OF A VECTOR MEASURE.euler.us.es/~olvido/thesis.pdfOlvido Delgado Garrido to obtain the Degree of...

FURTHER DEVELOPMENTS ON L1

OF A VECTOR MEASURE.

Olvido Delgado Garrido

UNIVERSITY OF SEVILLA

FURTHER DEVELOPMENTS ON L1

OF A VECTOR MEASURE.

Memoir presented byOlvido Delgado Garridoto obtain the Degree of Doctorin Mathematical Sciences.

V B Director

Olvido Delgado Garrido

D. Guillermo Curbera Costello.Prof. of the Department of

Mathematical Analysis of theUniversity of Sevilla.

Sevilla, September 2004.

TRIBUNAL OF THE PH. D. THESIS:

Prof. Dr. W. J. Ricker (Katholische Universitt Eichsttt - Ingolstadt, Germany).

Prof. Dr. Ben de Pagter (Delft University of Technology, The Netherlands).

Prof. Dr. S. Okada (Katholische Universitt Eichsttt - Ingolstadt, Germany).

Prof. Dr. A. Fernandez Carrion (Universidad de Sevilla, Spain).

Prof. Dr. L. Rodrıguez Piazza (Universidad de Sevilla, Spain).

After a hard way of a long run, an stage of my travel along the mathe-matical research concludes with this memoir, which has arrived to good endthanks to the guidance and support of my supervisor Professor GuillermoCurbera, to whom I express my sincere gratitude.

Also I would like to thank the people which encouraged me during therealization of this work, specially to my parents, Sonia and Emilio.

To Emilio

Index.

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xi

Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1

CHAPTER 1. Subspaces of L1(ν). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

CHAPTER 2. L1(ν) for ν defined on a δ–ring. . . . . . . . . . . . . . . . . . . . 25

Section 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Section 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

CHAPTER 3. Optimal domains for operators. . . . . . . . . . . . . . . . . . . 45

Section 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Section 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Introduction.

The integration of scalar functions with respect to a vector measure wasintroduced by Bartle, Dunford and Schwartz in 1955, when they extendedthe Riesz representation theorem to the vector case. The classical version ofthis theorem establishes that all positive linear functional T : C(K) → C ,where K is a compact Hausdorff space and C(K) is the space of continuousscalar functions on K, can be represented as an integration operator withrespect to a regular Borel measure µ on K, that is,

T (f) =

∫fdµ for all f ∈ C(K) .

In the vector version, continuous linear operators T : C(K) → X with valuesin a Banach space X are considered. Bartle, Dunford and Schwartz [BDS,Theorem 3.2] showed that for T weakly compact, there exists a vector mea-sure ν : B(K) → X defined on the σ–algebra B(K) of the Borel subsets ofK, such that

T (f) =

∫fdν for all f ∈ C(K) .

This result required the development of a theory of integration of scalarsfunctions with respect to a vector measure ν : Σ → X defined on a σ–algebra Σ and with values in a Banach space X. This theory was createdfrom a Lebesgue type definition for the integral

∫fdν of a function f with

respect to ν.

Later, in 1970, Lewis [L1] constructed a theory of integration of scalarfunctions with respect to a vector measure ν with values in a locally convexHausdorff linear topological space X, where the integral of a function f withrespect to ν,

∫fdν, is defined by duality. In the case when X is a Banach

space, this theory is equivalent to the Bartle, Dunford and Schwartz’s.

xii Introduction

The space L1(ν) of integrable functions with respect to a vector measureν with values in a Banach space X, has been thoroughly studied and itsproperties as well as the relations between these ones and the properties ofthe measure ν and the space X, are now well understood thanks to numerousworks. Within them are the followings: Kluvanek and Knowles [KK]; Ricker[R1], [R2]; Okada [O]; Curbera [C2], [C3], [C4]; Okada and Ricker [OR1],[OR2], [OR3]; Okada, Ricker and Rodrıguez-Piazza [ORR]; Sanchez Perez[S1], [S2], [S3].

In most of the works cited above, real functions and real vector spacesare considered, taking advantage in this case the simplicity of the latticestructure of L1(ν).

Recently, the works of Ricker [R3]; Curbera [C5]; Curbera and Ricker[CR1]; [CR2]; and Okada and Ricker [OR4], have shown the usefulness ofthe integration with respect to vector measures in the study of the optimaldomains for classic operators. In a general setting, given a Banach functionspace E ([LT, Definition 1.b.17]) with respect to a finite measure space, aBanach space X and a linear operator T : E → X , a classical procedure isto associate to T the set function ν defined by ν(A) = T (χA), for measurablesets A. Curbera and Ricker [CR1, Corollary 3.3] showed that under certainnatural condition on T , which makes ν be a vector measure, the space L1(ν)is the optimal domain for T within the class of order continuous Banachfunction spaces, that is, L1(ν) is the largest space in this class to whichT can be extended as a continuous operator, still with values in X. Thisidentification of the optimal domain of T is interesting in the sense that thespace L1(ν) is a Banach lattice with properties which can be deduced fromthe properties of ν (coming from T ) and the properties of X.

The finiteness of the measure space on which E is Banach function spaceis essential, since in other case the set function ν associated to T may notbe defined for sets with infinite measure. This happens for instance with theHilbert transform on the real line. However, this problem can be solved byconsidering ν defined on a structure weaker than σ–algebra, namely a δ–ring(i.e. ring closed under countable intersections). In the case of the Hilbert

Introduction xiii

transform, the associated set function is defined for Lebesgue measurable setsof finite measure, which constitute a δ–ring.

One of our aims is to extend the result of Curbera and Ricker for opera-tors T defined on Banach function spaces with respect to an infinite measurespace, for which the associated set function ν is defined on a δ–ring. Thisleads to the study of the integration with respect to a vector measure on aδ–ring and the related space of integrable functions.

The theory of integration for vector measures on δ–rings, began to con-solidate in 1972, when Lewis in [L2] extended the results obtained in [L1].The Lewis’ work was continued in 1989 by Masani and Niemi [MN1], [MN2].

This memoir consist of two research lines within the frame of the vectormeasures with values in a real Banach space. The first one (Chapter 1) isa contribution to the understanding of the space L1(ν) of a classical vectormeasure ν (defined on a σ–algebra), through the study of what we will callBanach function subspaces of L1(ν). In the second one, we work within themore extensive setting of the vector measures ν defined on a δ–ring. Westudy, on the one hand, the lattice properties of L1(ν) and the relation withthe analytic properties of ν (Chapter 2), and on the other hand, the problemof determining the optimal domain for operators between spaces of functionsrelated to infinite measure spaces, by applying integration with respect to ν

(Chapter 3).

The essence of Chapter 1 is illustrated with the simple fact that for afinite positive measure µ and p ∈ (1,∞), the space Lp(µ) is continuouslycontained in L1(µ) and it can be described in terms of µ as the space offunctions f such that f p is integrable with respect to µ. Consider the caseof a vector measure ν : Σ → X defined on a σ–algebra Σ and with values ina Banach space X. The space L1(ν) is an order continuous Banach functionspace with respect to certain finite measure space (Ω, Σ, λ), [C2, Theorem1]. Given a Banach function space Y with respect to (Ω, Σ, λ), continuouslycontained in L1(ν), which we call Banach function subspace of L1(ν), westudy the following problem: Is it possible to describe Y in terms of the

xiv Introduction

vector measure ν?

The tool for solving this problem is certain map ρ : X∗ ×M → [0,∞] ,where X∗ is the dual space of X and M is the space of measurable functions(with respect to Σ), which has natural properties related to ν. We say thatρ is a ν–norm function. From a ν–norm function ρ, we construct a Banachfunction subspace E(ρν) of L1(ν), as the closure of the simple functions inthe space of functions f in M such that

ρν(f) = supx∗∈BX∗

ρ(x∗, f) < ∞ ,

with respect to the norm ρν . This method allows to define Orlicz spacesLΦ(ν) with respect to the vector measure ν, as spaces E(ρν) for a suitableν–norm function ρ. In the case when Φ is an Orlicz function with the ∆2

property, we obtain that LΦ(ν) is the space of functions f in M such thatΦ(f) is integrable with respect to ν.

We end the chapter showing that every order continuous Banach functionsubspace Y of L1(ν), can be represented as a space E(ρν) generated by a ν–norm function ρ (Theorem 1.17). So, the answer to our question is positivefor order continuous Banach function spaces of L1(ν).

In Chapter 2, we work with vector measures ν : R → X defined on aδ–ringR and with values in a Banach space X. In the first section, we exposethe principal results known about the integration with respect to this typeof vector measures ν and the space L1(ν). Also, we comment the differencesexisting with the case of vector measures defined on σ–algebras. In particular,L1(ν) is an order continuous Banach lattice which may not have a weak unit.We include a result (Proposition 2.3) which gives an equivalent condition tothe integrability of a function with respect to ν. This condition extends thedefinition of integrable function given by Bartle, Dunford and Schwartz forvector measures defined on σ–algebras [BDS, Definition 2.5] and it does notappear in the works of Lewis [L2] and Masani and Niemi [MN2].

In the second section, we study the properties of strong additivity andσ–finiteness for a vector measure ν defined on a δ–ring R. We prove that

Introduction xv

ν is σ–finite if and only if L1(ν) has a weak unit g (Theorem 2.7), and inthis case L1(ν) is order isometric to L(νg), where νg is the vector measurewith density g defined by νg(A) =

∫A

gdν for all A in the σ–algebra Rloc ofsets which are locally in R (Theorem 2.9). In the case when ν is stronglyadditive, g = χΩ is a weak unit of L1(ν), νg is an extension of ν to the σ–algebra Rloc and L1(ν) coincides with L1(νg) (Theorem 2.6). When ν is notσ–finite, L1(ν) can be represented as an unconditional direct sum of a familyof disjoint ideals, each one of them being order isometric to a space L1(νA),where the vector measure νA is the restriction of ν to certain σ–algebra ofthe type A∩R with A ∈ R (Theorem 2.10). Other question we study is theexistence of a “Rybakov local control measure” for ν endowing L1(ν) withstructure of Banach function space.

The Chapter 3 (inspired by [CR1]) is divided in two sections. In the firstone we consider a linear operator T : S(R) → X defined over the simplefunctions with support in a δ–ring R of subsets of a set Ω and with values ina Banach space X. When T satisfies appropriate conditions, the space L1(ν),where ν : R → X is the vector measure defined by ν(A) = T (χA), is theoptimal domain for T within the class of Banach function spaces on (Ω,R, µ)(Definition 3.1) which are order continuous and such that µ is equivalent toν, that is, µ has the same null sets than ν (Theorem 3.5). Moreover, in thiscase the integration operator f → ∫

fdν extends T to L1(ν).

The second section is focused on the kernel operators T defined as

T (f) =

∫ ∞

0

f(y)K(· , y)dy ,

where K is a non negative measurable function on [0,∞)× [0,∞) such thatthe set function ν associated to T is well defined on the δ–ring Bb of boundedBorel subsets of [0,∞). We say that K is an admissible kernel.

For a Banach function space X such that ν : Bb → X is a vector mea-sure, we prove that L1(ν) is the optimal domain for T (considered withvalues in X) within the class of order continuous Banach function spaceson ([0,∞),Bb, µ) with µ equivalent to ν (Proposition 3.8). If the admissiblekernel K satisfies some integrability and monotonicity conditions, then the

xvi Introduction

associated set function ν : Bb → X is a vector measure for any rearrange-ment invariant space X on [0,∞). In this case, if X is also order continuous,we give a precise description of L1(ν) (i.e. the optimal domain for T ) asan interpolation space (L1

ω, L1ξ)X , where L1

ω, L1ξ are the spaces of integrable

functions with respect to the Lebesgue measure with density (defined fromthe admissible kernel K) ω and ξ respectively (Theorem 3.15).

Preliminaries.

In this section, we expose the concepts and results used throughout thememoir about Banach lattices, specially Banach functions spaces. Also, webriefly recall the theory of integration of real functions with respect to vectormeasures defined on σ–algebras.

All Banach spaces are considered over the field of real numbers R. Givena Banach space X, we denote by BX the unit ball and by X∗ the dual spaceof X.

1. Banach lattices. A Banach lattice is a Banach space E with norm ‖·‖E,endowed with an order relation ≤ satisfying

1) if x, y, z ∈ E with x ≤ y, then x + z ≤ y + z,

2) if x, y ∈ E and a ≥ 0 with x ≤ y, then ax ≤ ay,

3) for x, y ∈ E, there exists the supremum and the infimum of x and ywith respect to the order,

4) if x, y ∈ E with |x| ≤ |y|, then ‖x‖E ≤ ‖y‖E, where |x| = supx,−xis the modulo of x.

An ideal of E is a closed vector subspace F satisfying that y ∈ F whenevery ∈ E with |y| ≤ |x| for some x ∈ F . A weak unit of E is an element0 ≤ e ∈ E with the property that infx, e = 0 implies x = 0.

A set A contained in a Banach lattice is downward directed if for all x, y ∈A there exists z ∈ A such that z ≤ x and z ≤ y. A Banach lattice is ordercontinuous if every downward directed set A with infimum equal to zero,

2 Preliminaries

satisfies that inf‖x‖ : x ∈ A = 0 [LT, Definition 1.a.6]. A Banach latticeis order continuous if and only if every order bounded increasing sequence isnorm convergent [LT, Proposition 1.a.8].

A linear operator T : E → F between Banach lattices is positive ifT (x) ≥ 0 whenever x ≥ 0. An important result of the Banach lattice theorystates that every positive linear operator between Banach lattices is contin-uous [LT, p. 2].

Two Banach lattices E and F are order isomorphic if there exists abijective linear operator T : E → F which preserves the order structure,that is,

T (supx, y) = supT (x), T (y) , for all x, y ∈ E ,

or equivalently, since T is linear and bijective, Tx ≥ 0 if and only if x ≥ 0.If T is also an isometry, E and F are said to be order isometric.

2. Banach function spaces. Following Lindenstrauss and Tzafriri [LT,Definition 1.b.17], a Banach function space (also called Kothe function space)with respect to a σ–finite measure space (Ω, Σ, µ), is a Banach space E ofclasses of real functions which are integrable with respect to µ over sets withfinite measure, satisfying

1) if f is a measurable function, g ∈ E and |f | ≤ |g| µ–a.e., then f ∈ Eand ‖f‖E ≤ ‖g‖E,

2) χA ∈ E for all A ∈ Σ with µ(A) < ∞,

where functions which are equal almost every where with respect to µ (µ–a.e.) are identified. Note that E is a Banach lattice with the µ–a.e. order(f ≥ 0 if f ≥ 0 µ–a.e.) and convergence in norm of a sequence implies µ–a.e.convergence for some subsequence.

Given a Banach function space E with respect to (Ω, Σ, µ), throughoutthis memoir by Banach function subspace of E, we mean a Banach functionspace F with respect to (Ω, Σ, µ), contained in E (non necessarily with the

Preliminaries 3

same norm). This inclusion is automatically continuous since it is a positivelinear operator between Banach lattices.

The Kothe dual of a Banach function space E with respect to (Ω, Σ, µ)is the space E ′ of measurable functions g : Ω → R such that fg is integrablewith respect to µ, for all f ∈ E. The space E ′ is a vector subspace of E∗

(the dual space of E) and is endowed with the norm of E∗. The spaces E ′

and E∗ coincide if and only if E is order continuous [LT, p. 29]. The Kothebidual of E is the Kothe dual of E ′.

Let (Ω, Σ, µ) be a σ–finite measure space, M the space of all measurablereal functions with respect to Σ and M+ the cone of all positive functionsin M. A function norm is a map ρ : M+ → [0,∞] , satisfying

1) ρ(f) = 0 if and only if f = 0 µ–a.e.,

2) ρ(af) = aρ(f), for all a ≥ 0 and f ∈M+,

3) ρ(f + g) ≤ ρ(f) + ρ(g), for all f, g ∈M+,

4) if f, g ∈M+ and f ≤ g µ–a.e., then ρ(f) ≤ ρ(g),

5) if f, fn ∈M+ and fn ↑ f µ–a.e., then ρ(fn) ↑ ρ(f),

6) ρ(χA) < ∞ for all A ∈ Σ with µ(A) < ∞,

7) for every A ∈ Σ with µ(A) < ∞, there exists CA > 0 such that

∫

A

fdµ ≤ CA ρ(f) for all f ∈M+ .

Bennett and Sharpley define a Banach function space [BS, Definition I.1.3],as a space of the type E(ρ) = f ∈M : ρ(|f |) < ∞ for some function normρ.

The space E(ρ) endowed with the norm ‖f‖ = ρ(|f |) and the µ–a.e.order, is a Banach function space in the sense of Lindenstrauss and Tzafriri,[BS, Theorem I.1.7].

4 Preliminaries

Given a Banach function space E with respect to (Ω, Σ, µ) in the senseof Lindenstrauss and Tzafriri, the map ρ : M+ → [0,∞] defined by ρ(f) =‖f‖E if f ∈ E and ρ(f) = ∞ if f /∈ E, satisfies conditions 1)–7) requiredto a function norm, except condition 5), called Fatou property, [LT, p. 30].Hence, the two definitions only differ in the Fatou property, in a way that theBanach function spaces in the sense of Bennett and Sharpley are a subclassof the Banach function spaces in the sense of Lindenstrauss and Tzafriri. Inboth cases we speak of Banach function spaces, specifying if it is necessarywhen we have one in the sense of Bennett and Sharpley.

Let E be a Banach function space with respect to (Ω, Σ, µ). A functionf ∈ E has absolutely continuous norm if ‖fχAn‖E → 0 for every sequence(An) ⊂ Σ such that χAn → 0 µ–a.e. Note that if (Ω, Σ, µ) is a finite measurespace then a function f ∈ E has absolutely continuous norm if and only if‖fχA‖E → 0 as µ(A) → 0.

A Banach function space E is order continuous if and only if every func-tion in E has absolutely continuous norm. This result can be deduced from[BS, Proposition I.3.5], of which proof does not use the Fatou property and soit holds for Banach function spaces in the sense of Lindenstrauss and Tzafriri.

3. Rearrangement invariant spaces. Let B be the σ–algebra of allBorel subsets of [0,∞) and m the Lebesgue measure on [0,∞). A rearrange-ment invariant space on [0,∞) is a Banach function space X with respectto ([0,∞),B,m) with the Fatou property (i.e. in the sense of Bennett andSharpley), satisfying that if f ∈ X then its decreasing rearrangement f ∗ ∈ X

with ‖f ∗‖X = ‖f‖X . The decreasing rearrangement of a function f with re-spect to m is the function defined by

f ∗(t) = infr ≥ 0 : mx ∈ [0,∞) : |f(x)| > r ≤ t

for t ≥ 0 .

Relevant rearrangement invariant spaces on [0,∞) are the sum and theintersection of the spaces L1 and L∞. The space L1 ∩ L∞ is endowed withthe norm ‖f‖L1∩L∞ = max‖f‖1, ‖f‖∞ and the space L1 + L∞ of functions

Preliminaries 5

f = g + h with g ∈ L1 and h ∈ L∞, is endowed with the norm

‖f‖L1+L∞ = inf‖f‖1+‖f‖∞ : f = g+h con g ∈ L1, h ∈ L∞

=

∫ 1

0

f ∗(t)dt

[BS, Theorem II.6.4].

If X is a rearrangement invariant space on [0,∞), then continuously wehave L1 ∩ L∞ ⊂ X ⊂ L1 + L∞, [BS, Theorem II.6.6].

Let (X0, X1) be Banach spaces continuously embedded in a commonHausdorff topological vector space. The K–functional of x ∈ X0 + X1 isdefined, for t > 0, by

K(t, x; X0, X1) = inf‖x0‖+ t‖x1‖ : x = x0 + x1 ; x0 ∈ X0 , x1 ∈ X1

.

Assume that X0∩X1 is dense in X0. The function K(·, x; X0, X1) is increasingand concave and its derivative K′(·, x; X0, X1) is non negative and decreasing.

Given a rearrangement invariant space X on [0,∞), the space (X0, X1)X

is the space of vectors x ∈ X0 + X1 such that K′(·, x; X0, X1) belongs to X,and it is endowed with the norm

‖x‖(X0,X1)X= ‖K′(·, x; X0, X1)‖X .

The space (X0, X1)X is an interpolation space between X0 and X1 (see [BS,Chp.V]), that is, if T : X0 + X1 → X0 + X1 is a linear operator such thattakes Xi into Xi continuously for each i = 0, 1, then T takes (X0, X1)X into(X0, X1)X continuously.

Every rearrangement invariant space X on [0,∞) can be obtained throughthe above procedure, called K–method of interpolation of Peetre, as X =(L1, L∞)X .

4. Orlicz spaces. Given a measure space (Ω, Σ, µ) and a convex continuousincreasing function Φ: [0,∞) → [0,∞) satisfying that Φ(t) = 0 if and only

6 Preliminaries

if t = 0, the Orlicz space

LΦ(µ) =

f : Ω → R measurable :

∫Φ(α|f |)dµ < ∞ for some α > 0

,

is a Banach space with the norm

‖f‖Φ = inf

k > 0 :

∫Φ

( |f |k

)dµ ≤ 1

.

Even more, LΦ(µ) is a Banach function space with respect to (Ω, Σ, µ), havingthe Fatou property. For results concerning Orlicz spaces, see [KR] and [RR].

5. L1(ν) spaces. Let Σ be a σ–algebra of subsets of a set Ω, X a Banachspace and ν : Σ → X a vector measure, that is, a countably additive setfunction:

∑ν(An) converges to ν(∪An) in X, whenever (An) ⊂ Σ is a

sequence of disjoint sets. Observe that in this case ν is strongly additive,that is, ν(An) → 0 for every sequence of disjoint sets (An) ⊂ Σ. Then ν isbounded, that is, the range of ν is a bonded set in X, [DU, Corollary I.1.19].

For each x∗ ∈ X∗, the dual space of X, the variation of the measurex∗ν : Σ → R is denoted by |x∗ν|. The semivariation of ν is the set functiondefined on Σ by

‖ν‖(A) = sup|x∗ν|(A) : x∗ ∈ BX∗ ,

where BX∗ is the unit ball of X∗. The map ‖ν‖ is monotone increasing,countably subadditive and for all A ∈ Σ satisfies

1

2‖ν‖(A) ≤ sup‖ν(B)‖X : B ∈ Σ ∩ 2A ≤ ‖ν‖(A) ,

[DU, Proposition I.1.11]. The boundedness of ν is equivalent to the finitenessof the semivariation of ν. A set A ∈ Σ is ν–null if ‖ν‖(A) = 0. A propertyholds almost everywhere with respect to ν (ν–a.e.) if it holds except on aν–null set.

A countably additive measure λ : Σ → [0,∞) is a control measure for ν

iflim

λ(A)→0‖ν(A)‖ = 0 .

Preliminaries 7

This implies that if λ(A) = 0 then ‖ν‖(A) = 0. Even more, this last conditionmeans that λ is a control measure for ν, since λ and ν are defined on a σ–algebra and they are countably additive.

Bartle, Dunford and Schwartz prove that every vector measure definedon a σ–algebra has a control measure [BDS, Corollary 2.4]. Later, Rybakovprove that such a control measure can be taken of the type |x∗0ν| for certainx∗0 ∈ BX∗ , [DU, Theorem IX.2.2]. In this case |x∗0ν| is called Rybakov controlmeasure for ν. Observe that a Rybakov control measure λ = |x∗0ν| satisfies‖ν‖(A) = 0 if and only if λ(A) = 0. The vectors x∗ ∈ X∗ such that |x∗ν| isa control measure for ν constitute a dense set in X∗ [DU, Corollary IX.2.3],which is also a Gδ set [BL].

For general results about vector measures see [DU] and [D].

Identifying functions which are equal ν–a.e., the space L1w(ν) of functions

f : Ω → R which are integrable with respect to x∗ν, for all x∗ ∈ X∗, is aBanach space with norm

‖f‖ν = supx∗∈BX∗

∫|f |d|x∗ν| ,

see [KK] and also [St, Theorem 9]. Moreover, L1w(ν) is a Banach function

space with respect to (Ω, Σ, λ), where λ is any Rybakov control measure forν. Note that L1

w(ν) has the Fatou property and so is a Banach function spacein the sense of Bennett and Sharpley.

A function f ∈ L1w(ν) is integrable with respect to ν (in the sense of

Lewis [L1, Definition 2.1]) if it satisfies that for each A ∈ Σ, there exists avector

∫A

fdν ∈ X such that

x∗( ∫

A

fdν)

=

∫

A

fdx∗ν , for all x∗ ∈ X∗ .

When A = Ω, we write∫

fdν instead of∫Ω

fdν.

The space L1(ν) of integrable functions with respect to ν with the norm‖ · ‖ν is a Banach space, [KK, II.2, Theorem IV.4.1, Theorem IV.7.1]. Every

8 Preliminaries

simple function ϕ =∑n

i=1 aiχAiwith ai ∈ R and Ai ∈ Σ, is integrable with

respect to ν and

∫

A

ϕdν =n∑

i=1

aiν(Ai ∩ A) , for all A ∈ Σ .

Even more, the simple functions are dense in L1(ν), [L2, Theorem 3.5].

The space L1(ν) satisfies the following theorem of dominated conver-gence: Given fn, g ∈ L1(ν) such that fn → f ν–a.e. and |fn| ≤ g for all n,we have that f ∈ L1(ν) and fn → f in L1(ν), [BDS, Theorem 2.8] and [L1,Theorem 2.2].

Given a Rybakov control measure λ for ν, since L1w(ν) is a Banach func-

tion space with respect to (Ω, Σ, λ) in the sense of Bennett and Sharpley, andL1(ν) is the closure of the simple functions in L1

w(ν), we have that L1(ν) is anideal of measurable functions with respect to λ, that is, if f is a measurablefunction such that |f | ≤ |g| λ–a.e. for some g ∈ L1(ν), then f ∈ L1(ν),[BS, Theorem I.3.11]. Thus, L1(ν) is a Banach function space with respectto (Ω, Σ, λ), which is also order continuous and has weak unit, [C2, Theo-rem 1]. Note that the order continuity of L1(ν) follows from the dominatedconvergence theorem.

Since L1(ν) is an ideal of measurable functions which contains the simplefunctions, every bounded measurable function f is in L1(ν) with ‖f‖ν ≤‖f‖∞‖ν‖(Ω).

The space L1(ν) is a Banach function subspace of L1w(ν), in this case

with the same norm. The spaces L1(ν) and L1w(ν) coincide when X does not

contain any subspace isomorphic to c0, [L2, Theorem 5.1].

CHAPTER 1: Subspaces of L1(ν)

The space L1(ν) of integrable functions with respect to a vector measureν defined on a σ–algebra and with values in a Banach space is, as notedin Preliminaries, an order continuous Banach function space with respectto certain measure space, having a weak unit. In this chapter, by Banachfunction subspace of L1(ν) we mean Banach function space on the samemeasure space that L1(ν), continuously contained in L1(ν). We prove thatevery order continuous Banach function subspace of L1(ν) can be describedthrough certain function closely related to ν, which we call ν–norm function.Each ν–norm function generates a Banach function subspace of L1(ν). Inparticular, taking the appropriate ν–norm function, we obtain some Orlicztype spaces LΦ(ν). When the Orlicz function Φ has the ∆2 property, LΦ(ν)can be described as the space of measurable functions f such that Φ(f) isintegrable with respect to ν.

Let Σ be a σ–algebra of subsets of an abstract set Ω and M the spaceof measurable functions f : Ω → R . Throughout this chapter different mea-sures on Σ are considered, therefore we identify functions which are equalalmost everywhere only after fixing a measure.

Let ν : Σ → X be a vector measure with values in a Banach spaceX. Recall that X∗ is the dual space of X and BX∗ is the unit ball of X∗.The following definition provides the function from which we will constructBanach function subspaces of L1(ν).

Definition 1.1. A map ρ : X∗ ×M → [0,∞] is a ν–norm function if thefollowing conditions hold:

a) For x∗ ∈ X∗, the map ρx∗ : M → [0,∞] given by ρx∗(f) := ρ(x∗, f)

10 Chapter 1

satisfies:

a1) ρx∗(f) = 0 if and only if f = 0 |x∗ν|–a.e.,

a2) ρx∗(af) = |a|ρx∗(f), for all a ∈ R and f ∈M,

a3) ρx∗(f + g) ≤ ρx∗(f) + ρx∗(g), for all f, g ∈M,

a4) if f, g ∈M and |f | ≤ |g| |x∗ν|–a.e., then ρx∗(f) ≤ ρx∗(g),

a5) if f, fn ∈M and 0 ≤ fn ↑ f |x∗ν|–a.e., then ρx∗(fn) ↑ ρx∗(f),

a6) ρx∗(χΩ) < ∞,

a7) there exists C = C(x∗) > 0 such that

∫|f |d|x∗ν| ≤ Cρx∗(f) for all f ∈M .

b) For f ∈ M, the map ρf : X∗ → [0,∞] given by ρf (x∗) := ρ(x∗, f)

satisfies:

b1) |a|ρf (x∗) ≤ ρf (ax∗), for all a ∈ R with |a| ≤ 1 and x∗ ∈ X∗,

b2) if f = χΩ then supx∗∈BX∗ ρf (x∗) < ∞.

Example 1.2. The map ρ : X∗ ×M→ [0,∞] defined by

ρ(x∗, f) =

∫|f |d|x∗ν| ,

is a ν–norm function. Indeed, the definition of ν–norm function is moti-vated by this map, and as we will see later, it is the ν–norm function whichgenerates the space L1(ν).

Given a ν–norm function ρ, for each x∗ ∈ X∗ we consider the space

Ex∗ = f ∈M : ρx∗(f) < ∞ ,

where functions which are equal |x∗ν|–a.e. are identified. Observe thatproperties a1)–a7) of the Definition 1.1, establish that ρx∗ restricted to thecone of all positive functions in M, is a function norm in the sense of Ben-nett and Sharpley (see Preliminaries) and since property a4) implies that

Chapter 1 11

ρx∗(f) = ρx∗(|f |) for all f ∈ M, we have that Ex∗ endowed with the normρx∗ is a Banach function space with respect to the measure space (Ω, Σ, |x∗ν|).

Note that if ρ is the ν–norm function of Example 1.2, then each spaceEx∗ is just the space L1(|x∗ν|).

Definition 1.3. Let ρ be a ν–norm function. We define ρν : M → [0,∞]by


ρx∗(f) ,

and denote Ew(ρν) = f ∈ M : ρν(f) < ∞, where functions which areequal ν–a.e. are identified.

For the ν–norm function ρ given in Example 1.2, we have that ρν(f) =‖f‖ν for all f ∈M and Ew(ρν) = L1

w(ν). Recall that L1w(ν) is a Banach func-

tion space with respect to (Ω, Σ, λ), where λ is a Rybakov control measurefor ν.

Proposition 1.4. Given a ν–norm function ρ, the space Ew(ρν) is Banachfunction subspace of L1

w(ν), with norm ρν.

PROOF. Let λ = |x∗0ν| be a Rybakov control measure for ν, with x∗0 ∈ BX∗ .Let us see that ρν satisfies conditions 1)–7) of the definition of function normwith respect to the measure λ. Since the λ–null and the ν–null sets coincide,and every ν–null set is |x∗ν|–null for all x∗ ∈ X∗, properties 1)–4) for ρν

follow from the corresponding properties a1)–a4) for each ρx∗ .

Let us prove property 5). Let fn, f ∈ M be such that 0 ≤ fn ↑ f λ–a.e.Since ρν satisfies property 4), then

ρν(fn) ↑ α := supn≥1

ρν(fn) ≤ ρν(f) .

On other hand, given x∗ ∈ BX∗ , since ρx∗ satisfies property a5), we have

ρx∗(f) = limn→∞

ρx∗(fn) ≤ limn→∞

ρν(fn) = α .

12 Chapter 1

Hence ρν(f) ≤ α. So, it follows that ρν(fn) ↑ ρν(f).

Property 6) for ρν is just property b2) for ρ and property 7) holds forthe constant C = C(x∗0) given by property a7) for ρx∗0 .

Since ρν is a function norm with respect to λ with ρν(f) = ρν(|f |) forall f ∈ M, we have that Ew(ρν) is a Banach function space with respectto (Ω, Σ, λ) in the sense of Bennett and Sharpley. In particular, Ew(ρν) is aBanach lattice.

Given x∗ ∈ BX∗ , from property a7) for ρx∗ it follows

∫|f |d|x∗ν| ≤ C(x∗)ρx∗(f) ≤ C(x∗)ρν(f) .

Thus, if f ∈ Ew(ρν) then f is integrable with respect to |x∗ν| for all x∗ ∈BX∗ and so for all x∗ ∈ X∗. Hence, Ew(ρν) is contained in L1

w(ν). Theinclusion is well defined since in both spaces the equivalence classes are takenfor functions which are equal ν–a.e., or equivalently λ–a.e. Moreover, theinclusion is continuous since it is a positive linear operator between Banachlattices. So, Ew(ρν) is a Banach function subspace of L1

w(ν) (with differentnorm).

The aim of property b1) of a ν–norm function ρ is to obtain that iff ∈ Ew(ρν) then f ∈ Ex∗ for all x∗ ∈ X∗, such as it happens with L1

w(ν)and L1(|x∗ν|). Given f ∈M and x∗ ∈ BX∗ , by definition of ρν we have thatρx∗(f) ≤ ρν(f). For x∗ ∈ X∗ with ‖x∗‖ > 1, applying property b1) we obtain

ρν(f) ≥ ρ x∗‖x∗‖

(f) = ρf

(x∗

‖x∗‖)≥ ρf (x

∗)‖x∗‖ ,

and so ρx∗(f) ≤ ‖x∗‖ρν(f). Then each x∗ ∈ X∗ satisfies

ρx∗(f) ≤ max1, ‖x∗‖ρν(f) for all f ∈M .

Since all ν–null set is |x∗ν|–null, we have that f = g ν–a.e. implies f = g

|x∗ν|–a.e. and so two functions in the same equivalence class of Ew(ρν), are

Chapter 1 13

in the same equivalence class of Ex∗ . Then, the map which takes a class ofEw(ρν) represented by f into the class of Ex∗ represented by f is well definedand continuous. Moreover, it is injective, that is, it is the identity map, ifand only if |x∗ν| is a control measure for ν.

In view of the above comments, we have the following containment ofsets

Ew(ρν) ⊂ f ∈M : ρx∗(f) < ∞ for all x∗ ∈ X∗ . (1.1)

The two vector spaces will coincide (after identifying functions which areequal ν–a.e.) if every f ∈ M such that the map ρf is finite, satisfies thatρf (BX∗) is a bounded set, since in this case if f ∈ M satisfies ρx∗(f) < ∞for all x∗ ∈ X∗, that is, ρf is finite, we have


ρx∗(f) = supx∗∈BX∗

ρf (x∗) < ∞

and so f ∈ Ew(ρν). When the containment in (1.1) is an equality we havea simpler description for the space Ew(ρν), so we are interested in obtainingconditions over the ν–norm function ρ which establish this equality.

It is known that the ν–norm function given in Example 1.2 satisfies theequality in (1.1). This fact is deduced by applying the Banach–SteinhausTheorem to the continuous linear operators given by

x∗ ∈ X∗ →∫

A

ϕndx∗ν ∈ R ,

with A ∈ Σ and (ϕn) sequence of positive simple functions increasing to |f |,for each f ∈M such that

∫ |f |d|x∗ν| < ∞ for all x∗ ∈ X∗.

From this idea and a version of the Banach–Steinhaus Theorem which canbe applied to functions of the type ρf : X∗ → [0,∞) , for ρ a ν–norm function,in Proposition 1.6 we obtain conditions which guarantee the equality in (1.1).The proof of the following lemma is similar to the proof of the Banach–Steinhaus’ classical result.

Lemma 1.5. Let X be a Banach space and Tαα∈I a family of continuousmaps Tα : X → [0,∞) satisfying

14 Chapter 1

1) Tα(x + y) ≤ Tα(x) + Tα(y) for all x, y ∈ X (i.e. Tα subadditive).

2) |a|Tα(x) ≤ Tα(ax) for all x ∈ X and a ∈ R with |a| ≤ 1.

If Tαα∈I is pointwise bounded (i.e. for each x ∈ X there exists Mx > 0such that Tα(x) ≤ Mx for all α ∈ I), then it is uniformly bounded over BX ,that is, there exists M > 0 such that supx∈BX

Tα(x) ≤ M for all α ∈ I.

PROOF. Given n ∈ N, we consider the set

Fn = x ∈ X : Tα(x) ≤ n for all α ∈ I =⋂α∈I

T−1α

([0, n]

).

The continuity of each Tα implies that Fn is a closed set in X. Since Tαα∈I

is pointwise bounded, we have that X = ∪n≥1Fn. From the Baire Theorem,there exists n0 ∈ N such that Fn0 has non empty interior. So, there existx0 ∈ Fn0 and 0 < r < 1 such that

B(x0, r) = x ∈ X : ‖x− x0‖X < r ⊂ Fn0 .

Given x ∈ BX and α ∈ I, applying conditions 1) and 2), we have

Tα(x) ≤ 1

rTα(rx) ≤ 1

r

(Tα(rx + x0) + Tα(−x0)

) ≤ 2n0

r,

since rx + x0 and x0 belong to B(x0, r) and from 2) Tα(−y) = Tα(y) for ally ∈ X. Hence, Tαα∈I is uniformly bounded over BX by M = 2n0/r.

Proposition 1.6. Let ρ be a ν–norm function. If every simple function ϕ

satisfies that the map ρϕ : X∗ → [0,∞) is subadditive and continuous, then

Ew(ρν) = f ∈M : ρx∗(f) < ∞ for all x∗ ∈ X∗ .

PROOF. Let f ∈ M be such that ρx∗(f) < ∞ for all x∗ ∈ X∗ and (ϕn) asequence of positive simple functions increasing to |f |. By hypothesis the mapρϕn : X∗ → [0,∞) is subadditive and continuous, and by property b1) of ρ,we have that |a|ρϕn(x∗) ≤ ρϕn(ax∗) for all a ∈ R with |a| ≤ 1 and x∗ ∈ X∗.Moreover ρϕn is a pointwise bounded family of maps, since ρϕn(x∗) =ρx∗(ϕn) ≤ ρx∗(f) for all n. Then, by Lemma 1.5, there exists a constant

Chapter 1 15

M > 0 such that supx∗∈BX∗ ρϕn(x∗) ≤ M for all n. From property a5) of ρ,for each x∗ ∈ BX∗ we have that ρx∗(f) = ρx∗(|f |) = limn→∞ ρx∗(ϕn) ≤ M .Hence,


ρx∗(f) ≤ M

and so f ∈ Ew(ρν).

Given a ν–norm function ρ we have constructed a Banach function sub-space Ew(ρν) of L1

w(ν). From Ew(ρν), we obtain a Banach function subspaceof L1(ν).

Definition 1.7. Let ρ be a ν–norm function. The space E(ρν) is defined asthe closure of the simple functions in the space Ew(ρν).

Observe that for the ν–norm function ρ of Example 1.2, the space E(ρν)is precisely L1(ν), as we had expected.

Proposition 1.8. Given a ν–norm function ρ, the space E(ρν) is a Banachfunction subspace of L1(ν), with norm ρν.

PROOF. By definition E(ρν) is a closed vector subspace of Ew(ρν), so E(ρν)is a Banach space. In the proof of Proposition 1.4 we saw that Ew(ρν) is aBanach function space with respect to (Ω, Σ, λ) in the sense of Bennett andSharpley, where λ is a Rybakov control measure for ν. Since E(ρν) is theclosure of the simple functions in Ew(ρν), we have that E(ρν) is an ideal ofmeasurable functions with respect to λ, that is, f ∈ E(ρν) whenever f ∈Mwith |f | ≤ |g| λ–a.e. for some g ∈ E(ρν), [BS, Theorem I.3.11]. Thus,E(ρν) is a Banach function space with respect to (Ω, Σ, λ). Indeed, E(ρν) isa Banach function subspace of Ew(ρν).

Given a function f ∈ E(ρν), by definition there exists a sequence ofsimple functions (ϕn) such that ϕn → f in the norm ρν . From Proposition1.4 we have that Ew(ρν) is continuously contained in L1

w(ν), then ϕn → f inthe norm ‖ · ‖ν . Since L1(ν) is the closure of the simple functions in L1

w(ν),it follows that f ∈ L1(ν). So, E(ρν) is continuously contained in L1(ν).

16 Chapter 1

The order continuity of a Banach function subspace Y of L1(ν), will beone of the keys for representing Y as a space E(ρν) generated by a ν–normfunction ρ. We have the following results concerning order continuity ofspaces of the type E(ρν).

Lemma 1.9. Let ρ be a ν–norm function and λ a Rybakov control measurefor ν. Then:

a) E(ρν) is order continuous if and only if ρν(χA) → 0 as ‖ν‖(A) → 0.

b) If E(ρν) is order continuous, then

E(ρν) =

f ∈M : lim‖ν‖(A)→0

ρν(fχA) = 0

.

PROOF. The space E(ρν) is order continuous if and only if all its functionshave absolutely continuous norm with respect to λ (see Preliminaries). SinceE(ρν) is the closure of the simple functions in the Banach function space (inthe sense of Bennett and Sharpley) Ew(ρν) with respect to (Ω, Σ, λ) and λ

is finite, from [BS, Theorem I.3.13] it follows that E(ρν) is order continuousif and only if χΩ has absolutely continuous norm, that is, ρν(χA) → 0 asλ(A) → 0, or equivalently ‖ν‖(A) → 0. So a) holds.

Let us prove b). Suppose that E(ρν) is order continuous. Then everyfunction f ∈ E(ρν) has absolutely continuous norm with respect to λ andsince λ is a control measure for ν, ρν(fχA) → 0 as ‖ν‖(A) → 0.

Conversely, let f ∈ M satisfy ρν(fχA) → 0 as ‖ν‖(A) → 0, or equiva-lently λ(A) → 0. Consider the functions fn = fχ|f |−1([0,n]). Since E(ρν) is anideal containing the simple functions and each fn is bounded, we have thatfn ∈ E(ρν). Moreover, since λ

(|f |−1((n,m])) → 0 as m,n → ∞, we have

ρν(fm− fn) = ρν

(fχ|f |−1((n,m])

) → 0. Then, there exists h ∈ E(ρν) such thatfn → h in E(ρν) and there exists a subsequence fnk

→ h λ–a.e. Observethat fn → f pointwise, so f = h ∈ E(ρν).

The spaces Lp(ν) = f ∈ M : fp ∈ L1(ν), studied by Sanchez Perez[S1], can be generated through a ν–norm function.

Chapter 1 17

Example 1.10. Given p ∈ [1,∞), we consider the map ρ : X∗×M→ [0,∞]defined by

ρ(x∗, f) =( ∫

|f |pd|x∗ν|)1/p

.

It can be checked that ρ is a ν–norm function. For each x∗ ∈ X∗, the spaceEx∗ coincides with the space Lp(|x∗ν|). Moreover, for all f ∈ M we havethat


ρx∗(f) = supx∗∈BX∗

( ∫|f |pd|x∗ν|

)1/p

= ‖fp‖1/pν .

So, Ew(ρν) = f ∈M : fp ∈ L1w(ν). In this case the equality in (1.1) holds.

The space Lp(ν), obviously contained in Ew(ρν), is a Banach space with thenorm ρν , in which the simple functions are dense, [S1, Proposition 4]. Then,E(ρν) coincides with Lp(ν). The order continuity of the space Lp(ν), provedby Sanchez Perez in [S1, Proposition 6], can be directly checked from Lemma1.9.a):

lim‖ν‖(A)→0

ρν(χA) = lim‖ν‖(A)→0

‖χA‖1/pν = lim

‖ν‖(A)→0‖ν‖(A)1/p = 0 .

We can try to construct ν–norm functions which would generate Banachfunction subspaces of L1(ν), through the norms of given Banach functionspaces. This is what we are going to do with the Orlicz spaces.

Example 1.11. Let Φ: [0,∞) → [0,∞) be a convex continuous increasingfunction such that Φ(t) = 0 if and only if t = 0, and ρ : X∗ ×M → [0,∞]the map defined by

ρ(x∗, f) = inf

k > 0 :

∫Φ

( |f |k

)d|x∗ν| ≤ 1

. (1.2)

Observe that if the infimum in (1.2) is finite and strictly positive, then it is aminimum. It can be directly checked that ρ is a ν–norm function. For eachx∗ ∈ X∗, the space Ex∗ = f ∈M : ρx∗(f) < ∞ is the classical Orlicz spaceLΦ(|x∗ν|).

From the ν–norm function given in Example 1.11, we construct the spacesEw(ρν) and E(ρν) which will be denoted by LΦ

w(ν) and LΦ(ν) respectively. We

18 Chapter 1

call Orlicz space with respect to the vector measure ν and the function Φ toLΦ(ν). Although by analogy with the classical Orlicz spaces this designationcould correspond to LΦ

w(ν), in view of next Remark 1.16, we think it is moreappropriate for LΦ(ν). The spaces LΦ(ν) generalize the spaces Lp(ν), whichcan be obtained by taking the function Φ(t) = tp.

Proposition 1.12. The space LΦw(ν) satisfies the equality in (1.1). Moreover,

the space LΦ(ν) is order continuous.

PROOF. Let us see that for all function f ∈M, the map ρf : X∗ → [0,∞]is subadditive. Set f ∈ M and x∗1, x

∗2 ∈ X∗. Suppose that ρf (x

∗i ) < ∞ for

i = 1, 2, then there exist k1, k2 > 0 such that∫

Φ( |f |

ki

)d|x∗i ν| ≤ 1 for i = 1, 2

and so∫

Φ( |f |

k1 + k2

)d|(x∗1 + x∗2)ν| ≤

∑i=1,2

∫Φ

( |f |k1 + k2

)d|x∗i ν|

≤∑i=1,2

ki

k1 + k2

∫Φ

( |f |ki

)d|x∗i ν| ≤ 1 .

Hence, from (1.2) it follows that ρf (x∗1 +x∗2) ≤ k1 +k2, where taking infimum

in k1 and k2 we obtain ρf (x∗1 + x∗2) ≤ ρf (x

∗1) + ρf (x

∗2). The case ρf (x

∗i ) = ∞

for some i = 1, 2 is trivial.

Let h ∈ M be bounded ν–a.e. by a constant C > 0. Then the functionρh : X∗ → [0,∞) is continuous at zero, since for ε > 0 we have

∫Φ

( |h|ε

)d|x∗ν| ≤ Φ

(C

ε

)|x∗ν|(Ω) ≤ Φ

(C

ε

)‖ν‖(Ω)‖x∗‖ ≤ 1

whenever ‖x∗‖ ≤ δε =(‖ν‖(Ω)Φ(C/ε)

)−1and so, by (1.2), ρh(x

∗) ≤ ε for allx∗ ∈ X∗ with ‖x∗‖ ≤ δε. From the subadditivity of ρh, it follows that it isalso continuous at every x∗ ∈ X∗.

In particular, for every simple function ϕ the map ρϕ : X∗ → [0,∞) issubadditive and continuous. Thus, by Proposition 1.6 we have that LΦ

w(ν)satisfies the equality in (1.1).

Chapter 1 19

The space LΦ(ν) is order continuous if and only if ρν(χA) → 0 as‖ν‖(A) → 0 (Lemma 1.9 a)). Given ε > 0, for all x∗ ∈ BX∗ we have

∫Φ

(χA

ε

)d|x∗ν| = Φ

(1

ε

)|x∗ν|(A) ≤ Φ

(1

ε

)‖ν‖(A) ≤ 1

whenever ‖ν‖(A) ≤ δε = Φ(

1ε

)−1and in this case, by (1.2), ρx∗(χA) ≤ ε.

Hence, ρν(χA) ≤ ε whenever ‖ν‖(A) ≤ δε and so LΦ(ν) is order continuous.

An important property of Orlicz functions is the ∆2–property, that is,the existence of a constant b > 0 such that Φ(2t) ≤ bΦ(t) for all t ≥ 0. Inour case, this property allows us to give a simple description of the spacesLΦ

w(ν) and LΦ(ν).

In the case when Φ has the ∆2–property, for each x∗ ∈ X∗ and f ∈ M,we have that ρx∗(f) < ∞ if and only if

∫Φ(|f |)d|x∗ν| < ∞. Then, from the

equality (1.1) for LΦw(ν), it follows

LΦw(ν) = f ∈M : Φ(|f |) ∈ L1

w(ν) .

For obtaining a description of LΦ(ν) in a similar way, we need the followingcharacterization of the convergence in LΦ

w(ν).

Proposition 1.13. Suppose that the function Φ has the ∆2–property. Then,a sequence (fn) converges to zero in the norm of LΦ

w(ν) if and only if thesequence (Φ(|fn|)) converges to zero in the norm of L1

w(ν).

PROOF. Let (fn) be a sequence convergent to zero in LΦw(ν). Then, for large

enough n, we have ρx∗(fn) ≤ ρν(fn) ≤ 1 for all x∗ ∈ BX∗ . If ρx∗(fn) > 0,from the convexity of Φ it follows

∫Φ(|fn|)d|x∗ν| ≤ ρx∗(fn)

∫Φ

( |fn|ρx∗(fn)

)d|x∗ν| ≤ ρx∗(fn) ≤ ρν(fn) .

If ρx∗(fn) = 0, then fn = 0 |x∗ν|–a.e. and so∫

Φ(|fn|)d|x∗ν| = 0. Hence,‖Φ(|fn|)‖ν ≤ ρν(fn) and so (Φ(|fn|)) converges to zero in L1

w(ν).

20 Chapter 1

Conversely, suppose that (Φ(|fn|)) converges to zero in the norm ‖ · ‖ν .Given ε > 0, we take kε ∈ N such that 1/2kε < ε. From the ∆2–property ofΦ, it follows

∫Φ(2kε|fn|)d|x∗ν| ≤ bkε

∫Φ(|fn|)d|x∗ν| ≤ bkε‖Φ(|fn|)‖ν ≤ 1

for all x∗ ∈ BX∗ and for large enough n (only depending on ε). Thus,ρν(fn) ≤ 1/2kε < ε for large enough n.

Remark 1.14. The necessity condition in Proposition 1.13 holds even whenΦ does not satisfy the ∆2–property, that is, if Φ is any Orlicz function thenfn → 0 in LΦ

w(ν) implies that Φ(|fn|) → 0 in L1w(ν). For the sufficiency

condition, we can weaken the hypothesis by requiring the ∆2–property inthe infinite, that is, there exist t0 > 0 and b > 0 such that if t ≥ t0 thenΦ(2t) ≤ bΦ(t). In this case, if ‖Φ(|fn|)‖ν → 0, there exists a subsequenceΦ(|fnj

|) → 0 ν–a.e. Since Φ is an increasing positive function vanishing onlyat zero, then fnj

→ 0 ν–a.e. Given ε > 0, let kε ∈ N satisfies 1/2kε < ε.From the ∆2–property (in the infinite) of Φ, for every x∗ ∈ BX∗ it follows

∫

|fnj |−1[t0,∞)

Φ(2kε |fnj|)d|x∗ν| ≤ bkε

∫Φ(|fnj

|)d|x∗ν| ≤ bkε‖Φ(|fnj|)‖ν .

So,

∫Φ(2kε|fnj

|)d|x∗ν| ≤ bkε‖Φ(|fnj|)‖ν + ‖Φ(2kε|fnj

|)χ|fnj |−1[0,t0)‖ν ≤ 1

for large enough j (only depending on ε), where we have used the domi-nated convergence theorem for the functions Φ(2kε |fnj

|)χ|fnj |−1[0,t0) which are

bounded by Φ(2kεt0)χΩ ∈ L1(ν) and converge to zero ν–a.e. since fnj→ 0

ν–a.e. Then, ρν(fnj) ≤ 1/2kε < ε for large enough j.

Therefore, if Φ has the ∆2–property in the infinite, the sufficiency con-dition in Proposition 1.13 is only satisfied for a subsequence (fnj

), but thisis enough to obtain the description of LΦ(ν) given in the next proposition.

Chapter 1 21

Proposition 1.15. If Φ has the ∆2–property in the infinite, then

LΦ(ν) = f ∈M : Φ(|f |) ∈ L1(ν) .

PROOF. The space LΦ(ν) is order continuous (Proposition 1.12), then byLemma 1.9.b), we have

LΦ(ν) =

f ∈M : lim‖ν‖(A)→0

ρν(fχA) = 0

.

If Φ has the ∆2–property in the infinite, then from Remark 1.14 it follows

LΦ(ν) =

f ∈M : lim

‖ν‖(A)→0‖Φ(|f |)χA‖ν = 0

= f ∈M : Φ(|f |) ∈ L1(ν) ,

where the last equality is deduced from [L1, Theorem 2.6], that is, Lemma1.9.b) applied to the space L1(ν).

Remark 1.16. Suppose Φ satisfies the ∆2–property in the infinite. Then,LΦ(ν) = LΦ

w(ν) if and only if L1(ν) = L1w(ν). So, there exist vector mea-

sures ν such that LΦ(ν) LΦw(ν). Since Φ has the ∆2–property, the simple

functions are dense in the classical Orlicz space LΦ(µ) for a finite positivemeasure µ. Then, in this case, the spaces LΦ(ν) resemble the classical Orliczspaces. Note that if X does not contain any copy of c0, then L1

w(ν) = L1(ν)and so LΦ(ν) = LΦ

w(ν).

We have seen that every ν–norm function ρ generates the Banach func-tion subspace E(ρν) of L1(ν). Now, the question is if any Banach functionsubspace of L1(ν) in the sense we have given to this concept, can be gen-erated through a ν–norm function. The answer will be positive for ordercontinuous subspaces.

Theorem 1.17. Let Σ be a σ–algebra of subsets of a set Ω, X a Banachspace, ν : Σ → X a vector measure and λ a Rybakov control measure forν. Consider an order continuous Banach function space Y with respect to(Ω, Σ, λ), continuously contained in L1(ν). Then there exists a ν–norm func-tion ρ such that Y = E(ρν) and ‖f‖Y = ρν(f) for all f ∈ Y .

22 Chapter 1

Let us introduce the tools necessary for the proof of Theorem 1.17.

Let ν : Σ → X be a vector measure, Y a Banach function subspace ofL1(ν) and λ = |x∗0ν| a Rybakov control measure for ν.

Definition 1.18. For each x∗ ∈ X∗, we define the space

Y ′x∗ =

g ∈ Y ′ : gχ[hx∗=0] = 0 λ–a.e.

⊂ Y ′ ,

where Y ′ is the Kothe dual of Y (with respect to λ) and hx∗ is the Radon-Nikodym derivative of the measure |x∗ν| with respect to λ. The space Y ′

x∗ isequipped with the norm and the order from Y ′.

Since Y is continuously contained in L1(ν) and every f ∈ L1(ν) is inte-grable with respect to |x∗ν|, we have that hx∗ ∈ Y ′

x∗ . The space Y ′x∗ is an

ideal in Y ′, that is, a closed vector subspace satisfying that f ∈ Y ′x∗ whenever

|f | ≤ |g| λ–a.e. for some g ∈ Y ′x∗ . Indeed, Y ′

x∗ is a band in Y ′ [LT, p. 3].

In general, the simple functions may not be included in Y ′x∗ . In the

case when this inclusion holds, we have that χA∩[hx∗=0] = 0 λ–a.e. for everyA ∈ Σ. Then, if |x∗ν|(A) = 0 (or equivalently hx∗χA = 0 λ–a.e.) we havethat λ(A) = 0, that is, λ is absolutely continuous with respect to |x∗ν|. Infact, the simple functions are contained in Y ′

x∗ if and only if λ is absolutelycontinuous with respect to |x∗ν|, or equivalently Y ′

x∗ = Y ′. This holds sinceif we suppose that λ(A) = 0 for all A ∈ Σ with |x∗ν|(A) = 0, then for g ∈ Y ′,the |x∗ν|–null set Sop (g) ∩ [hx∗ = 0] is λ–null, and so gχ[hx∗=0] = 0 λ–a.e.,that is, g ∈ Y ′

x∗ .

The spaces Y ′x∗ allow us to define a ν–norm function ρ such that the space

Ew(ρν) is just Y ′′, the Kothe bidual of Y .

Proposition 1.19. The map ρ : X∗ ×M→ [0,∞] defined by

ρ(x∗, f) = supg∈BY ′

x∗

∫|gf |dλ ,

is a ν–norm function.

Chapter 1 23

PROOF. Given x∗ ∈ X∗, let us see that the map ρx∗ : M→ [0,∞] satisfiespart a) of Definition 1.1. If f = 0 |x∗ν|–a.e., then its support set Sop(f)satisfies λ

(Sop(f) ∩ Sop(hx∗)

)= 0 and so fχSop(hx∗ ) = 0 λ–a.e. Hence,

fg = 0 λ–a.e. for every g ∈ Y ′x∗ and thus ρx∗(f) = 0. Conversely, suppose

that ρx∗(f) = 0. Since hx∗ ∈ Y ′x∗ , we have

∫ |f |d|x∗ν| =∫ |f |hx∗dλ = 0 and

so f = 0 |x∗ν|–a.e. Therefore, ρx∗ satisfies property a1). Properties a2)-a4)are also satisfied by ρx∗ and can be checked directly.

Let us prove property a5). Let fn, f ∈ M be such that 0 ≤ fn ↑ f

in Zc, the complement of a set Z ∈ Σ with |x∗ν|(Z) = 0, or equivalentlyλ(Z ∩ [hx∗ 6= 0]) = 0. Given g ∈ BY ′

x∗we have that 0 ≤ fn|g| ↑ f |g| λ–a.e.,

since λ(Z ∩ Sop (g)) ≤ λ(Z ∩ [hx∗ 6= 0]) + λ(Sop (g) ∩ [hx∗ = 0]) = 0. Fromthe monotone convergence theorem, it follows

∫|fg|dλ = lim

n→∞

∫|fng|dλ ≤ lim

n→∞ρx∗(fn) ≤ ρx∗(f) .

Taking supremum in g ∈ BY ′x∗

we have that ρx∗(fn) ↑ ρx∗(f).

For all A ∈ Σ we have that ρx∗(χA) ≤ ‖χA‖Y , so property a6) holds.Property a7) follows from

∫|f |d|x∗ν| =

∫|f |hx∗dλ ≤ ‖hx∗‖Y ′ρx∗(f) .

Set f ∈M. For all a ∈ R we have that hax∗ = |a|hx∗ λ–a.e. This impliesthat Y ′

ax∗ = Y ′x∗ for a 6= 0. Hence, ρf (ax∗) = ρf (x

∗) ≥ |a|ρf (x∗) whenever 0 <

|a| ≤ 1. So, the map ρf : X∗ → [0,∞] satisfies property b1) (the case whena = 0 is trivial). Property b2) holds, since supx∗∈BX∗ ρχΩ

(x∗) ≤ ‖χΩ‖Y .

Let ρ be the ν–norm function defined in Proposition 1.19. For eachx∗ ∈ X∗, we have

ρx∗(f) = supg∈BY ′

x∗

∫|fg|dλ ≤ sup

g∈BY ′

∫|fg|dλ = ρx∗0(f) ,

24 Chapter 1

since x∗0 ∈ BX∗ is such that |x∗0ν| is a Rybakov control measure for ν and soY ′

x∗0= Y ′. Then


ρx∗(f) = ρx∗0(f) = ‖f‖Y ′′

and thus Ew(ρν) = Y′′. Hence, the space E(ρν) is just the closure of the

simple functions in Y′′.

Finally we are in a position to prove Theorem 1.17. In the case when Y

is order continuous, we have that Y ′ is order isometric to Y ∗ and then

‖f‖Y = supg∈BY ′

∣∣∣∫

gfdλ∣∣∣ = ‖f‖Y ′′ for all f ∈ Y .

Moreover, the order continuity implies that the simple functions are dense inY , so Y is the closure of the simple functions in Y ′′. Therefore Y = E(ρν).

CHAPTER 2: L1(ν) for ν defined on a δ–ring.

SECTION 1. In this section we recall the integration theory with re-spect to vector measures defined on structures of sets weaker than σ–algebras,namely δ–rings. This integration theory, due to Lewis [L2] and Masani andNiemi [MN1], [MN2], extends the classical theory for vector measures definedon σ–algebras.

A δ–ring is a collection R of subsets of a set Ω, closed under relativecomplement, finite union and countable intersection of sets. Associated to aδ–ring R there is the σ–algebra

Rloc = A ⊂ Ω : A ∩B ∈ R for all B ∈ R .

The space of measurable real functions on (Ω,Rloc) is denoted by M. Thesimple functions are considered with respect to Rloc. An R–simple functionis a function of the type

∑ni=1 aiχAi

with ai ∈ R and Ai ∈ R. Note thatthe space of R–simple functions is just the space of simple functions withsupport in R.

Let R be a δ–ring and λ : R→ R a countably additive measure, that is,∑λ(An) converges to λ(∪An) whenever (An) is a sequence of disjoint sets in

R such that ∪An ∈ R. The variation of λ is the countably additive measure|λ| : Rloc → [0,∞] defined by

|λ|(A) = sup∑

|λ(Ai)| : (Ai) finite disjoint sequence in R∩ 2A

,

[MN1, Definition 2.3, Lemma 2.4]. A function f ∈ M is integrable withrespect to λ if |f |1,λ =

∫ |f |d|λ| < ∞, where |λ| is the variation of λ. Identi-fying functions which are equal |λ|–a.e., the space L1(λ) of integrable func-tions with respect to λ is a Banach space with norm | · |1,λ, in which the

26 Chapter 2

R–simple functions are dense [MN1, Triv. 2.15]. Given an R–simple func-tion ϕ =

∑ni=1 aiχAi

, the integral of ϕ with respect to λ over a set A ∈ Rloc

is defined by∫

Aϕ d λ =

∑ni=1 aiλ(Ai ∩ A). Given f ∈ L1(λ), the integral

of f with respect to λ over A ∈ Rloc is defined by∫

Af d λ = lim

∫A

ϕn d λ,where (ϕn) is a sequence of R–simple functions converging to f in L1(λ).

For every f ∈ L1(λ), the set function λf : Rloc → R defined by λf (A) =∫A

fdλ is a countably additive measure, which has finite variation satisfying|λf |(A) =

∫A|f |d|λ| for all A ∈ Rloc, [MN1, Lemma 2.21, Corollary 2.24,

Theorem 2.32].

Let R be a δ–ring, X a Banach space and ν : R→ X a vector measure,that is,

∑ν(An) converges to ν(∪An) in X whenever (An) is a sequence of

disjoint sets in R such that ∪An ∈ R. The semivariation of ν is the setfunction defined on Rloc by

‖ν‖(A) = sup|x∗ν|(A) : x∗ ∈ BX∗

,

where |x∗ν| is the variation of the measure x∗ν : R → R . The semivariationof ν is monotone increasing, countably subadditive, finite on R and for allA ∈ Rloc satisfies

1

2‖ν‖(A) ≤ sup

‖ν(B)‖X : B ∈ R ∩ 2A ≤ ‖ν‖(A) , (2.1)

[MN2, Lemma 3.4, Corollary 3.5]. In view of (2.1), the vector measure ν isbounded if and only if ‖ν‖(Ω) < ∞. A set A ∈ Rloc is ν–null if ‖ν‖(A) = 0,or equivalently if ν(B) = 0 for all B ∈ R ∩ 2A. A property holds ν–a.e. if itholds except on a ν–null set.

We denote by L1w(ν) the space of functions in M which are integrable

with respect to x∗ν for all x∗ ∈ X∗. Functions which are equal ν–a.e. areidentified. The space L1

w(ν) is a Banach space with the norm

‖f‖ν = supx∗∈BX∗

∫|f |d|x∗ν| ,

containing the R–simple functions and in which convergence in norm of asequence implies ν–a.e. convergence of some subsequence, [MN2, Lemma3.13].

Chapter 2 27

The space L1w(ν) is a Banach lattice for the ν–a.e. order (i.e. f ≥ 0 if

f ≥ 0 ν–a.e.). Moreover, L1w(ν) is an ideal of measurable functions, that is,

if f ∈M and g ∈ L1w(ν) with |f | ≤ |g| ν–a.e., then f ∈ L1

w(ν).

A function f ∈ L1w(ν) is integrable with respect to ν if for every A ∈ Rloc

there exists a vector denoted by∫

Afdν ∈ X, such that

x∗(∫

A

fdν

)=

∫

A

f dx∗ν for all x∗ ∈ X∗.

When A = Ω, we simply write∫

fdν for∫

Ωfdν.

We denote by L1(ν) the space of integrable functions with respect toν, where functions which are equal ν–a.e. are identified. Every R–simplefunction ϕ =

∑ni=1 aiχAi

is integrable with respect to ν with∫

Aϕ dν =∑n

i=1 aiν(Ai ∩ A) for all A ∈ Rloc. The space L1(ν) is a Banach space withthe norm ‖ · ‖ν , in which the R–simple functions are dense, [MN2, Theorem4.7]. Moreover, L1(ν) is a Banach lattice for the ν–a.e. order.

The space L1(ν) coincides with L1w(ν) whenever X does not contain any

subspace isomorphic to c0 [L2, Theorem 5.1].

The integration operator defined by f ∈ L1(ν) → ∫f dν ∈ X is linear,

continuous and satisfies

∥∥∥∫

fdν∥∥∥

X≤ ‖f‖ν . (2.2)

Given f ∈ L1(ν), the set function

A ∈ Rloc → νf (A) =

∫

A

f d ν ∈ X (2.3)

is a vector measure on the σ–algebra Rloc, with semivariation ‖νf‖(A) =‖fχA‖ν for all A ∈ Rloc, [L2, Theorem 3.2], [MN2, Theorem 4.4].

For every f ∈ L1(ν), applying the inequalities given in (2.1) to the vector

28 Chapter 2

measure νf and the total set Ω, we have

‖f‖ν

2≤ |‖f‖|ν = sup

∥∥∥∫

A

f d ν∥∥∥

X: A ∈ R

≤ ‖f‖ν . (2.4)

Therefore, |‖ · ‖|ν is a norm in L1(ν) equivalent to ‖ · ‖ν .

Lewis proves the following dominated convergence theorem [L2, Theorem3.3]:

Given a sequence (fn) in L1(ν) converging ν–a.e. to some function f ,such that |fn| ≤ g for all n and for some g ∈ L1(ν), we have that f ∈ L1(ν)and (fn) converges to f in L1(ν).

From this theorem it follows that L1(ν) is an order continuous Banach lattice.Moreover, L1(ν) is an ideal of measurable functions [MN2, Theorem 4.10].

Contrary to what happens with vector measures defined on σ–algebras,a vector measure defined on a δ–ring may be unbounded.

Example 2.1. Let R be the δ–ring of all Borel subsets of R having fi-nite Lebesgue measure which is denoted by m. For 1 ≤ p < ∞, the vec-tor measure ν : R → Lp(R) defined by ν(A) = χA is unbounded, since‖ν(A)‖p = m(A)1/p for all A ∈ R. In this case, ‖ϕ‖ν = ‖ϕ‖p for all R–simple function ϕ. Since the R–simple functions are dense in both L1(ν)and Lp(R), we have that L1(ν) = Lp(R). Moreover, the space Lp(R) doesnot contain any subspace isomorphic to c0 and so L1(ν) = L1

w(ν).

Given a vector measure ν defined on a δ–ring R of sets of Ω, since L1(ν)is an ideal of measurable functions, we have that the space of bounded mea-surable functions is contained in L1(ν) if and only if χΩ ∈ L1(ν). This failsto hold if ν is unbounded, in which this case χΩ is not even in L1

w(ν) since‖ν‖(Ω) = ‖χΩ‖ν = ∞. So, the bounded measurable functions may not beintegrable with respect to ν. Given a bounded measurable function f , if itssupport set Sop(f) has finite semivariation, or equivalently χSop(f) ∈ L1

w(ν),then f ∈ L1

w(ν) with

‖f‖ν ≤ ‖ν‖(Sop(f)) · ‖f‖∞ .

Chapter 2 29

If χSop(f) is also integrable with respect to ν, then f ∈ L1(ν).

Curbera proves in [C2, Theorem 8] that the class of order continuousBanach lattices having weak unit coincides with the class of spaces obtainedas L1 of a vector measure defined on a σ–algebra. The spaces L1(ν) for avector measure ν defined on a δ–ring, may not have a weak unit, as thenext example shows, although this spaces still are order continuous Banachlattices. Indeed, the characterization between these last classes of spacesholds, since Curbera proves in [C1, p. 22–23] that for every order continuousBanach lattice E, there exists a vector measure ν defined on a δ–ring suchthat E is order isometric to L1(ν).

Example 2.2. Let Γ be an abstract set and R the δ–ring of finite subsetsof Γ. In this case Rloc = 2Γ and M is the space of all functions f : Γ → R .Given p ∈ [1,∞], we denote Xp = `p(Γ) for p < ∞ and Xp = c0(Γ) forp = ∞. Consider the vector measure ν : R→ Xp defined by

ν(A) =∑γ∈A

eγ ,

where eγ is the characteristic function of the point γ ∈ Γ. The only ν–null setis the empty set. Each x∗ ∈ X∗

p is identified with an element (xγ)γ∈Γ ∈ `q(Γ),where 1/p+1/q = 1. Then, x∗ν(A) =

∑γ∈A xγ and |x∗ν|(A) =

∑γ∈A |xγ| for

all A ∈ R. For p < ∞, the space Xp = `p(Γ) does not contain any subspaceisomorphic to c0, so L1

w(ν) = L1(ν). In this case, given an R–simple functionϕ, for each x∗ = (xγ)γ∈Γ ∈ X∗

p we have

∫|ϕ|d|x∗ν| =

∑|ϕ(γ)||xγ| ≤

( ∑|ϕ(γ)|p

)1/p

‖x∗‖X∗p

and so

‖ϕ‖ν ≤( ∑

|ϕ(γ)|p)1/p

=∥∥∥

∑ϕ(γ)eγ

∥∥∥`p(Γ)

=∥∥∥

∫ϕdν

∥∥∥`p(Γ)

.

From this it follows by (2.2) that ‖ϕ‖ν = ‖ϕ‖`p(Γ). Since the R–simplefunctions are dense in `p(Γ), we have L1(ν) = `p(Γ). Suppose now thatp = ∞. Given f ∈ M, every x∗ = (xγ)γ∈Γ ∈ X∗

p has countable support and

30 Chapter 2

satisfies ∫|f |d|x∗ν| =

∑|f(γ)||xγ| ≤ ‖f‖∞‖x∗‖X∗

p.

Then,

|f(γ)| =∫|f |d|eγν| ≤ ‖f‖ν ≤ ‖f‖∞ for all γ ∈ Γ ,

that is, ‖f‖ν = ‖f‖∞. Hence, L1w(ν) = `∞(Γ). Moreover, L1(ν) = c0(Γ)

since the R–simple functions are dense in c0(Γ). In particular, L1(ν) is aproper closed subspace of L1

w(ν). For any p ∈ [1,∞], since every element ofXp has countable support, the space L1(ν) = Xp has a weak unit if and onlyif Γ is countable.

The following result provides an equivalent condition to the integrabilityof a function with respect to a vector measure defined on a δ–ring. Thisequivalent condition, which does not appear in the works of Lewis [L2], andMasani and Niemi [MN2], is an extension of the definition of integrable func-tion given by Bartle, Dunford and Schwartz for vector measures defined onσ–algebras, [BDS, Definition 2.5].

Proposition 2.3. Let R be a δ–ring, X a Banach space and ν : R → X avector measure. A function f ∈M is integrable with respect to ν if and onlyif there exists a sequence (ϕn) of R–simple functions satisfying

1) (ϕn) converges to f ν–a.e.

2)( ∫

Aϕndν

)converges in norm of X for all A ∈ Rloc.

PROOF. The necessity condition is obtained by taking a sequence of R–simple functions converging to f in the norm of L1(ν) and ν–a.e., and usingcontinuity of the integration operator.

Suppose that (ϕn) is a sequence of R–simple functions satisfying 1) and2). For obtaining that f is integrable with respect to ν, it suffices to provethat (ϕn) is a Cauchy sequence in L1(ν), since in this case (ϕn) converges tog in L1(ν) for some g ∈ L1(ν). Then, some subsequence of (ϕn) converges tog ν–a.e. and by 1) we have that f = g ν–a.e., so f ∈ L1(ν).

Chapter 2 31

For each n ∈ N, consider the vector measure νn : Rloc → X defined byνn(A) =

∫A

ϕndν. Let λn = |x∗nνn| a Rybakov control measure for νn, withx∗n ∈ BX∗ . Taking the non negative finite measure µ =

∑n≥1

λn

2n(λn(Ω)+1),

for each fixed n we have limµ(A)→0 ‖νn(A)‖X = 0. This fact together withcondition 2), are the hypothesis of Vitali-Hahn-Saks’s theorem (see [DU,Corollary I.5.6]) which ensures the uniformity with respect to n of the abovelimit. Hence, given ε > 0 there exists δ > 0 such that, for all n ≥ 1 and allA ∈ Rloc with µ(A) < δ, we have

∥∥∥∫

A

ϕndν∥∥∥

X= ‖νn(A)‖X < ε . (2.5)

Set Bm = ∩mj=1ϕ

−1j (0) and B = ∩m≥1Bm. Since µ(Bm\B) → 0 as

m →∞, there exists mδ such that µ(Bmδ\B) < δ/2.

On other hand, the condition 1) implies that ϕn → f µ–a.e., since everyν–null set is µ–null. By Egoroff’s theorem, we obtain a set Zδ in Rloc suchthat µ(Zδ) < δ/2 and ϕn → f uniformly in Zc

δ (the complement of Zδ). SinceBc

mδ∈ R and so ‖ν‖(Bc

mδ) < ∞, then there exists nδ such that

‖(f − ϕn)χZcδ‖∞ <

ε

2(1 + ‖ν‖(Bcmδ

)),

for all n ≥ nδ. So, for all m,n ≥ nδ and all D ∈ Rloc it follows∥∥∥

∫

D∩Zcδ∩Bc

mδ

(ϕn − ϕm)dν∥∥∥

X≤ ‖(ϕn − ϕm)χZc

δ∩Bcmδ‖ν

≤ ‖(ϕn − ϕm)χZcδ‖∞ · ‖ν‖(Bc

mδ) ≤ ε .

Set Hδ = Zδ ∪ Bmδ. Observe that ϕnχB = 0 for all n ∈ N and µ(Hδ\B) ≤

µ(Zδ) + µ(Bmδ\B) < δ

2+ δ

2= δ. Then, for all m,n ∈ N and all D ∈ Rloc,

from (2.5) it follows∥∥∥

∫

D∩Hδ


X=

∥∥∥∫

D∩(Hδ\B)


X≤ 2ε .

Hence, ‖ ∫D(ϕn−ϕm)dν‖X ≤ 3ε for all n,m ≥ nδ and all D ∈ Rloc. By (2.4)

we have ‖ϕn−ϕm‖ν ≤ 6ε for all n,m ≥ nδ. Thus, (ϕn) is a Cauchy sequencein L1(ν) and so f ∈ L1(ν).

32 Chapter 2

Remark 2.4. In the proof of Proposition 2.3, it is showed that if f ∈ Mand (ϕn) is a sequence of R–simple functions such that ϕn → f ν–a.e. and(∫

Aϕndν) converges in X for all A ∈ Rloc, then f ∈ L1(ν) and ϕn → f in

L1(ν). In particular,∫

Aϕdν → ∫

Afdν for all A ∈ Rloc.

SECTION 2. The properties of a vector measure ν defined on a δ–ringR influence the space L1(ν). In this section we study the effect on the latticeproperties of L1(ν), of the strong additivity and the σ–finiteness with respectto R of ν.

Let R be a δ–ring of subsets of Ω, X a Banach space and ν : R → X avector measure. The measure ν is strongly additive if (ν(An)) converges tozero whenever (An) is a disjoint sequence in R. Observe that ν is stronglyadditive if and only if

∑ν(An) is unconditionally convergent for all disjoint

sequence (An) in R, [BD, Section 1].

The measure ν : R → X is σ–finite with respect to R, or briefly σ–finite, if there exist a sequence (An) in R and a ν–null set N ∈ Rloc suchthat Ω = (∪An)∪N . Obviously, every vector measure defined on a σ–algebra,is strongly additive and σ–finite.

Every strongly additive vector measure defined on a δ–ring is σ–finite,[BD, Lemma 1.1]. The converse does not hold as Example 2.1 shows.

Let R be a δ–ring and ν : R → X a vector measure, following [BD,Section 2] we will say that a countably additive measure λ : R→ [0,∞] is acontrol measure for ν if it satisfies

1) limλ(A)→0 ‖ν(A)‖X = 0,

2) every ν–null set in Rloc is λ–null.

Observe the difference between this definition and the corresponding to thecase of σ–algebras.

Chapter 2 33

The space L1 of a vector measure defined on a σ–algebra, can be endowedwith structure of Banach function space through any Rybakov control mea-sure. Unfortunately, this fact does not hold for vector measures defined onδ–rings, since this measures may not have Rybakov control measure. In Ex-ample 2.2, for p > 1 and Γ uncountable, for any x∗ = (xγ)γ∈Γ ∈ X∗

p we sawthat |x∗ν|(A) =

∑γ∈A |xγ| for all A ∈ R. Since x∗ ∈ `q(Γ) and q < ∞, we

have that I = γ ∈ Γ : xγ 6= 0 is countable. Then, every non empty setA ⊂ Γ\I satisfies |x∗ν|(A) = 0, while A is not ν–null. Hence, |x∗ν| is not aRybakov control measure for ν. However, in the same example for p = 1, tak-ing x∗0 = (1)γ∈Γ ∈ `∞(Γ) we have |x∗0ν|(A) =

∑γ∈A 1 for all A ∈ R, that is,

|x∗0ν| is the counting measure. So, |x∗0ν| is a Rybakov control measure for ν.In this case, L1(ν) is a Banach function space with respect to (Γ,Rloc, |x∗0ν|).Note that we consider Banach function spaces with respect to a non σ–finitemeasure space defined in the sense of Lindenstrauss and Tzafriri, in a similarway that for σ–finite measures.

Actually, we can not even ensure (for non strongly additive measures)the existence of a control measure. However, this problem can be solved byconsidering the limit at condition 1) in the definition of control measure asa local limit.

A countably additive measure λ : R → [0,∞] is a local control measurefor ν ([BD, Section 2]) if it satisfies

1’) limA⊂B, λ(A)→0 ‖ν(A)‖X = 0 for every B ∈ R.

2) Every ν–null set in Rloc is λ–null.

The dependence on each set B ∈ R of the limit at 1’) is justified in [MN1,p. 231–232]. Condition 1’) is just the definition given in [MN1, Definition2.36] for ν absolutely continuous with respect to λ on R, which is equivalentto ν(A) = 0 whenever A ∈ R with λ(A) = 0 [MN2, Proposition 3.6]. By(2.1), this happens if and only if every λ–null set inRloc is ν–null. Conditions1) and 1’) coincide when ν is defined on a σ–algebra.

Every strongly additive vector measure has a control measure. Indeed,this holds for vector measures defined even on a ring of sets. The next propo-

34 Chapter 2

sition collects several results showing equivalent conditions to the strong ad-ditivity of a vector measure defined on a ring.

Proposition 2.5. Let R be a ring of subsets of Ω, X a Banach space andν : R→ X a vector measure. The following conditions are equivalent:

a) The measure ν is strongly additive.

b) There exists a σ–algebra Σ and a vector measure ν : Σ → X such thatR ⊂ Σ and ν(A) = ν(A) for all A ∈ R (i.e. ν extends to ν).

c) There exists a bounded control measure for ν.

If these conditions hold, then we can take a bounded control measure for ν ofthe type |x∗0ν| for certain x∗0 ∈ BX∗.

PROOF. The equivalence between a) and c) is proved by Brooks in [B,Theorem 2].

Kluvanek shows that a) is equivalent to the existence of a vector measureν : σ(R) → X which extends ν, where σ(R) is the σ–ring generated by R,[K, Theorem on Extrension]. In this case, applying Zorn’s lemma, it is provedin [BD, Lemma 1.1] that ν is σ–finite, that is, there exists a sequence (An)contained in R such that N = Ω\ ∪ An is a ν–null set, in the sense thatν(A) = 0 for all A ∈ R ∩ 2N . Consider the σ–algebra Σ = A ∪ B : A ∈σ(R) and B ⊂ N. Noting that ν(A) = 0 for every ν–null set A ∈ σ(R), itfollows that the set function ν : Σ → X given by ν(A ∪ B) = ν(A) is welldefined and is a vector measure extending ν. So a) and b) are equivalent.

Suppose conditions a)–c) hold. Let |x∗0ν| be a Rybakov control measurefor the vector measure ν given in b). Since |x∗0ν| ≤ |x∗0ν| with equality onR, then |x∗0ν| is a control measure for ν. From the boundedness of |x∗0ν|, wehave that |x∗0ν| is also bounded.

Applying Proposition 2.5 to the case of vector measures defined on aδ–ring we obtain the following results.

Chapter 2 35

Theorem 2.6. Let R be a δ–ring of subsets of Ω, X a Banach space andν : R→ X a vector measure.

a) If ν is strongly additive, then L1(ν) coincides with the space L1(ν),where ν : Rloc → X is a vector measure which extends ν.

b) The vector measure ν is strongly additive if and only if χΩ ∈ L1(ν).

c) If ν is strongly additive, then L1(ν) is a Banach function space withrespect to the measure space (Ω,Rloc, |x∗0ν|), where |x∗0ν| is a boundedcontrol measure for ν, for a certain x∗0 ∈ BX∗.

PROOF. Suppose ν is strongly additive. In the proof of Proposition 2.5,since R is a δ–ring and so Rloc is a σ–algebra, we have N ∈ Rloc, σ(R) ⊂Rloc ⊂ Σ and ν : Rloc → X is a vector measure extending ν. Note thatthe ν–null and ν–null sets coincide. For every R–simple function ϕ we havethat ‖ϕ‖ν = ‖ϕ‖ν . Then L1(ν) coincides with L1(ν) provided the R–simplefunctions are dense in L1(ν). In the proof of Proposition 2.5 we obtain thatΩ = (∪An)∪N with An ∈ R and N ∈ Rloc ν–null, that is, ν is σ–finite. Set0 ≤ f ∈ L1(ν) and (ψn) a sequence of simple functions such that 0 ≤ ψn ↑ f .For each fixed n we have that ϕn

m = ψnχ∪mj=1Aj

↑ ψn ν–a.e., or equivalently

ν–a.e. From the order continuity of L1(ν), it follows that for ε > 0 thereexist nε and mε = mε(nε), such that

‖f − ϕnεmε‖ν ≤ ‖f − ψnε‖ν + ‖ψnε − ϕnε

mε‖ν < ε .

Note that ϕnm is an R–simple function for every m, n, so the R–simple func-

tions are dense in L1(ν). Hence a) holds.

Let us prove b). If ν is strongly additive, by condition a) we have thatthere exists a vector measure ν : Rloc → X which extends ν and such thatL1(ν) = L1(ν). Since ν is defined on a σ–algebra, then χΩ ∈ L1(ν) and soχΩ ∈ L1(ν). Conversely, if χΩ ∈ L1(ν), by (2.3) the set function

A ∈ Rloc → ν(A) =

∫

A

χΩ dν ∈ X

is a vector measure which extends ν. Then, from Proposition 2.5 it followsthat ν is strongly additive.

36 Chapter 2

Observe that if ν is strongly additive, by Proposition 2.5, there existsa bounded control measure for ν of the type |x∗0ν| with x0 ∈ BX∗ , whichis defined on Rloc since it is the variation of the measure x∗0ν : R → R .Moreover, L1(ν) is continuously contained in L1(|x∗0ν|) and it is an ideal ofmeasurable functions with respect to the |x∗0ν|–a.e. order, which is equivalentto the ν–a.e. order. By b) we have that χΩ ∈ L1(ν) and so χA ∈ L1(ν) forall A ∈ Rloc. Hence c) holds.

As a consequence of Theorem 2.6, we obtain that for a vector measureν defined on a δ–ring, the bounded measurable functions are contained inL1(ν) if and only if ν is strongly additive.

Every strongly additive vector measure ν is bounded, since in this caseχΩ ∈ L1(ν), in particular ‖ν‖(Ω) = ‖χΩ‖ν < ∞. A classical proof of thisfact holds, that is, supposing ‖ν‖(Ω) = ∞, since the semivariation of ν issubadditive and finite over sets in R, we can take a sequence (An) ⊂ R ofdisjoint sets with ‖ν(An)‖ ≥ n. Example 2.2 for p = ∞ exhibits a boundedmeasure ν which is not strongly additive, since ‖ν(A)‖∞ = 1 for every nonempty set A ∈ R.

As noted before, from (2.1) it follows that a vector measure ν is boundedif and only if ‖ν‖(Ω) = ‖χΩ‖ν < ∞, or equivalently χΩ ∈ L1

w(ν). Hence, ifX is a Banach space which does not contain any subspace isomorphic to c0,then every vector measure ν : R → X defined on a δ–ring R satisfies thatL1(ν) = L1

w(ν) and so ν is bounded if and only if it is strongly additive.

The vector measure ν given in Example 2.2 for any p ∈ [1,∞], is σ–finiteif and only if Γ is countable. In the case p = ∞, ν is bounded. For p < ∞and Γ infinite, ν is unbounded. This shows that there is no relation betweenboundedness and σ–finiteness.

Observe that if ν is strongly additive, then χΩ ∈ L1(ν) is a weak unit ofL1(ν). The following result shows that the σ–finiteness of ν is equivalent tothe existence of weak unit for L1(ν). In this case, we only can guarantee theexistence of a local control measure for ν.

Chapter 2 37

Theorem 2.7. Let R be a δ–ring of subsets of Ω, X a Banach space andν : R→ X a vector measure. The following conditions are equivalent:

a) The measure ν is σ–finite.

b) The space L1(ν) has a weak unit.

c) There exists a bounded local control measure for ν.

PROOF. If ν is σ–finite, then there exist An ∈ R and N ∈ Rloc ν–null suchthat Ω = (∪An) ∪N . Then, the series

∑ (2n(‖ν‖(An) + 1)

)−1χAn converges

in L1(ν) and its sum g is a weak unit in L1(ν), since g−1(0) = N is ν-null.Hence a) implies b).

Let us prove that b) implies c). Let g be a weak unit in L1(ν). Sinceg ∈ L1(ν), the set function νg : Rloc → X defined by νg(A) =

∫A

g dν is avector measure. Let λ = |x∗0νg| be a Rybakov control measure for νg, wherex∗0 ∈ BX∗ . A set A ∈ Rloc is λ–null if and only if ‖gχA‖ν = ‖νg‖(A) = 0,that is gχA = 0 ν–a.e. or equivalently A is ν–null, since g−1(0) is ν–null.Therefore, λ is a bounded local control measure for ν.

Suppose λ : R→ [0,∞) is a bounded local control measure for ν. Then|λ|(Ω) = supA∈R |λ|(A) < ∞. Thus, there exists a sequence (An) of sets inR such that |λ|(Ω\An) < 1/n and so Ω = (∪An) ∪ N where N = Ω\(∪An)is λ–null or equivalently ν–null. Hence, ν is σ–finite. This proves that c)implies a).

Remark 2.8. Let ν : R→ X be a σ–finite vector measure and g a weak unitof L1(ν). For the same element x∗0 obtained in the proof of Theorem 2.7, wehave that |x∗0νg|(A) =

∫A

gd|x∗0ν| = 0 if and only if |x∗0ν|(A) = 0, and so |x∗0ν|is also a local control measure for ν, although in this case (if ν is unbounded)we cannot guarantee that it is bounded. Moreover, L1(ν) will be a Banachfunction space with respect to (Ω,Rloc, |x∗0ν|) provided that every A ∈ Rloc

with |x∗0ν|(A) < ∞ satisfies χA ∈ L1(ν). Unfortunately, there exist vectormeasures ν which do not satisfy this property. In Example 2.2 for p = ∞and Γ countable and infinite, we have that |x∗ν|(A) =

∑γ∈A |xγ| < ∞ for

38 Chapter 2

all x∗ = (xγ) ∈ `1(Γ) and all A ∈ Rloc, while χΓ /∈ L1(ν) = c0(Γ). Therefore,L1(ν) is not a Banach function space with respect to any (Γ,Rloc, |x∗ν|).However, for any p ∈ [1,∞] (even for Γ uncountable) the counting measureλ : R → [0,∞) is a control measure for ν such that L1(ν) is a Banachfunction space with respect to (Γ,Rloc, |λ|).

Given a σ–finite vector measure ν : R → X which is not strongly addi-tive, we cannot ensure that L1(ν) is a Banach function space with respect tosome measure space (Ω,Rloc, λ), with λ a local control measure for ν. How-ever, by Theorem 2.7, L1(ν) is an order continuous Banach lattice having aweak unit and so there exist a measure space (S, Σ, µ) and a Banach functionspace Y with respect to (S, Σ, µ) such that L1(ν) is order isometric to Y , [LT,Theorem 1.b.14]. Even more, L1(ν) is order isometric to the space L1(ν) forthe vector measure ν : Σ → Y defined by ν(A) = χA, [C2, Theorem 8].The measure ν is rather uncertain, like Y and the measure space (S, Σ, µ).The following result gives a representation of L1(ν) as an space L1 of a moreconcrete vector measure defined on the σ–algebra Rloc and with values in X.

Theorem 2.9. Under the conditions of Theorem 2.7, if ν is σ–finite, thenL1(ν) is order isometric to L1(νg), where νg : Rloc → X is the vector measuredefined by νg(A) =

∫A

g dν and g is a weak unit in L1(ν).

PROOF. Given a simple function ϕ =∑n

i=1 aiχAiwith (Ai) disjoint sets in

Rloc, we have that gϕ ∈ L1(ν) and

∫|ϕ|d|x∗νg| =

n∑i=1

|ai||x∗νg|(Ai) =n∑

i=1

|ai|∫

Ai

gd|x∗ν| =∫

g|ϕ|d|x∗ν|

for all x∗ ∈ X∗. So ‖ϕ‖νg = ‖gϕ‖ν . Consider f ∈ L1(νg) and (ϕn) a sequenceof simple functions converging to f in the norm of L1(νg) and νg–a.e. Then(gϕn) is a Cauchy sequence in L1(ν) and so it converges in norm to somefunction h ∈ L1(ν). By taking a subsequence (gϕnk

) that converges to h ν–a.e., or equivalently νg–a.e., we see that gf = h ∈ L1(ν) and ‖gf‖ν = ‖f‖νg .

Let us show that if h ∈ L1(ν), then h/g ∈ L1(νg). Suppose h ≥ 0 and let(ϕn) be a sequence of positive simple functions increasing to h/g. Then (gϕn)

Chapter 2 39

increases to h and, by order continuity, (gϕn) converges to h in L1(ν). Thisimplies that (ϕn) is a Cauchy sequence in L1(νg), so it converges in norm tosome function f ∈ L1(νg) and there exists a subsequence (ϕnk

) converging tof νg–a.e. Thus f = h/g ∈ L1(νg).

Therefore, the operator T : L1(νg) → L1(ν) given by T (f) = gf is anorder isometry. The injectivity and the preservation of the order are deducedfrom the fact that the ν–null and νg–null sets coincide.

Brooks and Dinculeanu show that every vector measure ν defined on a δ–ring (with no further properties) has a local control measure λ : R → [0,∞] ,[BD, Theorem 3.2]. By Theorem 2.7, if ν is not σ–finite, then λ must beunbounded.

Since L1(ν) is an order continuous Banach lattice, it can be represented asan unconditional direct sum of a family of disjoint ideals, each one of themhaving a weak unit, [LT, Proposition 1.a.9]. Moreover, by [C2, Theorem8], each one of these ideals is the space L1 of some vector measure definedon a σ–algebra. The next result gives a concrete representation of such adecomposition.

Theorem 2.10. Let R be a δ–ring of subsets of Ω, X a Banach space andν : R → X a vector measure. The space L1(ν) can be decomposed into anunconditional direct sum of a family of disjoint ideals, each one of them orderisometric to a space L1(νA), where each νA is the vector measure ν restrictedto a σ–algebra of the type A ∩R for some A ∈ R.

PROOF. In the proof of [BD, Theorem 3.2], it is shown that there exists amaximal family Aα : α ∈ ∆ of non ν–null sets in R, such that Aα ∩ Aβ

is ν–null for α 6= β. For each α ∈ ∆, we consider the class of sets Rα =Aα ∩ R = B ∈ R : B ⊂ Aα which is a δ–ring, like R. Since Aα ∈ Rα,we have that Rα is a σ–algebra of subsets of Aα. Let να : Rα → X bethe restriction of ν to the σ–algebra Rα and λα = |x∗ανα| a Rybakov controlmeasure for να. For each B ∈ R, it is proved that λα(B ∩ Aα) = 0 for allα ∈ ∆ except on a countable set. Then, the set function λ : R → [0,∞]

40 Chapter 2

defined by λ(A) =∑

α∈∆ λα(A ∩ Aα) is a local control measure for ν.

Consider the bounded linear projections Pα : L1(ν) → L1(ν) given byPα(f) = fχAα . Observe that

(Pα(L1(ν))

)α∈∆

is a family of (closed) disjointideals of L1(ν). Let f ∈ L1(ν) and (ϕn) be a sequence of R–simple functionsconverging to f in the norm of L1(ν) and ν–a.e. For each n, we take In ⊂ ∆ asthe countable set such that ϕnχAα = 0 ν–a.e. for all α 6∈ In. Then fχAα = 0ν–a.e. for all α 6∈ I, where I = ∪nIn is countable. Hence f =

∑α∈I fχAα ν–

a.e. and the sum converges unconditionally in L1(ν) by the order continuityof L1(ν). Thus, f is uniquely represented as an unconditional direct sumof elements of

(Pα(L1(ν))

)α∈∆

. Each space Pα(L1(ν)) is order isometric toL1(να), via restriction to the set Aα.

We end this chapter with some relevant examples of vector measuresdefined on δ–rings. The first one is a generalization of Example 2.2.

Example 2.11. Let Γ be an abstract set, X a Banach space and (xγ)γ∈Γ afamily of non null elements of X. Consider the δ–ring R of finite subsets ofΓ and the vector measure ν : R→ X defined by

ν(A) =∑γ∈A

xγ .

Observe that Rloc = 2Γ and the only ν–null set is the empty set. So, ν

is σ–finite if and only if Γ is countable. If Γ is uncountable, then ν is notstrongly additive. In the case when Γ is countable, we have that ν is stronglyadditive if and only if

∑γ∈Γ xγ is unconditionally convergent. In any case (Γ

countable or uncountable), ν is bounded if and only if |x∗ν|(Γ) < ∞ for eachx∗ ∈ X∗ (i.e. χΓ ∈ L1

w(ν)), or equivalently∑ |x∗(xγ)| converges for each

x∗ ∈ X∗, that is,∑

xγ is weakly unconditionally Cauchy.

From the facts that f ∈ L1w(ν) if and only if

∫ |f |d|x∗ν| < ∞ for allx∗ ∈ X∗ and

∫ |f |d|x∗ν| ≥ ∑ |f(γj)||x∗(xγj)| for all (γj) ⊂ Γ, we deduce

that L1w(ν) is the space of functions f : Γ → R such that

∑f(γ)xγ is weakly

unconditionally Cauchy.

Chapter 2 41

Let us see that the space L1(ν) is the space of functions f : Γ → R suchthat

∑f(γ)xγ is unconditionally convergent. If ϕ is an R–simple function,

we have that∫

ϕdν =∑

ϕ(γ)xγ. Let f ∈ L1(ν) and (ϕn) be a sequence ofR–simple functions converging to f in L1(ν) and ν–a.e. The support set of f iscontained in the union of the support sets of ϕn, so it is a countable set (γj).Then f =

∑f(γj)eγj

, where eγ is the characteristic function of the pointγ. The order continuity of L1(ν) implies that the series is unconditionallyconvergent in L1(ν). For any subsequence (γkj

) we have

∥∥∥m∑

j=n

f(γkj)xγkj

∥∥∥X

=∥∥∥

∫ ( m∑j=n

f(γkj)eγkj

)dν

∥∥∥X≤

∥∥∥m∑

j=n

f(γkj)eγkj

∥∥∥ν

.

Hence, the series∑

f(γ)xγ is unconditionally convergent in X. Supposenow that

∑f(γ)xγ is unconditionally convergent in X. Since xγ is non null

for all γ, the support set of f is countable γj. The R–simple functionsϕn =

∑nj=1 f(γj)eγj

are pointwise convergent to f . For any A ∈ Rloc, the

sequence( ∫

Aϕndν

)converges in X, since

∥∥∥∫

A

(ϕm − ϕn)dν∥∥∥

X=

∥∥∥m∑

j=n+1, γj∈A

f(γj)xγj

∥∥∥X

.

Therefore, from Proposition 2.3 it follows that f ∈ L1(ν).

Finally, we observe that the space L1(ν) is a Banach function space withrespect to (Γ,Rloc, λ), where λ : Rloc → [0,∞] is the counting measure.

In the following examples we study vector measures associated to classicaloperators.

Example 2.12. Consider an isomorphism T : Lp(R) → Lp(R) for any p ∈[1,∞). Let R be the δ–ring of all Borel subsets of R having finite Lebesguemeasure m. Then Rloc is the σ–algebra of all Borel subsets of R. The setfunction ν : R → Lp(R) defined by ν(A) = T (χA) is a σ–finite vector mea-sure. Moreover, L1

w(ν) = L1(ν) since Lp(R) does not contain any subspaceisomorphic to c0. Since T is an isomorphism, for every R–simple function ϕ

42 Chapter 2

we have1

‖T−1‖‖ϕ‖p ≤ ‖T (ϕ)‖p =∥∥∥

∫ϕdν

∥∥∥p≤ ‖T‖‖ϕ‖p . (2.6)

Then, from (2.4) it follows that ‖T−1‖−1‖ϕ‖p ≤ ‖ϕ‖ν ≤ 2‖T‖‖ϕ‖p. Inparticular, the ν–null sets coincide with the m–null sets. Hence, the densityof theR–simple functions in both spaces L1(ν) and Lp(R), implies that L1(ν)is order isomorphic to Lp(R). By (2.6) we have ‖T−1‖−1‖χA‖p ≤ ‖ν(A)‖p

for all A ∈ R. So, ν is not bounded and hence, not strongly additive.

Some examples of isomorphism T : Lp(R) → Lp(R) are:

(I) The multiplication operator by some function ϕ ∈ L∞(R) for whichalso 1/ϕ ∈ L∞(R).

(II) The dilation operator by some factor α > 0. That is, Tf(x) = f(αx)for x ∈ R.

(III) The Hilbert transform H for p > 1 (see [Ste]), defined by the principal–value integral

H(f)(x) =1

π

∫ +∞

−∞

f(t)

x− tdt .

Example 2.13. Let φ : [0,∞) → [0,∞) be an integrable function non iden-tically equal to zero and V the Volterra convolution operator defined by

V(f)(x) =

∫ x

0

φ(x− y)f(y)dy ,

for f ∈ M, provided the integral exists m–a.e. x ∈ [0,∞), where m is theLebesgue measure. This operator has been studied by Curbera in [C5] forthe interval [0, 1]. Let R be the δ–ring of all Borel subsets of [0,∞) havingfinite Lebesgue measure and Rloc the σ–algebra of all Borel subsets of [0,∞).

Since φ is integrable, the operator V is well defined for f = χA withA ∈ Rloc. Moreover, if A ∈ R we have that V(χA) ∈ (L1 ∩ L∞)[0,∞), with

‖V(χA)‖1 =

∫ ∞

0

V(χA)(x)dx = m(A)‖φ‖1

Chapter 2 43

and‖V(χA)‖∞ = sup

x≥0V(χA)(x) ≤ ‖φ‖1 .

Consider the set function ν : R → (L1 ∩ L∞)[0,∞) associated to Vgiven by ν(A) = V(χA). Let us see that ν is a vector measure (i.e. countablyadditive). It suffices to prove that ‖ν(A)‖∞ → 0 as m(A) → 0, since in thiscase, for a sequence (An) of disjoint sets in R such that ∪An ∈ R, we havem(Hn) → 0 where Hn = ∪j>nAj and so

∥∥∥ν(∪An)−n∑

j=1

ν(Aj)∥∥∥

L1∩L∞= max

‖ν(Hn)‖1, ‖ν(Hn)‖∞

= maxm(Hn)‖φ‖1, ‖ν(Hn)‖∞

→ 0 .

From the integrability of φ it follows

‖ν(A)‖∞ = supx≥0

∫ x

0

φ(x− y)χA(y)dy = supx≥0

∫

(x−A)∩[0,x]

φ(s)ds → 0

as m(A) → 0, since m((x− A) ∩ [0, x]) ≤ m(A) for all x ≥ 0.

Let X be a rearrangement invariant space on [0,∞). Then (L1∩L∞)[0,∞)is continuously embedded in X and thus ν : R→ X is a σ–finite vector mea-sure. The boundedness of ν depend on the space X where ν takes values.The measure ν : R → L1[0,∞) is unbounded since ‖ν(A)‖1 = ‖φ‖1m(A)for all A ∈ R, while the measure ν : R → L∞[0,∞) is bounded since‖ν(A)‖∞ ≤ ‖φ‖1 for all A ∈ R.

Let us show that the vector measure ν : R → (L1 + L∞)[0,∞) is notstrongly additive, thus we will have that ν : R → X is never strongly addi-tive. Let K0 > 0 satisfy

∫ K0

0φ(s)ds > 0. Such a K0 exists since φ is not iden-

tically zero and so ‖φ‖1 > 0. Consider the disjoint sets Aj = [aj, aj+1) ∈ Rwith aj = j(2K0 + 1) for all j ≥ 1. Let us see that ‖ν(Aj)‖L1+L∞ does notconverge to zero. For each j ≥ 1 we have

ν(Aj)(x) =

∫ x

0

φ(x− y)χAj(y)dy =

∫ x

0

φ(s)χx−Aj(s)ds

≥( ∫ x−aj

0

φ(s)ds)χAj

(x)

44 Chapter 2

for all x ∈ [0,∞). Denoting hj(x) =( ∫ x−aj

0φ(s)ds

)χAj

(x) it follows thatν(Aj)

∗ ≥ h∗j , where f ∗ means the decreasing rearrangement of a function f ,that is f ∗(t) = infr ≥ 0 : mx ∈ [0,∞) : |f(x)| > r ≤ t for t ≥ 0 .

For each 0 < t < 1 we take δ(t) = (2K0 + 1 − t)/2. Since δ(t) > K0 we

have∫ δ(t)

0φ(s)ds ≥ ∫ K0

0φ(s)ds > 0. Then,

x ∈ Aj :

∫ x−aj

0φ(s)ds > 1

2

∫ δ(t)

0φ(s)ds

∪x ∈ Aj :

∫ x−aj

0φ(s)ds ≥ ∫ δ(t)

0φ(s)ds

∪x ∈ Aj : δ(t) < x− aj

and x ∈ Aj : δ(t) < x−aj = (δ(t)+aj, aj+1) since δ(t) < aj+1−aj. Hence,

m

x ∈ Aj :∫ x−aj

0φ(s)ds > 1

2

∫ δ(t)

0φ(s)ds

≥ aj+1 − aj − δ(t)

= K0 +t + 1

2> t ,

that is

1

2

∫ δ(t)

0

φ(s)ds /∈ r ≥ 0 : mx ∈ [0,∞) : hj(x) > r ≤ t ,

and so1

2

∫ δ(t)

0

φ(s)ds < h∗j(t) ≤ ν(Aj)∗(t) .

Therefore

‖ν(Aj)‖L1+L∞ =

∫ 1

0

ν(Aj)∗(t)dt ≥ 1

2

∫ 1

0

∫ δ(t)

0

φ(s)dsdt >1

2

∫ K0

0

φ(s)ds > 0

and it follows that ν : R→ (L1 + L∞)[0,∞) is not strongly additive.

CHAPTER 3: Optimal Domains for operators

SECTION 1. The vector measures defined on δ–rings, allow us toachieve a study about the optimal domains for operators defined on certainspaces of functions, among them the Banach function spaces with respectto a σ–finite measure space. This study follows a similar way to the workof Curbera and Ricker [CR1], where operators defined on Banach functionspaces with respect to a finite measure space ore considered.

Let R be a δ–ring of subsets of a set Ω. We denote by M the spaceof functions f : Ω → R which are measurable with respect to Rloc and byS(R) the space of the R–simple functions. Let X be a Banach space andT : S(R) → X a linear operator. The optimal domain for T within a familyof spaces F is a space Y ∈ F to which T can be extended as a continuouslinear operator, still with values in X, and such that if T is extended to someF ∈ F then F is continuously contained in Y .

Associated to the operator T we have the finitely additive set functionν : R→ X given by ν(A) = T (χA). We prove that if T satisfies appropriateconditions, then the set function ν is a vector measure and T can be extendedto L1(ν) as a continuous linear operator, still with values in X. Even more,L1(ν) is the optimal domain for T within the spaces of functions of “its sameclass”, that is, spaces of functions which behave as L1(ν).

Curbera and Ricker consider operators T for which the associated setfunction ν is a vector measure defined on a σ–algebra, obtaining that thespace L1(ν) is the optimal domain for T within the class of the order contin-uous Banach function spaces, [CR1, Corollary 3.3]. In our setting, the vectormeasure ν associated to T is defined on a δ–ring, so we cannot ensure that

46 Chapter 3

L1(ν) is a Banach function space, as seen in Chapter 2. Hence, we look forthe optimal domain of T within a larger class of spaces. The following defi-nition extends the concept of Banach function space by considering a δ–ringin the role played by the sets of finite measure in the classical definition.

Definition 3.1. Let R be a δ–ring of subsets of a set Ω and µ : R→ [0,∞]a countably additive measure. A Banach function space with respect to(Ω,R, µ) is a Banach space E of classes of measurable functions with respectto Rloc, satisfying

1) if f ∈M, g ∈ E and |f | ≤ |g| µ–a.e., then f ∈ E and ‖f‖E ≤ ‖g‖E,

2) χA ∈ E for all A ∈ R,

where functions which are equal except on a set N in Rloc with |µ|(N) = 0(i.e. µ–a.e.) are identified.

Contrary to the case of Banach function spaces with respect to a measurespace on a σ–algebra, a Banach function space E with respect to (Ω,R, µ)with R δ–ring, is not required to satisfy that f is integrable with respect toµ over A for each f ∈ E and A ∈ R, that is χA ∈ E∗ for each A ∈ R, whereχA takes f ∈ E into

∫A

fdµ. The reason of this is that we look for spaceswith the properties of L1(ν) for a vector measure ν : R → X . We knowthat there exists λ : R → [0,∞] local control measure for ν ([BD, Theorem3.2]), but we do not know if each f ∈ L1(ν) and A ∈ R satisfy fχA ∈ L1(λ).When ν is σ–finite, we have seen that there exists a Rybakov local controlmeasure λ for ν, that is, λ = |x∗0ν| for certain x∗0 ∈ BX∗ (Remark 2.8). Then,for this measure λ the above condition holds, since for all f ∈ L1(ν) we havethat f ∈ L1(λ).

In any way, every Banach function space E on (Ω,R, µ) is a Banachlattice for the µ–a.e. order, in which convergence in norm of a sequenceimplies µ–a.e. convergence for some subsequence. Let us show that for(fn) ⊂ E such that ‖fn‖E → 0, by taking ‖fnj

‖E ≤ 12j we obtain that

∪N≥1∩k≥1∪j≥kω ∈ Ω : |fnj(ω)| > 1/N is a µ–null set in Rloc, thus fnj

→ 0µ–a.e. will hold. Fixed N ≥ 1, we take AN,j = ω ∈ Ω : |fnj

(ω)| > 1/N

Chapter 3 47

with j ≥ 1. For all k ≥ 1 it follows

χ∩k≥1∪j≥kAN,j≤ χ∪j≥kAN,j

≤∑

j≥k

χAN,j≤ N

∑

j≥k

|fnj| .

Since∑

j≥1 fnjconverges in E we have N

∑j≥k |fnj

| ∈ E and so, by condition1) of Definition 3.1, χ∩k≥1∪j≥kAN,j

∈ E and

‖χ∩k≥1∪j≥kAN,j‖E ≤ N

∥∥∥∑

j≥k

|fnj|∥∥∥

E≤ N

∑

j≥k

‖fnj‖E ≤ N

2k

for all k ≥ 1. Then, ‖χ∩k≥1∪j≥kAN,j‖E = 0 or equivalently χ∩k≥1∪j≥kAN,j

= 0µ–a.e., that is ∩k≥1 ∪j≥k AN,j is µ–null. Hence, ∪N≥1 ∩k≥1 ∪j≥kAN,j is alsoµ–null.

Example 3.2.

(I) A Banach function space with respect to a σ–finite measure space(Ω, Σ, µ) with Σ σ–algebra, is a Banach function space with respectto (Ω,R, µ), where R is the δ–ring of the sets in Σ with finite measureµ. In this case, Rloc = Σ.

(II) Let R be a δ–ring of subsets of Ω, X a Banach space, ν : R → X avector measure and λ : R → [0,∞] a local control measure for ν. InChapter 2 we saw that L1(ν) is an ideal of measurable functions withrespect to the ν–a.e. order which is equivalent to the λ–a.e. order.Also, L1(ν) contains the R–simple functions. So, L1(ν) is a Banachfunction space on (Ω,R, λ).

From now to the end of this section, we fix a δ–ring R of subsets of anabstract set Ω.

Proposition 3.3. Let E be a Banach function space with respect to (Ω,R, µ),X a Banach space and T : E → X a linear operator satisfying:

1) If fn, f ∈ E with 0 ≤ fn ↑ f µ–a.e., then Tfn converges weakly in X

to Tf .

48 Chapter 3

2) If A ∈ Rloc with χA ∈ E and x∗ ∈ X∗, we have

supB∈R∩2A

|x∗T (χB)| = 0 =⇒ x∗T (χA) = 0 .

Then, the set function ν : R→ X given by ν(A) = T (χA) is a vector measureand for all f ∈ E we have that f ∈ L1(ν) with

∫fdν = Tf . Even more,

if µ and ν are equivalent, that is, |µ|(B) = 0 if and only if ‖ν‖(B) = 0with B ∈ Rloc, then E is continuously contained in L1(ν) and the integrationoperator with respect to ν extends T .

PROOF. Let (An) be a sequence of disjoint sets inR such that ∪An ∈ R. Forany subsequence (Anj

), from 1) it follows that T (χ∪Nj=1Anj

) =∑N

j=1 ν(Anj)

converges weakly in X to T (χ∪Anj) = ν(∪Anj

). Then, by the Orlicz-Pettistheorem,

∑ν(An) is unconditionally convergent to ν(∪An), [DU, Corollary

I.4.4]. So, ν is a vector measure on R.

Firstly, suppose that every simple function (with respect to Rloc) ψ ∈ E,satisfies ψ ∈ L1(ν) and

∫ψdν = T (ψ).

Consider 0 ≤ f ∈ E and let (ψn) be a sequence of simple functions suchthat 0 ≤ ψn ↑ f . Then ψn ∈ E and so, by supposition, ψn ∈ L1(ν) with∫

ψndν = T (ψn). Since the R–simple functions are dense in L1(ν), we cantake ϕn ∈ S(R) such that ‖ψn − ϕn‖ν ≤ 1

n. Then ψn − ϕn → 0 in L1(ν) and

there exists a subsequence such that ψnk− ϕnk

→ 0 ν–a.e. Thus, ϕnk→ f

ν–a.e.

Fixed x∗ ∈ X∗, let us show that f is integrable with respect to x∗ν.Applying Proposition 2.3 to the measure x∗ν : R → R , we only have to

prove that( ∫

Aϕnk

dx∗ν)

converges in R for every A ∈ Rloc.

Given A ∈ Rloc, from condition 1) of the hypothesis it follows thatT (ψnk

χA) converges weakly in X to T (fχA), so x∗T (ψnkχA) → x∗T (fχA).

We have supposed that T (ψnkχA) =

∫A

ψnkdν, then

x∗( ∫

A

ψnkdν

)=

∫

A

ψnkdx∗ν → x∗T (fχA) .

Chapter 3 49

Moreover, since∣∣∣∫

A

ϕnkdx∗ν−

∫

A

ψnkdx∗ν

∣∣∣ ≤∫

A

|ψnk−ϕnk

|d|x∗ν| ≤ ‖x∗‖‖ψnk−ϕnk

‖ν → 0 ,

we have ∫

A

ϕnkdx∗ν → x∗T (fχA) .

Hence, f is integrable with respect to x∗ν and by Remark 2.4 it satisfies∫

A

fdx∗ν = limn→∞

∫

A

ϕnkdx∗ν = x∗T (fχA) .

This shows that f ∈ L1(ν) and∫

fdν = T (f).

Observe that every µ–null set is ν–null. This follows from the linearityof T , since if B ∈ Rloc is such that |µ|(B) = 0, for any A ∈ R ∩ 2B we haveµ(A) = 0 or equivalently χA = 0 in E, then ν(A) = T (χA) = 0 and so B isν–null. Given f ∈ E and any function g with f = g µ–a.e., we have thatf = g ν–a.e. and so f and g belong to the same equivalence class in L1(ν).Thus, the map which takes a class of E represented by a function f into theclass of L1(ν) represented by f , is well defined. In the case when µ and ν

are equivalent, since f = g ν–a.e. implies f = g µ–a.e., we have that theprevious map is injective, that is, it is the identity map. So, in this case E iscontained in L1(ν) and the inclusion is continuous since it is a positive linearoperator between Banach lattices.

Therefore, we only have to prove that every A ∈ Rloc with χA ∈ E

satisfies χA ∈ L1(ν) with∫

χAdν = T (χA). Let us see |x∗ν|(A) < ∞ for allx∗ ∈ X∗. Suppose that there exists x∗ ∈ X∗ with |x∗ν|(A) = ∞, then we cantake Bn ∈ R ∩ 2A such that |x∗ν(B1)| > 1 and |x∗ν(Bn)| > n|x∗ν|(∪n−1

j=1 Bj)for each n ≥ 2. Then, for Hn = ∪n

j=1Bj we have

|x∗ν(Hn)| = |x∗ν(Bn) + x∗ν(Hn\Bn)|≥ |x∗ν(Bn)| − |x∗ν(Hn\Bn)|≥ n|x∗ν|(Hn−1)− |x∗ν|(Hn−1)

≥ (n− 1)|x∗ν|(B1)

≥ n− 1 .

50 Chapter 3

On other hand, since χHn increases to χ∪Hn ∈ E, by condition 1) we have thatT (χHn) is weakly convergent to T (χ∪Hn), in particular x∗T (χHn) = x∗ν(Hn)converges. So, the contradiction establishes that |x∗ν|(A) < ∞ for all x∗ ∈X∗, what it means χA ∈ L1

w(ν).

Given B ∈ Rloc and x∗ ∈ X∗, since

|x∗ν|(A ∩B) = sup|x∗ν|(H) : H ∈ R ∩ 2A∩B < ∞ ,

we can take an increasing sequence (Hn) of sets in R ∩ 2A∩B such that|x∗ν|((A ∩ B)\Hn) → 0. Then, χHn ↑ χA∩B |x∗ν|–a.e. Noting

∫χHndx∗ν =

x∗ν(Hn) = x∗T (χHn), by the dominated convergence theorem and by condi-tion 1) we have∫

B

χAdx∗ν = limn→∞

∫χHndx∗ν = lim

n→∞x∗T (χHn) = x∗T (χ∪Hn) = x∗T (χA∩B) ,

where the last equality is obtained from condition 2), since (A ∩ B)\(∪Hn)is a |x∗ν|–null set and so x∗T (χ(A∩B)\(∪Hn)) = 0. Hence, χA ∈ L1(ν) and∫

BχAdν = T (χA∩B) for all B ∈ Rloc.

Under conditions of Proposition 3.3, we always have that a µ–null setis ν–null. The µ–null and ν–null sets coincide if and only if the operator T

satisfies the following: if B ∈ Rloc is such that T (χA) = 0 for all A ∈ R∩2B,then B is µ–null. If every A ∈ R with T (χA) = 0 satisfies µ(A) = 0, thenthe above claim holds for T . In particular, in the case when T is injective wehave that µ and ν are equivalent. Note that µ and ν are equivalent if andonly if µ is a local control measure for ν (see Chapter 2, Section 2).

As a consequence of Proposition 3.3, we obtain the following result.

Corollary 3.4. Let E be an order continuous Banach function space withrespect to (Ω,R, µ), X a Banach space and T : E → X a continuous linearoperator satisfying that B is µ–null whenever B ∈ Rloc with T (χA) = 0 for allA ∈ R ∩ 2B. Then, the set function ν : R → X associated to T is a vectormeasure, E is continuously included in L1(ν) and the integration operatorwith respect to ν extends T .

Chapter 3 51

PROOF. Since the condition required to T means that µ and ν are equiva-lent, we only have to prove that conditions 1) and 2) in Proposition 3.3 hold.From the order continuity of E and the continuity of T it follows that condi-tion 1) holds. If we show that the R–simple functions are dense in E, thencondition 2) can be deduced from this density and from the continuity of T .By Zorn’s lemma, we obtain a maximal family Aα : α ∈ ∆ of sets Aα ∈ R,with µ(Aα) > 0 and µ(Aα∩Aβ) = 0 for α 6= β. Given f ∈ E, for any sequence(αj) ⊂ ∆ we have

∑nj=1 |f |χAαj

= |f |χ∪nj=1Aαj

increases to |f |χ∪Aαjµ–a.e.

Since E is order continuous, the series∑

j≥1 |f |χAαjconverges in E. Thus,∑

α∈∆ |f |χAα satisfies the Cauchy condition and so fχAα = 0 µ–a.e. excepton a countable set αj. Suppose f ≥ 0 and let (ψn) be a sequence of simplefunctions with respect to Rloc such that 0 ≤ ψn ↑ f . Then, ϕn = ψnχ∪n

j=1Aαj

are R–simple functions such that 0 ≤ ϕn ↑ fχ∪Aαj= f µ–a.e. Again by

order continuity, (ϕn) converges to f in E.

From the previous result, we identify the optimal domain for a linear op-erator T : S(R) → X satisfying a very mild condition, namely its associatedset function is a vector measure.

Theorem 3.5. Let X be a Banach space and T : S(R) → X a linear op-erator satisfying that T (χAn) is weakly convergent to T (χA) in X, for everyincreasing sequence (An) of sets in R such that A = ∪An ∈ R. Then, the setfunction ν : R → X given by ν(A) = T (χA) is a vector measure and L1(ν)is the optimal domain for T within the class of the order continuous Banachfunction spaces on (Ω,R, µ) with µ equivalent to ν.

PROOF. Observe that the required condition to T is necessary and sufficientfor ν being a vector measure. Then, the space L1(ν) is an order continuousBanach function space on (Ω,R, λ) with λ a local control measure for ν,that is, λ is equivalent to ν (see Example 3.2.(II)). Moreover, the integrationoperator extends to T .

Suppose T : F → X is a continuous linear extension of T , where F isan order continuous Banach function space on (Ω,R, µ) with µ equivalentto ν. Let B ∈ Rloc be such that every A ∈ R ∩ 2B satisfies T (χA) = 0.

52 Chapter 3

Since T (χA) = T (χA) = ν(A), it follows that B is ν–null or equivalentlyµ–null. Hence, the hypothesis of Corollary 3.4 holds for T and so F iscontinuously included in L1(ν), where ν : R → X is the vector measuregiven by ν(A) = T (χA). Since ν is equal to ν we have L1(ν) = L1(ν).

As a direct consequence of Theorem 3.5, we obtain the following result.

Corollary 3.6. Let E be an order continuous Banach function space withrespect to (Ω,R, µ), X a Banach space and T : E → X a continuous linearoperator such that B is µ–null whenever B ∈ Rloc with T (χA) = 0 for allA ∈ R ∩ 2B. Then, the optimal domain for T within the class of the ordercontinuous Banach function spaces on (Ω,R, λ) with λ equivalent to ν, is thespace L1(ν), where ν is the vector measure associated to T .

The condition satisfied by the continuous linear operator T in Corollary3.6 is necessary for the integration operator being an extension of T to L1(ν),that is, for E being injected in L1(ν).

Example 3.7. Let R be the δ–ring of all Borel subsets of R with finiteLebesgue measure m and Rloc the σ–algebra of all Borel subsets of R. Givenp ∈ [1,∞), the space Lp(R) is an order continuous Banach function spaceon (R,R,m). For any isomorphism T : Lp(R) → Lp(R) the hypothesis ofCorollary 3.6 hold, in particular m–null and ν–null sets coincide. Then, L1(ν)for the vector measure ν : R→ Lp(R) given by ν(A) = T (χA), is the optimaldomain of T within the order continuous Banach function spaces on (R,R, µ)with µ equivalent to ν, in particular µ = m. For these operators T we havea precise description of the optimal domain, since we saw in Example 2.12that L1(ν) is order isomorphic to Lp(R).

SECTION 2. An important class of operators between function spacesis the class of operators defined by a kernel. In this section we consider func-tions defined on [0,∞). We obtain precise descriptions for the optimal do-mains of operators with kernels satisfying appropriate conditions. Note that

Chapter 3 53

similar conclusions can be obtained by considering kernel operators betweenspaces of functions defined on any σ–finite measure space.

Let B be the σ–algebra of all Borel subsets of [0,∞) and Bb the δ–ringof all bounded sets in B. Then Bloc

b = B. We denote by m the Lebesguemeasure on [0,∞).

Let K : [0,∞)×[0,∞) → [0,∞) a measurable function satisfying that forevery x ∈ [0,∞), the function Kx defined by Kx(y) = K(x, y) is integrablewith respect to m over sets in Bb. We say that a function K satisfyingthe previous conditions is an admissible kernel. Associated to an admissiblekernel K we have the finitely additive set function ν defined on Bb by

ν(A) =

∫

A

K(· , y) dy ,

and the linear operator T defined by

T (f) =

∫ ∞

0

f(y)K(· , y) dy

whenever the integral exists for every x ∈ [0,∞). Observe that ν is the setfunction associated to T , that is, ν(A) = T (χA) for all A ∈ Bb.

Given a set function ν associated to an admissible kernel K, the firstproblem we raise is to obtain Banach spaces X for which ν : Bb → X isa vector measure (i.e. well defined and countably additive). Our attentionwill be focused on finding Banach function spaces on [0,∞), that is, on([0,∞),B,m).

Proposition 3.8. Let X be a Banach function space on [0,∞) and K anadmissible kernel with associated operator T and associated set function ν.Suppose that ν : Bb → X is well defined.

a) If X is order continuous, then ν : Bb → X is countably additive.

b) If ν : Bb → X is countably additive, then every f ∈ L1(ν) satisfies thatT (f) ∈ X and T (f) =

∫fdν.

54 Chapter 3

c) If ν : Bb → X is countably additive, then L1(ν) is the optimal domainfor T : S(Bb) → X within the class of the order continuous Banachfunction spaces on ([0,∞),Bb, µ) with µ equivalent to ν.

PROOF. Let us prove a). Let (An) ⊂ Bb be such that ∪An ∈ Bb. Since K

is non negative, we have that ν(∪nj=1Aj) ↑ ν(∪Aj) ∈ X and since X is order

continuous,∑n

j=1 ν(Aj) = ν(∪nj=1Aj) converges in X to ν(∪Aj).

Suppose that ν : Bb → X is countably additive. Set 0 ≤ f ∈ L1(ν)and let (ψn) be a sequence of simple functions with 0 ≤ ψn ↑ f . Thefunctions ϕn = ψnχ[0,n] are Bb–simple and 0 ≤ ϕn ↑ f . Since L1(ν) isorder continuous, ϕn converges to f in L1(ν) and so

∫ϕndν → ∫

fdν inX. Consider a subsequence

∫ϕnk

dν = T (ϕnk) converging m–a.e. to

∫fdν.

Since K is non negative, 0 ≤ T (ϕnk) =

∫∞0

ϕnk(y)K(· , y)dy increases to∫∞

0f(y)K(· , y)dy. Hence, T (f) =

∫fdν ∈ X. So, b) holds.

If ν : Bb → X is countably additive, then T : S(Bb) → X satisfies thehypothesis of Theorem 3.5. Hence, c) also holds.

Let X be a Banach function space on [0,∞) and K an admissible kernelsuch that the associated set function ν : Bb → X is a vector measure. TheLebesgue measure m is equivalent to ν if and only if K satisfies the followingcondition: if A ∈ Bb is such that

∫A

K(x, y)dy = 0 m–a.e. x ∈ [0,∞), thenm(A) = 0. A sufficient condition for the above condition holds is that foralmost everywhere x, Kx > 0 (a.e.).

In view of Proposition 3.8, once we find Banach function spaces X on[0,∞) for which the set function ν : Bb → X associated to some admissiblekernel K is a vector measure, the problem of finding the optimal domain forthe operator T associated to K, is equivalent to identify the space L1(ν).

Let X be a rearrangement invariant space on [0,∞). The space L1∩L∞ iscontinuously contained in X, then any vector measure with values in L1∩L∞

is also a vector measure with values in X. Thus, we look for conditions onK guaranteeing that its associated set function ν : Bb → L1∩L∞ is a vector

Chapter 3 55

measure.

Proposition 3.9. Given an admissible kernel K, the associated set functionν : Bb → L1 ∩ L∞ is a vector measure if and only if K satisfies:

1) The function Ky defined by Ky(x) = K(x, y) is integrable for m–a.e.y ∈ [0,∞).

2) The function y → ∫∞0

Ky(x)dx is integrable over sets in Bb.

3) supx≥0

∫A

K(x, y)dy < ∞, for all A ∈ Bb.

4) limm(B)→0

∫B∩A

Kx(y)dy = 0 uniformly in x ∈ [0,∞), for all A ∈ Bb.

PROOF. Clearly ν(A) ∈ L∞ for all A ∈ Bb if and only if K satisfies condition3). On other hand, ν(A) ∈ L1 if and only if

∫ ∞

0

∫

A

K(x, y)dydx =

∫

A

∫ ∞

0

K(x, y)dxdy < ∞ ,

and this happens for every A ∈ Bb if and only if K satisfies conditions 1)and 2). Thus, ν is well defined if and only if K satisfies 1)–3). Since L1 isorder continuous, from Proposition 3.8.a) it follows that ν : Bb → L1 ∩ L∞

is countably additive if and only if ν : Bb → L∞ is countably additive.

Suppose K satisfies 4). Let (An) be a sequence of disjoint sets in Bb suchthat A = ∪An ∈ Bb. Since m(A) < ∞, we have m(∪j>nAj) → 0 and from 4)it follows

∥∥∥ν(A)−n∑

j=1

ν(Aj)∥∥∥∞

= ‖ν( ∪j>n Aj

)‖∞ = supx≥0

∫

∪j>nAj

K(x, y)dy → 0 .

So, ν : Bb → L∞ is countably additive.

Conversely, suppose ν : Bb → L∞ is countably additive. If K does notsatisfy 4) for some A ∈ Bb, then there exist δ > 0 and sets (Bn) withm(Bn) < 1/2n such that

δ < supx≥0

∫

Bn∩A

Kx(y)dy = ‖ν(Bn ∩ A)‖∞ ≤ ‖χBn∩A‖ν ≤ ‖χHn‖ν ,

56 Chapter 3

where Hn = ∪j≥nBj∩A. Since χHn are Bb–simple functions decreasing to zerom–a.e. and so ν–a.e., the order continuity of L1(ν) implies that ‖χHn‖ν → 0.This makes a contradiction.

Example 3.10. Let φ : [0,∞) → [0,∞) be a measurable function. ConsiderK : [0,∞) × [0,∞) → [0,∞) defined by K(x, y) = φ(x − y)χ[0,x](y). Thefunction K is an admissible kernel satisfying conditions 1)–4) in Proposition3.9 if and only if φ is integrable. In this case, the associated set functionν : Bb → L1 ∩ L∞ is a vector measure. This was seen in Example 2.13,since ν is the set function associated to the Volterra convolution operatorby φ. Hence, if φ is integrable and X is any rearrangement invariant spaceon [0,∞), then the space L1(ν) with ν : Bb → X is the optimal domain forthe operator T : S(Bb) → X associated to K, within the order continuousBanach function spaces on ([0,∞),Bb, µ) with µ equivalent to ν.

Let us see that the Lebesgue measure m is equivalent to ν. Let A ∈ Bb

be such that∫

Aφ(x − y)χ[0,x](y)dy = 0 m–a.e. x ∈ [0,∞). Taking integral

with respect to x in [0,∞) we have

0 =

∫ ∞

0

∫ ∞

0

φ(x− y)χ[0,x](y)χA(y)dydx

=

∫ ∞

0

χA(y)

∫ ∞

y

φ(x− y)dxdy

=

∫ ∞

0

χA(y)

∫ ∞

0

φ(s)dsdy

= ‖φ‖1m(A) .

The case φ = 0 m–a.e. is not interesting, so we assume the converse. Then,‖φ‖1 > 0 and so m(A) = 0.

In order to facilitate the task of identifying the space L1(ν) of a vectormeasure ν associated to an admissible kernel K for which conditions 1)-4) ofProposition 3.9 hold, K will be required to satisfy the following monotoneproperty.

Chapter 3 57

An admissible kernel K is said to be monotone decreasing if it satisfiesthat Kx1(y) ≥ Kx2(y) m–a.e. y ∈ [0,∞) whenever x1 ≤ x2. A monotonedecreasing admissible kernel K satisfies conditions 3) and 4) in Proposition3.9, since for all x ≥ 0 we have Kx ≤ K0 m–a.e. and K0 is integrable over setsof Bb. Thus, for these kernels, the associated set function ν : Bb → L1∩L∞ isa vector measure if and only if K satisfies conditions 1) and 2) in Proposition3.9. In this case, if K also satisfies that Kxn ↑ K0 m–a.e. whenever xn ↓ 0,we have that ν and m are equivalent if and only if K0 > 0 m–a.e.

Observe that the monotone increasing admissible kernels (i.e. Kx1(y) ≤Kx2(y) m–a.e. y ∈ [0,∞) for x1 ≤ x2) are incompatible with condition 1) inProposition 3.9, since m–a.e. y ∈ [0,∞), the function Ky is increasing andpositive on [0,∞).

Example 3.11. Let K : [0,∞) × [0,∞) → [0,∞) be the admissible kerneldefined by K(x, y) = exp

(− λ(y − x))χ[x,∞)(y) with λ ∈ R. It can be

checked directly that K satisfies conditions 1)–4) in Proposition 3.9. Thus,the set function ν : Bb → L1 ∩L∞ associated to K is a vector measure, withall its implications. Moreover, ν is equivalent to m, since if

∫A

K(x, y)dy =eλx

∫A

e−λyχ[x,∞)(y)dy = 0 m–a.e. then∫

Ae−λydy = 0 and so m(A) = 0. In

the case λ ≤ 0 we have that K is monotone decreasing.

For each monotone decreasing admissible kernel K satisfying conditions1) and 2) in Proposition 3.9, we consider the functions ω and ξ defined by

ω(y) =

∫ ∞

0

Ky(x)dx and ξ(y) = K(0, y) for y ∈ [0,∞) .

Note that ω and ξ are integrable functions over sets in Bb, since K satisfiescondition 2) in Proposition 3.9 and ξ = K0. Denote by L1

ω the space ofintegrable functions with respect to the Lebesgue measure with density ω

and by ‖ · ‖ω its norm. Similarly for L1ξ .

Theorem 3.12. Let K be a monotone decreasing admissible kernel satisfyingconditions 1) and 2) in Proposition 3.9. Then, the associated set functionν : Bb → L1∩L∞ is a vector measure and the space L1(ν) is order isomorphicto L1

ω ∩ L1ξ.

58 Chapter 3

PROOF. The hypothesis implies by Proposition 3.9 that ν : Bb → L1 ∩L∞

is a vector measure.

The space L1ω ∩L1

ξ endowed with the norm ‖f‖L1ω∩L1

ξ= max‖f‖ω, ‖f‖ξ

and the mψ–a.e. order, where mψ is the Lebesgue measure with densityψ = maxω, ξ, is an order continuous Banach function space with respectto ([0,∞),B,mψ). Moreover, by order continuity, the Bb–simple functionsare dense in L1

ω ∩ L1ξ .

From Proposition 3.8.b) we have that∫

fdν =∫

f(y)K(·, y)dy ∈ L1∩L∞

for all f ∈ L1(ν). Thus, for every Bb–simple function ϕ we have

‖ϕ‖ω =

∫ ∞

0

|ϕ(y)|ω(y)dy

=

∫ ∞

0

∫ ∞

0

|ϕ(y)|K(x, y)dydx (3.1)

=

∫ ∞

0

( ∫|ϕ|dν

)(x)dx

=∥∥∥

∫|ϕ|dν

∥∥∥1

,

and since K is monotone decreasing,

‖ϕ‖ξ =

∫ ∞

0

|ϕ(y)|ξ(y)dy = supx≥0

∫ ∞

0

|ϕ(y)|K(x, y)dy =∥∥∥

∫|ϕ|dν

∥∥∥∞

. (3.2)

Then, ‖ϕ‖L1ω∩L1

ξ= ‖ ∫ |ϕ|dν‖L1∩L∞ . In particular, the mψ–null and ν–null

sets coincide. From (2.4) it follows 12‖ϕ‖ν ≤ ‖ϕ‖L1

ω∩L1ξ≤ ‖ϕ‖ν . So, by the

density of the Bb–simple functions in both spaces, we conclude that L1(ν) isorder isomorphic to L1

ω ∩ L1ξ .

Remark 3.13. Under the same conditions of Theorem 3.12, the set functionν : Bb → L1 is a vector measure. For x∗ ∈ (L1)∗ to which a function g ∈L∞ is associated, we have that x∗ν(A) =

∫g(x)ν(A)(x)dx and |x∗ν|(A) ≤∫ |g(x)|ν(A)(x)dx for all A ∈ Bb. Thus, every Bb–simple function ϕ satisfies

∫|ϕ|d|x∗ν| ≤

∫|g(x)|

( ∫|ϕ|dν

)(x)dx .

Chapter 3 59

Then,

∥∥∥∫|ϕ|dν

∥∥∥1≤ ‖ϕ‖ν = sup

x∗∈B(L1)∗

∫|ϕ|d|x∗ν|

≤ supg∈BL∞

∫|g(x)|

( ∫|ϕ|dν

)(x)dx

=∥∥∥

∫|ϕ|dν

∥∥∥1

,

and so, by (3.1) we have that ‖ϕ‖ν = ‖ϕ‖ω. Hence, L1(ν) coincides with L1ω

and ‖f‖ν = ‖f‖ω = ‖ ∫ |f |dν‖1 for all f ∈ L1ω.

Consider now the vector measure ν : Bb → L∞ . From (2.4) and (3.2) itfollows that 1

2‖ϕ‖ν ≤ ‖ϕ‖ξ ≤ ‖ϕ‖ν for all Bb–simple function ϕ. So, L1(ν) is

order isomorphic to L1ξ and ‖f‖ξ = ‖ ∫ |f |dν‖∞ for all f ∈ L1

ξ .

We identify the space L1(ν) of the vector measure ν : Bb → L1 + L∞

associated to an admissible kernel K as in Theorem 3.12 by adding a furtherproperty for K.

Theorem 3.14. Let K be a monotone decreasing admissible kernel satisfyingconditions 1) and 2) in Proposition 3.9. Suppose K satisfies the followingcondition: there exists a constant C > 0 such that

∫ 1

0

Ky(x)dx ≥ C min ∫ ∞

0

Ky(x)dx , K(0, y)

, for all y ≥ 0 . (3.3)

Then, the associated set function ν : Bb → L1 + L∞ is a vector measure andthe space L1(ν) is order isomorphic to L1

ω + L1ξ.

PROOF. The space L1ω + L1

ξ of measurable functions f such that f = g + h

for some g ∈ L1ω and h ∈ L1

ξ , is endowed with the norm

‖f‖L1ω+L1

ξ= inf

‖g‖ω + ‖h‖ξ : f = g + h with g ∈ L1ω, h ∈ L1

ξ

=

∫ ∞

0

|f(y)|minω(y), ξ(y)dy

60 Chapter 3

(see [BK, (3.1.39) p. 307]) and the mη–a.e. order, where mη is the Lebesguemeasure with density η = minω, ξ. With this structure, L1

ω + L1ξ is an

order continuous Banach function space on ([0,∞),B,mη). Moreover, theBb–simple functions are dense in L1

ω + L1ξ .

Let ϕ be a Bb–simple function. Given g ∈ L1ω and h ∈ L1

ξ such thatϕ = g + h, by Remark 3.13 we have that ‖g‖ω = ‖ ∫ |g|dν‖1 and ‖h‖ξ =‖ ∫ |h|dν‖∞. Then

‖g‖ω + ‖h‖ξ ≥∥∥∥

∫(|g|+ |h|)dν

∥∥∥L1+L∞

≥∥∥∥

∫|ϕ|dν

∥∥∥L1+L∞

,

where by taking infimum in g and h we have ‖ϕ‖L1ω+L1

ξ≥ ‖ ∫ |ϕ|dν‖L1+L∞ .

On other hand, noting that∫

A|f(x)|dx ≤ ∫ m(A)

0f ∗(x)dx for any A ∈ B and f

measurable function with decreasing rearrangement f ∗ [BS, Theorem II.2.2],by condition (3.3) we obtain

∥∥∥∫|ϕ|dν

∥∥∥L1+L∞

=

∫ 1

0

( ∫|ϕ|dν

)∗(x)dx

≥∫ 1

0

( ∫|ϕ|dν

)(x)dx

=

∫ ∞

0

|ϕ(y)|∫ 1

0

Ky(x)dxdy

≥ C

∫ ∞

0

|ϕ(y)|minω(y) , ξ(y)dy

= C‖ϕ‖L1ω+L1

ξ.

Then, ‖ ∫ |ϕ|dν‖L1+L∞ ≤ ‖ϕ‖L1ω+L1

ξ≤ 1

C‖ ∫ |ϕ|dν‖L1+L∞ . Thus, from (2.4) it

follows 12‖ϕ‖ν ≤ ‖ϕ‖L1

ω+L1ξ≤ 1

C‖ϕ‖ν . In particular, mη and ν have the same

null sets. Hence, L1(ν) and L1ω + L1

ξ are order isomorphic.

Finally, we give a description of the space L1(ν) for the vector measureν : Bb → X associated to an admissible kernel K as in the previous theorems,where X is any order continuous rearrangement invariant space on [0,∞).For this, K is required to satisfy a stricter condition than (3.3). Also, we usethe K–method of interpolation of Peetre.

Chapter 3 61

Theorem 3.15. Let X be an order continuous rearrangement invariant spaceon [0,∞) and K a monotone decreasing admissible kernel satisfying condi-tions 1) and 2) in Proposition 3.9. Suppose K satisfies the following condi-tion: there exists a constant C > 0 such that

∫ t

0

Ky(x)dx ≥ C min ∫ ∞

0

Ky(x)dx , t K(0, y)

, for all t, y ≥ 0 . (3.4)

Then, the associated set function ν : Bb → X is a vector measure and thespace L1(ν) is order isomorphic to (L1

ω, L1ξ)X .

PROOF. For every function f ∈ L1ω + L1

ξ , by [BK, (3.1.39) p. 307] we have

K(t, f ; L1ω, L1

ξ) =

∫ ∞

0

|f(y)|minω(y), tξ(y)dy . (3.5)

In next Lemma 3.16, we will show that (L1ω, L1

ξ)X is a Banach function spaceon ([0,∞),Bb,mη) with η = minω, ξ, to which X transmits the ordercontinuity property. Thus, the Bb–simple functions are dense in (L1

ω, L1ξ)X .

Let ϕ be a Bb–simple function. Recall that ω(y) =∫∞0

Ky(x)dx andξ(y) = K(0, y). From conditions (3.4) and (3.5) it follows

K(t, ϕ; L1ω, L1

ξ) ≤ 1

C

∫ ∞

0

|ϕ(y)|∫ t

0

Ky(x)dxdy

=1

C

∫ t

0

∫ ∞

0

|ϕ(y)|K(x, y)dydx

=1

C

∫ t

0

( ∫|ϕ|dν

)(x)dx

≤ 1

C

∫ t

0

( ∫|ϕ|dν

)∗(x)dx .

Moreover, K(t, ϕ; L1ω, L1

ξ) ≥∫ t

0(∫ |ϕ|dν)∗(x)dx, since for every g ∈ L1

ω

62 Chapter 3

and h ∈ L1ξ with ϕ = g + h, by Remark 3.13 we have

‖g‖ω + t‖h‖ξ =∥∥∥

∫|g|dν

∥∥∥1+ t

∥∥∥∫|h|dν

∥∥∥∞

≥ K(t,

∫(|g|+ |h|)dν; L1, L∞

)

≥ K(t,

∫|ϕ|dν; L1, L∞

)

=

∫ t

0

( ∫|ϕ|dν

)∗(x)dx ,

where we have used that K(t, f ; L1, L∞) =∫ t

0f ∗(x)dx for all f ∈ L1 + L∞,

[BS, Theorem V.1.6].

Given f1, f2 ∈ X such that∫ t

0f ∗1 (x)dx ≤ ∫ t

0f ∗2 (x)dx for all t > 0, we have

that ‖f1‖X ≤ ‖f2‖X , [BS, Corollary II.4.7]. Since ϕ is a Bb–simple function,then

∫ |ϕ|dν, K′(·, ϕ; L1ω, L1

ξ) ∈ X. We have seen that

∫ t

0

( ∫|ϕ|dν

)∗(x)dx ≤ K(t, ϕ; L1

ω, L1ξ) =

∫ t

0

K′(x, ϕ; L1ω, L1

ξ)dx

≤ 1

C

∫ t

0

( ∫|ϕ|dν

)∗(x)dx , for all t > 0 .

Then, since K′(·, ϕ; L1ω, L1

ξ) is a decreasing function and so it coincides withits decreasing rearrangement, we have

∥∥∥∫|ϕ|dν

∥∥∥X≤ ‖ϕ‖(L1

ω ,L1ξ)X

= ‖K′(·, ϕ; L1ω, L1

ξ)‖X ≤ 1

C

∥∥∥∫|ϕ|dν

∥∥∥X

.

The order continuity of X implies that X∗ coincides with the Kothe dualX ′, then we can follow the argument of Remark 3.13 to obtain that ‖ϕ‖ν =‖ ∫ |ϕ|dν‖X . Hence, ‖ϕ‖ν ≤ ‖ϕ‖(L1

ω ,L1ξ)X

≤ 1C‖ϕ‖ν . In particular, mη and ν

have the same null sets. Thus, L1(ν) is order isomorphic to (L1ω, L1

ξ)X .

Let us show the technical lemma used in the proof of the previous the-orem. We keep the notation L1

ω for the space of integrable functions with

Chapter 3 63

respect to the Lebesgue measure with density ω and ‖·‖ω for its norm, whereω : [0,∞) → [0,∞) is any measurable function.

Lemma 3.16. Let ω, ξ : [0,∞) → [0,∞) be functions which are integrableover sets of Bb and X an order continuous rearrangement invariant spaceon [0,∞). Then (L1

ω, L1ξ)X is an order continuous Banach function space on

([0,∞),Bb,mη), where mη denotes the Lebesgue measure with density η =minω, ξ.

PROOF. To simply notation let K(t, f) = K(t, f ; L1ω, L1

ξ). For any functionf ∈ L1

ω + L1ξ , we can write K′(t, f) as

K′(t, f) = limh→0+

1

h

(K(t + h, f)−K(t, f)

). (3.6)

Denoting Φ(t, y) = minω(y), tξ(y), from (3.5) (held for any ω, ξ) it follows

1

h

(K(t + h, f)−K(t, f)

)=

1

h

∫ ∞

0

|f(y)|(Φ(t + h, y)− Φ(t, y))dy . (3.7)

For all A ∈ Bb we have that χA ∈ L1ω ∩ L1

ξ ⊂ (L1ω, L1

ξ)X . Let us see that(L1

ω, L1ξ)X is an ideal of measurable functions. Let f be a measurable function

and g ∈ (L1ω, L1

ξ)X such that |f | ≤ |g| mη–a.e. Note that mη(A) = 0 if andonly if m(A ∩ Sop(η)) = 0, and Sop(η) = Sop(ω) ∩ Sop(ξ), then

|f |(Φ(t + h, ·)− Φ(t, ·)) ≤ |g|(Φ(t + h, ·)− Φ(t, ·)) m–a.e.

for all t, h > 0, since Φ(t + h, ·) ≥ Φ(t, ·). Hence, from (3.6) and (3.7) itfollows that K′(t, f) ≤ K′(t, g) for all t > 0. Since g ∈ (L1

ω, L1ξ)X , that is

K′(·, g) ∈ X, we have K′(·, f) ∈ X and so f ∈ (L1ω, L1

ξ)X . Moreover,

‖f‖(L1ω ,L1

ξ)X= ‖K′(·, f)‖X ≤ ‖K′(·, g)‖X = ‖g‖(L1

ω ,L1ξ)X

.

Thus, (L1ω, L1

ξ)X is a Banach function space on ([0,∞),Bb,mη).

Let us prove now that (L1ω, L1

ξ)X is order continuous. If we show thatfn, f ∈ (L1

ω, L1ξ)X with 0 ≤ fn ↑ f mη–a.e. implies K′(·, fn) ↑ K′(·, f), then

64 Chapter 3

by the order continuity of X, we will have that K′(·, fn) converges to K′(·, f)in X. Since 0 ≤ fn ≤ f mη–a.e., from (3.6) and (3.7) it follows

K′(t, f)−K′(t, fn) = K′(t, f − fn)

and so fn converges to f in (L1ω, L1

ξ)X .

So, let us see that K′(·, fn) ↑ K′(·, f) whenever 0 ≤ fn ↑ f mη–a.e. SinceK′ is decreasing (see Preliminaries), for any function g ∈ L1

ω + L1ξ we have

1

h

(K(t + h, g)−K(t, g)

)=

1

h

∫ t+h

t

K′(s, g)ds ≤ K′(t, g) . (3.8)

If 0 ≤ fn ↑ f mη–a.e., we have seen before that K′(t, fn) ≤ K′(t, fn+1) ≤K′(t, f) for all t > 0. From the monotone convergence theorem applied to(3.5), it follows that K(t, f) = limn→∞K(t, fn) for all t > 0. Then, by (3.8)we have

1

h

(K(t + h, f)−K(t, f)

)= lim

n→∞1

h

(K(t + h, fn)−K(t, fn)

)

≤ limn→∞

K′(t, fn) ≤ K′(t, f) .

Taking limit as h → 0+, we obtain K′(t, fn) ↑ K′(t, f) for all t > 0.

Consider the admissible kernel K given in Example 3.11 for λ ≤ 0, inwhich case it is monotone decreasing and satisfies properties 1) and 2) inProposition 3.9. Recall that K(x, y) = exp

(−λ(y−x))χ[x,∞)(y). Let us see

that K satisfies condition (3.4) in Theorem 3.15, that is, there exists C > 0such that

∫ t

0

K(x, y)dx ≥ C minω(y), tξ(y) , for all t, y ≥ 0 , (3.9)

where ω(y) =∫∞0

K(x, y)dx and ξ(y) = K(0, y).

If λ = 0 we have that ω(y) = y and ξ(y) = 1. Then

minω(y), tξ(y) = miny, t =

∫ t

0

K(x, y)dx ,

Chapter 3 65

and so (3.9) holds.

Suppose λ < 0. In this case we have ω(y) = 1−e−λy

λ, ξ(y) = e−λy and

∫ t

0

K(x, y)dx =

1−e−λy

λif y ≤ t

e−λy

λ(eλt − 1) if y > t

If y ≤ t then minω(y), tξ(y) ≤ ω(y) =∫ t

0K(x, y)dx. For y > t we distin-

guish two cases. If t ≥ 1, since λ < 0 we have 1 − eλt ≥ 1 − eλ > 0 andso

∫ t

0

K(x, y)dx =e−λy

(−λ)(1− eλt)

≥ ω(y)(1− eλ)

≥ (1− eλ) minω(y), tξ(y) .

Suppose t < 1. The function g(s) = 1− e−s− eλs is increasing in [0,−λ] andsince 0 ≤ (−λ)t ≤ −λ we have 0 = g(0) ≤ g(−λt) and so (−λ)teλ ≤ 1− eλt.Then,

minω(y), tξ(y) ≤ tξ(y) ≤ 1

(−λ)eλ(1− eλt)ξ(y) = e−λ

∫ t

0

K(x, y)dx .

In any case we have

minω(y), tξ(y) ≤ max

e−λ,1

1− eλ

∫ t

0

K(x, y)dx ,

so (3.9) holds.

Therefore, by Theorem 3.15, the set function ν : Bb → X defined byν(A) =

∫A

K(·, y)dy and with values in any order continuous rearrangementinvariant space X on [0,∞), is a vector measure and L1(ν) is order isomorphicto (L1

ω, L1ξ)X . Moreover, by Proposition 3.8, (L1

ω, L1ξ)X is the optimal domain

within the order continuous Banach function spaces on ([0,∞),Bb, µ) with µ

equivalent to ν (in particular µ = m), of the operator T : S(R) → X definedby the kernel K as T (f) =

∫∞0

f(y)K(·, y)dy.

66 Chapter 3

We can obtain a more precise description of the space (L1ω, L1

ξ)X with thefollowing result.

Lemma 3.17. Let ω, ξ : [0,∞) → [0,∞) be integrable functions over sets ofBb. For every function f ∈ L1

ω + L1ξ we have

K′(t, f ; L1ω, L1

ξ) =

∫

y≥0 : tξ(y)≤ω(y)|f(y)|ξ(y)dy , for all t > 0 . (3.10)

PROOF. Set K(t, f) = K(t, f ; L1ω, L1

ξ). We will describe K′(t, f) by using(3.6) and (3.5). Denoting Φ(t, y) = minω(y), tξ(y), we observe

Φ(t + h, ·)− Φ(t, ·) = (ω − tξ)χ[tξ<ω≤(t+h)ξ] + hξχ[(t+h)ξ<ω] .

Let f1 ∈ L1ω and f2 ∈ L1

ξ be such that f = f1 + f2. Since |f |ξχ[tξ≤ω] ≤1t|f1|ω+ |f2|ξ ∈ L1[0,∞), from the dominated convergence theorem it follows

1

h

∫

[tξ<ω≤(t+h)ξ]

|f(y)|(ω(y)− tξ(y))dy ≤∫

[tξ<ω≤(t+h)ξ]

|f(y)|ξ(y)dy → 0

and ∫

[(t+h)ξ<ω]

|f(y)|ξ(y)dy →∫

[tξ<ω]

|f(y)|ξ(y)dy

as h → 0+. Then,

K′(t, f) = limh→0+

1

h

(K(t + h, f)−K(t, f)

)

= limh→0+

1

h

∫ ∞

0

|f(y)|(Φ(t + h, y)− Φ(t, y)

)dy

=

∫

[tξ<ω]

|f(y)|ξ(y)dy .

Consider again the admissible kernel of Example 3.11. For λ = 0 we have

K′(t, f ; L1ω, L1

ξ) =

∫ ∞

t

|f(y)|dy

and for λ < 0,

K′(t, f ; L1ω, L1

ξ) =

∫ ∞

ln(1+λt)1/λ

|f(y)|e−λydy .

Chapter 3 67

Taking for instance the Lorentz space Λφ of measurable functions f suchthat

∫∞0

f ∗(s)φ′(s)ds < ∞, for the case λ = 0 we have that f ∈ (L1ω, L1

ξ)Λφif

∫ ∞

0

K′(s, f ; L1ω, L1

ξ)φ′(s)ds =

∫ ∞

0

φ′(s)∫ ∞

s

|f(y)|dyds

=

∫ ∞

0

|f(y)|∫ y

0

φ′(s)dsdy

=

∫ ∞

0

|f(y)|φ(y)dy < ∞ ,

so (L1ω, L1

ξ)Λφ= L1

φ. In the case when λ < 0 we have that f ∈ (L1ω, L1

ξ)Λφif

∫ ∞

0

K′(s, f ; L1ω, L1

ξ)φ′(s)ds =

∫ ∞

0

φ′(s)∫ ∞

ln(1+λs)1/λ

|f(y)|e−λydyds

=

∫ ∞

0

|f(y)|e−λyφ(eλy − 1

λ

)dy ,

so (L1ω, L1

ξ)Λφ= L1

g where g(y) = e−λyφ(

eλy−1λ

).

Referencias.

[BDS] R. G. Bartle, N. Dunford and J. Schwartz, Weak compactness andvector measures, Canad. J. Math., 7 (1955) 289–305.

[BS] C. Bennett and R. Sharpley, Interpolation of operators (Academic Press,Boston, 1988).

[BL] R. G. Bilyeu and P. W. Lewis, Differentiability of convex functions andRybakov’s theorem, Proc. Amer. Math. Soc., 86 (1982) 186–187.

[B] J. K. Brooks, On the existence of a control measure for strongly boundedvector measures, Bull. Amer. Math. Soc., 77 (1971) 999–1001.

[BD] J. K. Brooks and N. Dinculeanu, Strong additivity, absolute continuityand compactness in spaces of measures, J. Math. Anal. and Appl., 45(1974) 156–175.

[BK] Yu. A. Brudnyı and N. Ya. Krugljak, Interpolation Functors andInterpolation Spaces (North–Holland, Amsterdam, 1991).

[C1] G. P. Curbera, El espacio de funciones integrables respecto de una me-dida vectorial (Ph. D. Thesis, Univ. of Sevilla, 1992).

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