-negligible for all A ∈ A and (ii) if C ∈ Rloc and A \ C is μ-negligible for all A ∈ A, then B \ C is also μ-negligible. Consider the measure space (Ω, R, μ) and denote by N (μ) the ideal of μ-negligible sets in Rloc . Define an equivalence relation ∼ on Rloc by putting A1 ∼ A2 whenever A1 A2 ∈ N (μ). Let B(μ) stands for the Boolean algebra quotient Rloc /N (μ). The coset of a set A ∈ Rloc we will denote by A˜ ∈ B(Ω). Then B(μ) is a Dedekind σ-complete Boolean algebra and the canonical map A → A˜ from Rloc onto B(μ) is an order ˜ B ˜ ∈ B(μ) we have A˜  B ˜ if and only if there exist σ-continuous Boolean homomorphism. Note that for A,   A1 ∈ A and B1 ∈ B such that A1 ⊂ B1 . Lemma 3.9. A measure μ : R → X+ is localizable if and only if the Boolean algebra B(μ) is Dedekind complete. Proof. The proof given in [10, 322B (d,e)] for scalar measures carries over verbatim. 2 Theorem 3.10. For a vector measure μ : R → X+ the following are equivalent: (1) μ : R → X+ is localizable. (2) B(μ) is complete Boolean algebra. (3) L0 (μ) is universally complete vector lattice. Proof. The equivalence (1) ⇐⇒ (2) follows from Lemma 3.9. To ensure that (2) ⇐⇒ (3) it suffices to observe that L0 (μ) is a universally σ-complete vector lattice with the constant function one on Ω taken as an order unit 1, the Boolean algebras B(μ) and C(1) are isomorphic, and L0 (μ) is Dedekind complete if and only so is C(1), see [28, Theorem V.4.3]. 2 Corollary 3.11. If the measure μ : R → X+ is localizable then L1o (μ) and L1ow (μ) are Dedekind complete and order dense ideals of L0 (μ). Proof. This is immediate from Theorem 3.10, since L1o (μ) and L1ow (μ) are order dense ideals of L0 (μ).

2

Theorem 3.12. Let X be Dedekind complete vector lattice and μ : R → X+ a localizable measure. Assume that an operator Iˆμo : L1ow (μ) → X is order continuous. If (fα )α∈A is an increasing net in L1ow (μ) and there exists y := supα∈A Iˆμo (fα ) in X+ then there is f ∈ L1ow (μ) such that supα fα = f and Iˆμo (f ) = y.

562

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Proof. According to Corollary 3.11 and Theorem 3.10 L1ow (μ) is an order dense ideal of universally complete vector lattice L0 (μ). By Theorem 2.15 and Lemma 3.3(2) the operator Iˆμo is order continuous and strictly positive. Hence, we can apply Lemma 3.6 to Iˆμo . 2 4. Direct sums of vector measures Now we introduce a basic construction of the direct sums of a family of vector measure spaces. Everywhere in this section X is a Dedekind σ-complete vector lattice.   Consider an indexed family Ωα , Σα , μα ) α∈A of measure spaces. We will assume that Ωα are pairwise disjoint. (Otherwise we replace Ωα by Ωα × {α}.) Moreover, for simplicity, we assume that Σα is a σ-algebra for all α ∈ A.   Definition 4.1. Say that a triple (Ω, R, μ) is the direct sum of the family of measure spaces (Ωα , Σα , μα ) α∈A or μ is the direct sum of the family of vector measures (μα )α∈A , whenever it satisfies the following conditions:  (1) Ω = α∈A Ωα ;  (2) the collection R ⊂ 2Ω comprises the sets of the form α∈A Aα ⊂ Ω, where Aα ∈ Σα for all α ∈ A and μα (Aα ) = 0 except for at most a finite set of α ∈ A;   (3) μ(A) = α∈A μα (Aα ) for any A = α∈A Aα ∈ R with Aα ∈ Σα for all α ∈ A.  Remark 4.2. Every set A ∈ R has a unique representation A = α∈A Aα with Aα ∈ Σα for all α ∈ A, since (Ωα )α∈A is a family of mutually disjoint subsets of Ω. In particular, the mapping μ : R → X+ in 4.1 (3) is well defined. The next lemma asserts that the direct sum (Ω, R, μ) in Definition 4.1 is a vector measure space. Lemma 4.3. The collection R is a δ-ring and the mapping μ is a measure.   Proof. Consider two sets A1 = α∈A A1,α , A2 = α∈A A2,α ∈ R. There exist finite sets θ1 , θ2 ⊂ A such that μα (Ai,α ) = 0 for all α ∈ A \ θi , i = 1, 2. Consequently, A1 ∪ A 2 =

 α∈A

  A1,α ∪

α∈A

  A2,α =

α∈A

(A1,α ∪ A2,α ) ∈ R,

since {α ∈ A : μα (A1,α ∪ A2,α ) = 0} = θ1 ∪ θ2 . At the same time A 1 \ A2 =

 α∈A

(A1α \ A2α ) ∈ R,

since A1,α \ A2,α ∈ Σα for all α ∈ A, {α ∈ A : μα (A1,α \ A2,α ) = 0} ⊂ θ1 , and Ωα ∩ Ωβ = ∅ for all α = β. Moreover, A1 ∩ A2 = A1 \ (A1 \ A2 ) implies A1 ∩ A2 ∈ R.  Assume now that for n ∈ N a set An ∈ R has the representation An := α∈A An,α with An,α ∈ Σα for all n ∈ N and α ∈ A. Then the equality ∞  n=1

An =

∞  

An,α ∈ R

α∈A n=1

∞ holds with the right-hand side in R. Indeed, if x ∈ n=1 An , then there exists a function ν : N → A ∞ such that x ∈ n=1 An,ν(n) . Since An,ν(n) ∩ Am,ν(m) = ∅ for all ν(n) = ν(m), there is n ¯ ∈ N such that ∞ x ∈ n=1 An,ν(¯n) . This proves the inclusion ⊂ and the converse inclusion is trivial. It follows that R is a δ-ring.

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It remains to prove that μ : R → X+ is a measure. First note that μ(∅) = 0. Let (An ) be the above ∞ sequence of R with the additional assumptions that the An are disjoint sets and n=1 An ∈ R.  ∞ ∞ ∞ Then n=1 An = α∈A n=1 An,α and, as n=1 An ∈ R, there exists a finite subset θ ⊂ A such that ∞ μα ( n=1 An,α ) = 0 for all α ∈ A \ θ. Consequently,  ∞

μ

 An

n=1

 α∈θ

o-

∞ 

   ∞

=μ

 An,α

α∈A n=1 ∞  

μα (An,α ) = o-

n=1

n=1 α∈θ

=

 ∞   μ An,α = α∈θ

n=1

∞  μα (An,α ) = oμ(An ). n=1

Thus, μ is countable additive, as claimed. 2 Recall that the σ-algebra Rloc comprises all sets A ⊂ Ω such that A ∩ B ∈ R for all B ∈ R. Lemma 4.4. A subset A ⊂ Ω belongs to Rloc if and only if the representation A = Aα ∈ Σα for all α ∈ A. Moreover such representation is unique.

 α∈A

Aα holds with

 Proof. Let C stand for the collection of sets representable as α∈A Aα with Aα ∈ Σα for all α ∈ A. If    A = α∈A Aα ∈ C and B = β∈A Bβ ∈ R, then A ∩ B = α∈A (Aα ∩ Bα ), since Aα ∩ Bβ = ∅ for α = β. By 4.1 (2) the relation B ∈ R implies the existence of a finite set θ ⊂ A such that μα (Bα ) = 0 for α ∈ A \ θ. From this we see that μα (Aα ∩ Bα ) = 0 for all α ∈ A \ θ, whence A ∩ B ∈ R. As B ∈ R is arbitrary, A ∈ Rloc and we get the inclusion C ⊂ Rloc . To ensure the converse inclusion, note that A ∈ Rloc implies A ∩ Ωα ∈ R,  since Ωα ∈ R. Thus, the relation A ∈ C follows from the representation A = A ∩ Ω = α∈A (A ∩ Ωα ).  Consequently, C = Rloc . The uniqueness of the representation A = α∈A Aα follows from the fact that (Ωα )α∈A is a family of mutually disjoint sets. 2  Lemma 4.5. Let the equality A = α∈A Aα hold with Aα ∈ Σα for all α ∈ A. Then A is μ-negligible if and only if Aα is μα -negligible for all α ∈ A. Proof. Let μ(Aα ) = 0 for all α ∈ A and ensure that then A is μ-negligible. Indeed, take an arbitrary B ∈ R  with B ⊂ A and note that, as B = α∈A (Aα ∩ B), in view of Remark 4.2 there exists a finite subset θ ⊂ A  such that μ(B) = α∈θ μ(B ∩ Aα ) = 0. By Definition 2.3 A is μ-negligible.  Conversely, if A = α∈A Aα ∈ Rloc is μ-negligible, then taking B := Aα in Definition 2.3 yields μ(Aα ) = 0 for all α ∈ A. 2 Remark 4.6. For a function f ∈ L0 (Ω, Rloc μ) denote by f |α the restriction of f to Ωα . Observe that if n  n g = k=1 ak χAk and Ak = α∈A Ak,α ∈ Rloc , then g|α = k=1 ak χAk,α , i.e. g|α is a Σα -simple function for all α ∈ A. Lemma 4.7. A function f : Ω → R ∪ {±∞} belongs to L0 (Ω, Rloc μ) if and only if f |α belongs to L0 (Ωα , Σα , μα ) for all α ∈ A. Proof. It is immediate from Lemma 4.4 that f ∈ L0 (Ω, Rloc μ) implies f |α ∈ L0 (Ωα , Σα , μα ) for all α ∈ A. Conversely, assuming f |α ∈ L0 (Ωα , Σα , μα ) for all α ∈ A, we may apply Lemma 4.4 again, taking the  representation f −1 (B) = α∈A (f |α )−1 (B) into account, to get f ∈ L0 (Ω, Rloc μ). 2 Lemma 4.8. For f ∈ L0 (Ω, Rloc , μ) and α ∈ A we have f χΩα ∈ L1o (Ω, Rloc , μ) if and only if f |α ∈



L1o (Ωα , Σα , μα ). Moreover, f χΩα dμ = f |α dμα .

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Proof. It suffices to verify the claim for positive f ∈ L0 (Ω, Rloc , μ). Suppose that f χΩα ∈ L1o (Ω, Rloc , μ). n  Take an R-simple function g = i=1 ai χAi such that 0  g  f χΩα μ-a.e. and Ak = α∈A Ak,α ∈ R for all k = 1, . . . , n. Because {g = 0} ⊂ Ωα up to a μ-negligible set, μ(Ak ) = μα (Ak,α ) for all k = 1, . . . , n. It follows that   n n   ai μ(Ak ) = ai μα (Ak,α ) = g|α dμα . (2) g dμ = k=1

k=1

Consider now a sequence of positive R-simple functions (gn )n∈N converging to f χΩα μ-a.e. Then the se

∞

quence (gn |α ) increases and converges to f |α μα -a.e., so by (2) we have f χΩα dμ = n=1 gn dμ =





∞

∞ 1 f |α dμα = n=1 gn |α dμα = f χΩα dμ. n=1 gn |α dμα . Therefore, f |α ∈ Lo (Ωα , Σα , μα ) and Conversely, assume that f |α ∈ L1o (Ωα , Σα , μα ) and pick an increasing sequence of positive Σα -simple functions (fn )∞ n=1 converging to f |α μα -a.e. Define a function gn by writing gn (t) := fn (t) for t ∈ Ωα and gn (t) := 0 for t ∈ Ω \ Ωα . Then (gn )n∈N is an increasing sequence of positive R-simple functions converging

to f χΩα μ-a.e. Moreover, gn |Ωα = fn for all n ∈ N and, according to (2), the equalities f |α dμα =



∞

∞

1 loc , μ) and f χΩα dμ = f |α dμα . 2 n=1 fn dμα = n=1 gn dμ hold. Consequently, f χΩα ∈ Lo (Ω, R   Theorem 4.9. Let (Ω, R, μ) be the direct sum of the family of vector measure spaces (Ωα , Σα , μα ) α∈A . Then f ∈ L0 (Ω, Rloc , μ) belongs to L1o (Ω, Rloc , μ), if and only if there is a function ν := νf : N → A

∞

depending on f such that |f | = n∈N |f |χΩν(n) μ-a.e. and o- n=1 |f |χΩν(n) dμ exists in X. In this event,

∞

|f | dμ = o- n=1 |f |χΩν(n) dμ. Proof. Obviously, it suffices to check this statement for a positive f ∈ L0 (Ω, Rloc , μ). If 0  f ∈ L1o (Ω, Rloc , μ) then there exists a sequence of R-simple functions (gn )∞ n=1 such that 0  gn ↑ f μ-a.e.

∞

and f dμ = n=1 gn dμ. Since {gn = 0} ∈ R up to a μ-negligible set, there exists a finite subset θn ⊂ A  ∞ such that {gn = 0} ⊂ α∈θ Ωα up to a μ-negligible set for all n ∈ N. Clearly, θn ↑ and Θ := n=1 θn is a countable subset of A. Pick an appropriate mapping ν from N onto Θ (there exist an increasing sequence

m of naturals (nk ) with n0 = 0 such that ν sends {nk−1 + 1, . . . , nk } onto θk ) and denote fm := n=1 f χΩν(n) for all m ∈ N. Then 0  fm ↑m f μ-a.e., because for each n ∈ N there is m ∈ N such that gn  fm μ-a.e.

Taking into account that the inequalities fm  n∈N f χΩν (n) are true μ-a.e. for all m ∈ N we get the desired

m representation f = n∈N f χΩν(n) μ-a.e. Moreover, the relations fm = n=1 f χΩν(n) and 0  fm ↑m f hold μ-a.e., so that making use of Theorem 2.14 we deduce  ∞  ∞  m    f dμ = fm dμ = f χΩν(n) dμ = of χΩν(n) dμ m=1 n=1

m∈N

n=1



∞

whence f dμ = o- n=1 f χΩν (n) dμ.

Conversely, assume that 0  f ∈ L0 (Ω, Rloc , μ) admits a representation f = n∈N f χΩν(n) μ-a.e., where ∞

m ν : N → A is an into mapping and o- n=1 f χΩν(n) dμ exists in X+ . Put fm := n=1 f χΩν(n) μ-a.e. for every m ∈ N. Then 0  fm ↑ μ-a.e. and f=



f χΩν(n) =

∞ m

f χΩν(n) =

m=1 n=1

n∈N

∞

fm μ-a.e.,

m=1

that is 0  fm ↑ f μ-a.e. Now, making use of Theorem 2.12 and the equality fm = see that 

∞ 

f dμ =

m=1

Thus, f ∈ L1o (Ω, Rloc , μ) and



fm dμ =

∞  m 

m n=1

∞   f χΩν(n) dμ = of χΩν(n) dμ.

m=1 n=1

∞

f dμ = o- n=1 f χΩν(n) dμ.

n=1

2

f χΩν(n) μ-a.e. we

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Remark 4.10. Theorem 4.9 remains valid when Σα is a δ-ring for all α ∈ A but in this generality the result is not used in the present work. 5. Representation of Dedekind complete vector lattices In this section we demonstrate that, given an arbitrary Dedekind complete vector lattice, there exists an order dense ideal in it which is lattice isomorphic to the vector lattice of equivalence classes of integrable functions with respect to a vector measure. Moreover, the latter admits a natural extension which is lattice isomorphic to the whole vector lattice. We start with the following preliminary result. Theorem 5.1. Let X be a Dedekind σ-complete vector lattice with a weak order unit e > 0 and let Q be the Stone representation space of a σ-algebra C(e) of components of e. Then there exists a Baire measure μ on

Q with values in X+ such that the integration operator Iμo : f˜ → f dμ is a lattice isomorphism of L1o (μ) onto X. Proof. Denote by Σ the Baire σ-algebra of Q and observe that Σ coincides with the σ-algebra of subsets of Q generated by the set Clop(Q) of all clopen subsets of Q. Let Δ stand for the collection of all meager subsets (= sets of first category) of Q contained in Σ. Then by Loomis–Sikorski Theorem [14, Theorem 1.2.6] there exists a Boolean isomorphism h from the quotient algebra Σ/Δ onto C(e). Note that the quotient mapping ϕ : Σ → Σ/Δ is Boolean σ-homomorphism. Define μ : Σ → X+ as μ := h ◦ ϕ. Then μ is a Baire measure on Q. It is immediate from the definition of μ that Δ coincides with the collection of μ-negligible sets.

By Theorem 2.15 the integration operator Iμo : f˜ → f dμ from the Dedekind σ-complete vector lattice L1o (Q, Clopσ (Q), μ) to X is strictly positive and order σ-continuous. Moreover, Iμo is injective lattice homomorphism, since μ is Boolean homomorphism. To complete the proof we have to show that Iμo is onto. Recall that an e-step element in X is any n vector x ∈ X representable as x = k=1 λk ek where e1 , . . . , en are pairwise disjoint components of e with e = e1 + · · · + en and λ1 , . . . , λn are arbitrary reals. Note that for every e-step element x ∈ X there exists a simple function f ∈ L1o (μ) such that Iμo (f ) = x, since the restriction μ|Clop(Q) is an isomorphism of Clop(Q) onto C(e). Now, take an element x ∈ X+ and put xn := x ∧(ne) for all n ∈ N. Then the sequence (xn )n∈N lies in the order ideal Xe generated by e. By Freudenthal’s Spectral Theorem [1, Theorem 2.8] for every n ∈ N there exists an e-step element un satisfying 0 ≤ xn − un ≤ e/n. For n ∈ N pick a simple function gn ∈ L1o (μ) such that un = Iμo (gn ) and put fn := g1 ∨· · ·∨gn and vn := u1 ∨· · ·∨un . Clearly, (fn ) and (vn ) are increasing sequences connected by the formula vn = Iμo (fn ) (n ∈ N). Moreover, 0 ≤ xn − vn ≤ xn − un ≤ e/n and

∞

∞ hence x = n=1 vn . By Monotone convergence Theorem 2.12 there exists f ∈ L1o (μ) such that f = n=1 fn μ-a.e. and x = Iμo (f ). Thus Iμo is onto and the proof is complete. 2 Lemma 5.2. Let X be a Dedekind σ-complete vector lattice. There exists a disjoint family of nonzero positive

elements Γ ⊂ X such that each element x ∈ X+ admits a unique representation x = γ∈Γ xγ with 0  xγ ∈ Xγ := {γ}⊥⊥ for all γ ∈ Γ. Proof. Each disjoint collection Γ of nonzero positive elements in X with X = Γ⊥⊥ is suitable, see Vulikh [28, Lemma IV.7.1, Theorems IV.5.2 and IV.5.3]. 2 Let X, Γ, and (Xγ )γ∈Γ be as in Lemma 5.2. By Theorem 5.1 for each γ ∈ Γ there exist a nonempty set Ωγ , a σ-algebra Σγ of its subsets, and a measure μγ : Σγ → Xγ such that the integration operator

Iμγ : f˜ → f dμγ is a lattice isomorphism of L1o (Ωγ , Σγ , μγ ) onto Xγ . Make the direct sum (Ω, R, μ) of the family (Ωγ , Σγ , μγ )γ∈Γ , see Definition 4.1.

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Definition 5.3. Define XΓ as the set of elements x ∈ X, whose representation in Lemma 5.2 has at most countable nonzero projections; in symbols   ∞ XΓ := x ∈ X : (∃ν : N → Γ) |x| = πν(n) |x| , n=1

(3)

where πγ is a band projection in X onto the band Xγ := {γ}⊥⊥ . Clearly, the inclusion Γ ⊂ XΓ implies that XΓ is an order dense ideal of X. If Γ is at most countable, then XΓ = X. Below μ stands for the direct sum of the family of vector measures (μγ )γ∈Γ . Note that μ : R → XΓ . Theorem 5.4. Let X be a Dedekind σ-complete vector lattice and Γ is chosen as in Lemma 5.2. Then the

integration operator Iμo : f → f dμ is a lattice isomorphism from L1o (Ω, Rloc , μ) onto XΓ . Proof. In virtue of Theorem 2.15 the integration operator Iμo : L1o (Ω, Rloc , μ) → XΓ is strictly positive. Show that Iμo is a lattice homomorphism. To do this, it suffices to ensure that Iμo (f ) ∧ Iμo (g) = 0 for all f, g ∈ L1o (Ω, Rloc , μ)+ with f ∧ g = 0 μ-a.e., see [1, Theorem 2.14]. By Theorem 4.9 and Lemma 4.8 there are functions ν, ξ : N → Γ such that

f=

 f χΩν(n) μ-a.e.,

n∈N



g=



f dμ = 

gχΩξ(n) μ-a.e.,

f χΩν(n) dμ,

n∈N



g dμ =

n∈N

f χΩξ(n) dμ.

n∈N

Pick m, n ∈ N and observe that if ν(n) = ξ(m) then by Definition 4.1 and Lemma 4.8 we have 

 f χΩν(n) dμ ∧



 f |ν(n) dμν(n) ∧

gχΩξ(m) dμ =

g|ξ(m) dμξ(m) ∈ Xν(n) ∩ Xξ(m) = {0}.

If ν(n) = ξ(m) then by Theorem 5.1 and Lemma 4.8 we have 

 f χΩν(n) dμ ∧ It follows that



 gχΩξ(m) dμ =

f χΩν(n) dμ ∧ 



 g|ν(n) dμν(n) =

f |ν(n) ∧ g|ν(n) dμν(n) = 0.

gχΩξ(m) dμ = 0 for all n, m ∈ N. From this we can deduce

 f dμ ∧

 f |ν(n) dμν(n) ∧

g dμ =

 

    f χΩν(n) dμ ∧ gχΩξ(m) dμ =

n∈N



f χΩν(n) dμ ∧



m∈N

 gχΩξ(m) dμ = 0,

n∈N m∈N

whence Iμo is a lattice homomorphism. The fact that Iμo is one-to-one follows by observing that if f˜ ∈ L1o (Ω, Rloc , μ) and Iμo (f˜) = 0, then 0 = |Iμo (f˜)| = Iμo (|f˜|) and f = 0 μ-a.e. or f˜ = 0 by strictly positivity. It remains to show that Iμo is onto. For an arbitrary 0  x ∈ XΓ there exists a mapping ν : N → Γ such

that x = n∈N xν(n) , where 0  xν(n) ∈ Xν(n) for all n ∈ N. By definition, for every n ∈ N one can chose

0  fν(n) ∈ L1 (Ων(n) , Σν(n) , μν(n) ) such that Iμν(n) (fν(n) ) = fν(n) dμν(n) = xν(n) . Put fγ (t) = 0 for all t ∈ Ωγ and γ ∈ Γ \ ν(N). Define a function f : Ω → R ∪ {∞} by putting f (t) := fγ (t) for t ∈ Ωγ and γ ∈ Γ. Then the restriction f |γ of f onto Ωγ belongs to L0 (Ωγ , Σγ , μγ ) for all γ ∈ Γ so that f ∈ L0 (Ω, Rloc , μ)+

by Lemma 4.7. Moreover, f = n∈N f χΩν(n) μ-a.e. and by Lemma 4.8 we have

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∞    oxν(n) = x. f χΩν(n) dμ = f |ν(n) dμν(n) = n=1

n∈N

By Theorem 4.9 it follows that f˜ ∈ L1o (Ω, Rloc , μ)+ and Iμo (f˜) = x.

n∈N

2

Lemma 5.5. Let X be a Dedekind complete vector lattice. The direct sum of the family (Ωγ , Σγ , μγ )γ∈Γ is a vector measure space with a localizable measure μ : R → X+ . Proof. By virtue of Theorem 3.10 it suffices to ensure Dedekind completeness of B(Ω) := Rloc /N , where N := {A ∈ Rloc : μ(A) = 0}. Denote by B(Ωγ ) the quotient algebra Σγ /Nγ , where Nγ is the collection of μγ -negligible sets for all γ ∈ Γ. Then Lemmas 4.4 and 4.5 imply that the Boolean algebra B(Ω) is isomorphic  to the product γ∈Γ B(Ωγ ) of a family of Dedekind complete Boolean algebras (B(Ωγ ))γ∈Γ . Therefore, B(Ω) is Dedekind complete. 2 Recall that the lattice isomorphism Iμo of L1o (μ) onto XΓ in Theorem 5.4 has the smallest extension Iˆμo whose domain L1ow (μ) is taken as the space of weakly integrable functions (see Definition 3.7). Say that a disjoint set Γ ⊂ X+ is complete if X = Γ⊥⊥ . Theorem 5.6. Let X be a Dedekind complete vector lattice and Γ a complete disjoint set in X+ . Then there exists a vector measure space (Ω, R, μ) with localizable μ such that the integration operator Iμo is a lattice isomorphism of L1o (μ) onto XΓ . Moreover, the smallest extension Iˆμo of Iμo with respect to L0 (μ) is a lattice isomorphism of L1ow (μ) onto X. Proof. The existence of a vector measure space (Ω, R, μ) as well as the fact that the integration operator Iμo is a lattice isomorphism of L1o (μ) onto XΓ follows from Theorem 5.4. Consequently, Iμo : Loμ (μ) → X is order continuous, since XΓ is order dense ideal of X. It follows from Lemmas 2.11(2), 3.2 and 3.3 that Iˆμo : L1ow (μ) → X is order continuous one-to-one lattice homomorphism. Moreover, by Lemma 5.5 the measure μ is localizable. To complete the proof we have to show that Iˆμo is onto. Given arbitrary x ∈ X+ , there exists an increasing net (xα )α∈A in XΓ order convergent to x, since XΓ is an order dense ideal of X. Put fα := Iμo−1 (xα ) and observe that (fα )α∈A is an increasing net in L1o (μ) with x = supα Iμo (fα ). Hence, in view of Theorem 3.12 there exists f ∈ L1ow (μ) such that 0  fα ↑ f and Iˆμo (f ) = x. 2 The authors thank the referee for many helpful comments that improved the quality of the paper. References [1] C.D. Aliprantis, O. Burkinshaw, Positive Operators, Acad. Press Inc., London, 1985. [2] R.G. Bartle, N. Dunford, J. Schwartz, Weak compactness and vector measures, Canad. J. Math. 7 (1955) 289–305. [3] J.M. Calabuig, O. Delgado, M.A. Juan, E.A. Sánchez Pérez, On the Banach lattice structure of L1w of a vector measure on a δ-ring, Collect. Math. 65 (2014) 67–85. [4] G.P. Curbera, W.J. Ricker, Vector Measures, Integration, Applications, Positivity (Trends Math.), Birkhäuser, Basel, 2007, pp. 127–160. [5] O. Delgado, Optimal extensions for positive order continuous operators on Banach function spaces, Glasg. Math. J. 56 (2014) 481–501. [6] O. Delgado, M.A. Juan, Representation of Banach lattices as L1w spaces of a vector measure defined on a δ-ring, Bull. Belg. Math. Soc. Simon Stevin 19 (2) (2012) 239–256. [7] O. Delgado, E.A. Sánchez Pérez, Strong extensions for q-summing operators acting in p-convex Banach function spaces for 1 ≤ p ≤ q, Positivity 20 (2016) 999–1014. [8] O. Delgado, E.A. Sánchez Pérez, Optimal extensions for pth power factorable operators, Mediterr. J. Math. 13 (2016) 4281–4303. [9] D.H. Fremlin, Measure Theory, vol. 2. Broad Foundation, Cambridge University Press, 2001. [10] D.H. Fremlin, Measure Theory, vol. 3. Measure Algebras, Cambridge University Press, 2002.

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