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Integration of Functions with Values in a Banach Lattice INTEGRATION OF FUNCTIONS WITH VALUES IN A BANACH LATTICE G. A. M.JEURNINK INTEGRATION OF FUNCTIONS WITH VALUES IN A BANACH LATTICE PROMOTOR: PROF. DR. А. С. M. VAN ROOIJ INTEGRATION OF FUNCTIONS WITH VALUES IN A BANACH LATTICE PROEFSCHRIFT TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE WISKUNDE EN NATUURWETENSCHAPPEN AAN DE KATHOLIEKE UNIVERSI­ TEIT TE NIJMEGEN, OP GEZAG VAN DE RECTOR MAGNIFICUS PROF DR Ρ G А В WIJDEVELD, VOLGENS BESLUIT VAN HET COI LEGE VAN DEKANEN IN HET OPENBAAR TE VERDEDIGEN OP VRIJDAG 18 JUNI 1982, DES MIDDAGS TE 2 00 ULR PRECIES DOOR GERARDUS ALBERTUS MARIA JEURNINK GEBOREN TE DIEPENVEEN И krips repro meppel 1982 Aan mijn ошіелі Voor hun medewerking aan dit proefschrift ben ik Trees van der Eem-Mijnen en Ciaire Elings-Mesdag zeer dankbaar. CONTENTS INTRODUCTION AND SUMMARY CONVENTIONS AND NOTATIONS CHAPTER I PRELIMINARIES 1 §1 Measurability and integrability of functions with values in a Banach space 1 §2 Banach lattices 13 §3 Summability of sequences in Banach lattices 23 CHAPTER II INTEGRATION 35 51 Integration of functions with values in a Banach lattice 35 §2 Riesz spaces of integrable functions 47 §3 Banach lattice theory for spaces of integrable functions 58 §4 Examples 68 CHAPTER III SPECIAL CLASSES OF OPERATORS AND TENSOR PRODUCTS 75 51 Induced maps between spaces of integrable functions 75 §2 θ-operators 79 5 3 Δ-operators 85 5 4 Tensor products of Banach lattices 92 55 Tensor products of Banach spaces and Banach lattices 103 56 Examples of tensor products 108 CHAPTER IV VECTOR MEASURES 115 §1 Vector measures with values in a Banach lattice 115 §2 Weakly equivalent functions 124 §3 The Radon-Nikodym property 136 54 Weak measurable functions 147 CHAPTER V DANIELL INTEGRATION 153 §1 An extension of the Pettis integral 153 §2 The extension of the integral on S(μ. Χ) 158 §3 Convergence theorems for the Danieli integral 162 §4 Equivalence of Danieli integrable functions 167 REFERENCES 172 INDEX OF SYMBOLS 176 INDEX OF TERMS 177 SAMENVATTING 179 CURRICULUM VITAE 181 INTRODUCTION AND SUMMARY We will study the integrationtheories for functions which are defined on a finite measure space (Ω, Σ, μ) and which take on values in a Banach lattice. In the general case of a Banach space we have the no- * tions Bochner, Pettis and Gelfand (= weak ) integrability. When an order structure in a Banach space is given, which turns that Banach space into a Riesz space or even a Banach lattice, we will define the order Pettis and order Gelfand integrability. The advantage of this is, that the thus created spaces of integrable functions are Riesz spaces for the natural ordering of functions. It is true that the Bochner integrable functions form a Riesz space, but the Pettis and the Gelfand integrable functions do not generally do so. In order to end up with even larger Riesz spaces of Pettis integrable functions, we will generalize the concept order Pettis integrability into Danieli integrability. The space |p|(μ, X) of all order Pettis integrable functions on Ω with values in a Banach lattice X generally appears not to be com­ plete with respect to its natural norm. In this it therefore differs from the space Β(μ, X) of Bochner integrable functions, which is even a Banach lattice. For the completion of |p|(μ, X) we will in­ troduce the order tensor product of two Banach lattices. The order Gelfand integrable functions also form a normed Riesz space, which is generally not complete. But here we prefer to compare the functions with vector measures rather than with tensors. Every order Gelfand inte­ grable function corresponds with a regular vector measure of σ-finite variation. In relation to vector measures we can also * define the (weak, weak ) equivalence of vector valued functions. In that case positive measures correspond with positive functions, which is proved for Bochner, Danieli and Gelfand integrable func­ tions, but yet remains the question for Pettis integrable functions. vii We proceed by giving a summary of the various chapters. In chapter I we will give the most important definitions and theorems from the theories of vector valued integration and of Banach lattices. For sequences in Banach lattices we will introduce the notion "order suimnable". Chapter II is simply titled "Integration" and gives the various integra­ tion concepts Bochner (B), order Pettis (|p|) and order Gelfand (|G|). After classification in equivalence classes we will then get the inte­ gration spaces Β(μ,Χ), |ρ|(μ,Χ) and JGJCPrX ), in which X is a Banach * lattice with norm dual X . It turns out that they are all Riesz spaces for the natural ordering of equivalence classes of functions. Every separably valued Pettis (Gelfand) integrable function is order Pettis (order Gelfand) integrable if and only if the Banach lattice, in which the function takes on its values, is isomorphic to an AM-space. Now ι ι * II |G| (μ,Χ ) is always σ-Dedekind complete, while Β(μ,Χ) and |P| (μ,Χ) are (a-)Dedekind complete if and only if the norm on X is (a-)order conti­ nuous and X itself is (a-)Dedekind complete. If we provide the integra­ tion spaces with their natural norms, then Β(μ,Χ) is complete, while |ρ|(μ,Χ) (respectively |G|(P,X*)) is complete if and only if X (respect- it ively X ) is isomorphic to an AL-space. The norm duals of Β (μ,Χ) and |p|(μ,Χ) are characterized as the spaces of the bounded and of the cone 1 * absolutely summing operators from L (μ) to X . When comparing the several integration spaces m chapter III, we will introduce the notions Θ- and Δ-operator. Now it is certain that θ-operators (respectively A-operators) transform order Pettis integra­ ble functions (respectively essentially separably valued Pettis inte­ grable functions) into order Pettis integrable functions. Here again the AL- and AM-spaces play a special part. Every bounded operator which takes on values in an AM-space is both a θ-operator and a Δ-operator. Every operator defined on an AL-space is a θ-operator. These assertions are also reversible. The second part of this chapter deals with ordered tensor products. We will prove that the completion of |p|(μ,Χ) is characterized as such a tensor product. The duality between the order projective tensor norm |π| and the order injective tensor norm |ε| has not completely been solved. We show that | π | = |ε| and |e| <_ | ττ | , but we do not know if |ε| = |π| holds. vin Vector measures are the subject of chapter IV. In short we will mention the integration of real valued functions for a given vector measure. The equivalence of vector valued functions f can be translated into the equality of the induced vector measures m . Here we will also mention the recent results on Banach spaces, especially those of G.A. Edgar and R.F. Geitz. Our attention is called to functions with values in a Banach lattice X. A weakly measurable function f : Ω •+ X is weakly equivalent to a strongly measurable function if one of the following conditions is satisfied. (i) f is positive and Pettis integrable. (11) f is Pettis integrable and с ^ X. (ill) с ^ X and f (Si) has IR-nonmeasurable cardinality. The condition с ¿ X is necessary after all so that X has the (weak) * Radon-Nikodym property. If X has this property or is weakly compactly * * * generated, then every weak measurable function Ω •+• X is weak equiva­ lent to a strongly measurable function. If f is Gelfand integrable and m is positive, then f is equivalent to a positive function. If the analogous case with Pettis integrability instead of Gelfand mtegrabl- lity is correct, is as yet an open question. Chapter V finally extends the Pettis integral, viewed as an X-valued function on |pl (μ,Χ). Let us assume that X is σ-Dedekind complete. The Danieli procedure will then produce a Riesz space of Pettis integrable functions. It is a proper extension if and only if the norm on X is not σ-order continuous. Therefore we will give special attention to CD the space Я (Г), with Γ any infinite set. We will investigate if the theorems on monotonie and dominated convergence are valid for these spaces. We will end with the theorem that states that isomorphic mea­ sure spaces will lead to isomorphic Danieli spaces. This is not the case for Pettis spaces. ix CONVENTIONS AND NOTATIONS The scalar field for a Banach space will always be Ш. The cardinality of a set A will be denoted by #A. The cardinality of И is ÎV) » while К stands for the first uncountable cardinal. If an equivalence rela­ tion is given in A we will denote by [a] the equivalence class repre­ sented by the element a of A. The characteristic function of A will be denoted by χ . If a word is typed in italics, then the meaning of that word will be explained in the sentence in which it has appeared. We use the for­ mula χ := у when χ is the same as у by definition. The symbol » will express equality by approximation. The abbreviation "iff" will stand for "if and only if". In references we will first indicate the number of the chapter, next the section and finally the theorem, example ... etc. If we refer to some statement in this thesis we will omit the number of the chapter (and the section) when it is done in that chap­ ter (section). The end of a statement will be marked by D.
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