Operator-valued Measures and Integrals for Cone-valued Functions (Lecture Notes in Mathematics)

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 1964 Walter Roth O...

Author: Walter Roth

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Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1964

Walter Roth

Operator-Valued Measures and Integrals for Cone-Valued Functions

ABC

Walter Roth Department of Mathematics University of Brunei Darussalam BE 1410 Gadong Brunei Darussalam [email protected]

ISBN: 978-3-540-87564-2 e-ISBN: 978-3-540-87565-9 DOI: 10.1007/978-3-540-87565-9 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008938191 Mathematics Subject Classification (2000): 28B20, 46A13, 46E40, 46G10 c 2009 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publishing Services Printed on acid-free paper 987654321 springer.com

Preface

The aim of this book is twofold: Firstly, to introduce the developing theory of locally convex cones to a wider audience. This theory generalizes locally convex topological vector spaces and permits many additional and substantially diﬀerent examples and applications. In the aspects of the theory that have been developed so far, the increase in generality does not lead to any compromises with respect to the depth of its results. The main diﬀerence to vector spaces is the presence of inﬁnity-type unbounded elements and the general non-availability of the cancellation law. Some important mathematical models, while close to the structure of vector spaces are of this type. They do not allow subtraction of their elements or multiplication by negative scalars. Examples are certain classes of set-valued or extended real-valued functions that may take inﬁnite values. These arise naturally in integration theory, potential theory and in a variety of other settings and do not form vector spaces. Therefore many results and techniques from classical functional analysis can not be immediately applied. Locally convex cones carry a reﬂexive and transitive order relation, and their (convex semiuniform) topology is deﬁned using this order structure. The ﬁrst part of this book contains a review and summary of the aspects of the theory of locally convex cones that have been developed so far, sometimes without detailed proofs, but references to the sources instead. The theory is then developed further, adding some (hopefully) interesting new features. This leads to the second objective: Locally convex cones are used to provide the setting for a novel approach to integration theory. The generality of their structure allows to deal simultaneously with a wide variety of situations, including extended real-valued, vector-valued, operator-valued and cone-valued measures and functions. Topological limits from the classical theory are replaced by approximations in terms of the order structure of a locally convex cone. The main results include convergence theorems for measures and functions and an integral representation theorem for continuous linear operators on certain cones of functions. The latter establishes that a given operator can be expressed as an integral with respect to some unique measure. This is a

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Preface

very technical result and requires a lengthy proof. It is followed by a comprehensive collection of special cases and applications. Some of these lead to known representation results for compact and weakly compact operators on Banach space-valued functions, but the more general cases are new. The insertions of a special case yield the classical Spectral Representation Theorem for normal linear operators on a complex Hilbert space.

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I

1

Locally Convex Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1. Locally Convex Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Locally Convex Cones via Convex Quasiuniform Structures . . 13 2. Continuous Linear Operators and Functionals . . . . . . . . . . . . . . 17 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3. Weak Local and Global Preorders . . . . . . . . . . . . . . . . . . . . . . . . 23 4. Boundedness and the Relative Topologies . . . . . . . . . . . . . . . . . . 26 The Weak Topology σ(P, P ∗ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Boundedness Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Locally Convex Cones with Uniform Boundedness Components 42 Bounded Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Closed Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5. Locally Convex Lattice Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Locally Convex Lattice Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Locally Convex Complete Lattice Cones . . . . . . . . . . . . . . . . . . . 58 Zero Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Order Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Order Continuous Linear Operators and Functionals . . . . . . . . 77 Lattice Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Functionals Supporting the Separation Property . . . . . . . . . . . . 79 Almost Order Convergent Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Order Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Weak Order Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Extension of Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 The Standard Lattice Completion of a Locally Convex Cone . 97 Simpliﬁed Standard Lattice Completion . . . . . . . . . . . . . . . . . . . 104 6. Quasi-Full Locally Convex Cones . . . . . . . . . . . . . . . . . . . . . . . . . 107 Quasi-Full Locally Convex Cones . . . . . . . . . . . . . . . . . . . . . . . . . 107 The Standard Full Extension of a Quasi-Full Cone . . . . . . . . . . 109 vii

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Contents

7. 8. II

Cones of Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Cones of Linear Functionals. The Second Dual . . . . . . . . . . . . . 114 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Measures and Integrals. The General Theory . . . . . . . . . . . . . 1. Measurable Cone-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . Weak σ-Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Inductive Limit Neighborhoods for Cone-Valued Functions . . . Inﬁnity as a Neighborhood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inductive Limit Neighborhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . The Cone FR (X, P) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Operator-Valued Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Modulus of a Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bounded Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extension of a Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Composition of Measures and Continuous Linear Operators . . Strong Additivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weak Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Integrals for Cone -Valued Functions . . . . . . . . . . . . . . . . . . . . . . Integrals for P-Valued Step Functions . . . . . . . . . . . . . . . . . . . . Integrals for Functions in FR (X, P) . . . . . . . . . . . . . . . . . . . . . . Sets of Measure Zero and Properties Holding Almost Everywhere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integrability over a Set E ∈ R . . . . . . . . . . . . . . . . . . . . . . . . . . . Integrability over a Set F ∈ AR . . . . . . . . . . . . . . . . . . . . . . . . . . 5. The General Convergence Theorems . . . . . . . . . . . . . . . . . . . . . . Families of Measures and Properties Holding Almost Everywhere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equibounded Families of Measures . . . . . . . . . . . . . . . . . . . . . . . . Integrability with Respect to Equibounded Families of Measures . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Locally Convex Cone F(F,Θ) (X, P), V(F, Θ) . . . . . . . . . . Subcone-Based Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sums, Multiples and Order for Measures . . . . . . . . . . . . . . . . . . Convergence of Sequences of Measures . . . . . . . . . . . . . . . . . . . . Residual Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strongly Integrable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convergence of Sequences in F(X, P) . . . . . . . . . . . . . . . . . . . . . Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strong Additivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weakly Sequentially Compact Sets of Measures . . . . . . . . . . . . 6. Examples and Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The case Q = R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extended Positive-Valued Functions and Measures . . . . . . . . . .

119 119 120 120 127 127 127 128 131 132 134 136 137 138 141 141 143 147 149 150 150 159 159 159 159 160 161 171 174 176 178 182 192 194 197 207 208 208

Contents

7. 8.

ix

Extended Real-Valued Functions and Positive-Valued Measures . . . . . . . . . . . . . . . . . . . . . . Real- or Complex-Valued Functions and Measures . . . . . . . . . . The Case that Q Is the Standard Lattice Completion of Some Subcone Q0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compact and Weakly Compact Measures . . . . . . . . . . . . . . . . . The Case that P Is a Locally Convex Vector Space . . . . . . . . Algebra Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lattice Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cone-Valued Functions and Positive Real-Valued Measures . . Vector-Valued Functions and Real- or Complex-Valued Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operator-Valued Functions and Operator-Valued Measures . . Positive, Real or Complex-Valued Functions and Operator-Valued Measures . . . . . . . . . . . . . . . . . . . . . . . . . Operator-Valued Functions and Cone-Valued Measures . . . . . . Positive, Real or Complex-Valued Functions and Cone- or Vector-Valued Measures . . . . . . . . . . . . . . . Positive Linear Operators on Cones of R-Valued Functions . . Bounded Linear Operators on Spaces of Real- or Complex-Valued Functions . . . . . . . . . . . . . . Extended Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

III Measures on Locally Compact Spaces . . . . . . . . . . . . . . . . . . . . . 1. Relatively Continuous Cone-Valued Functions . . . . . . . . . . . . . . Elementary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Cone-Valued Functions on Locally Compact Spaces . . . . . . . . . Inductive Limit Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functions that Vanish at Inﬁnity . . . . . . . . . . . . . . . . . . . . . . . . . The Cones E(X, P) and E0 (X, P) . . . . . . . . . . . . . . . . . . . . . . . . The Cones FV (X, P) and FV0 (X, P) . . . . . . . . . . . . . . . . . . . . r r (X, P) and CV (X, P) . . . . . . . . . . . . . . . . . . . . . The Cones CV 0 3. Continuous Linear Operators on Cones of Functions . . . . . . . . 4. Measures on Locally Compact Spaces . . . . . . . . . . . . . . . . . . . . . Regularity of Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measures as Continuous Linear Operators . . . . . . . . . . . . . . . . . 5. Integral Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Special Cases and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . The Case that Q Is the Standard Lattice Completion of Some Subcone Q0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compact and Weakly Compact Operators . . . . . . . . . . . . . . . . . Locally Convex Topological Vector Spaces . . . . . . . . . . . . . . . . . Algebra Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lattice Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

209 209 211 214 216 220 224 228 229 231 234 238 239 242 244 245 246 249 249 256 257 257 258 258 258 259 266 273 274 282 286 307 307 307 314 318 325

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7.

The Case P = R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Case P = R+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Case Q = R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sequence Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Convergence Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Case that Q Is the Standard Lattice Completion of Some Operator Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . The Spectral Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

330 330 330 331 331 332 338 338

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

Introduction

Integration theory was originally developed for real-valued functions with respect to real-valued measures. There are a great number of expositions devoted to this, most notably the classical treatises by Lebesgue [116], Carath´eodory [30], Radon [158] and Daniell [36]. More recent treatments in the works of Bourbaki [25], Hahn and Rosenthal [80], Halmos [83] and Saks [182] contain excellent historical notes and references on the subject. Vector integration was introduced in the ﬁrst half of the last century, and exhaustive discussions of the ﬁeld can for example be found in the works of Dunford and Schwartz [54], [55], [56], Diestel and Uhl [43] or Graves [75]. The aim of this book is to develop a general theory of integration which simultaneously deals with extended real-valued, vector-valued, operator-valued and cone-valued measures and functions. All except the last of these topics have been explored extensively, and integration theory as presented in the available standard texts uses diﬀerent approaches in each of these cases. Both ﬁnitely and countably additive measures have been considered. However, ﬁnitely additive measures yield only limited results and are therefore not widely used in analysis. As a consequence only countably additive measures shall be considered in this book. The order structure of the extended real number system R = R∪{+∞} is extensively and indispensably used for the integration of R-valued functions with respect to R-valued measures. Integrals are deﬁned using suprema and inﬁma in R. However, an order structure is generally not available in all of the above-mentioned cases and diﬀerent techniques need to be applied in integration theory, often replacing suprema and inﬁma with topological limits. The literature on these subjects is of course vast, and the approaches chosen by diﬀerent authors vary to some degree. Some of the more popular texts shall be cited throughout this book and many more are mentioned in the bibliography. In very general terms, integration theory deals with measures whose values are in some set, say L, integrating certain functions with values in a second set P, and resulting in integrals in a third set Q. In order to obtain a W. Roth, Operator-Valued Measures and Integrals for Cone-Valued Functions, Lecture Notes in Mathematics 1964, c Springer-Verlag Berlin Heidelberg 2009

1

2

Introduction

meaningful theory, one needs to impose some linear structure on the sets L, P and Q and a bilinear form from L×P into Q that determines how measures and functions interact with each other. Choosing (topological) vector spaces for L, P and Q does however severely restrict the use of unbounded, that is inﬁnity-type, elements. In classical extended real-valued measure theory, the use of the element +∞ is indeed essential. In more general settings, for example in function spaces, the availability of unbounded and inﬁnite-valued functions is equally desirable. They do however not ﬁt into vector spaces, since subtraction and multiplication by negative scalars may not be available. In order to apply vector space techniques, these inﬁnity-type elements have to be removed only to be (sometimes rather awkwardly) re-adjoined at later stages in building the theory. A diﬀerent and, in the context of integration theory, novel approach will therefore be followed in this book: The main idea is to use more general structures, called locally convex cones, which are modeled by cones of convex sets or cones of set-valued functions. The theory of locally convex cones was ﬁrst developed by the author together with K. Keimel [100] in 1992, mainly in the context of Korovkin-type linear approximation theory. Other than vector spaces, and this signiﬁes the main divergence and generalization in terms of their algebraic structure, cones do not necessarily satisfy the cancellation law, namely the rule that a + c = b + c for given elements a, b and c implies that a = c. The validity of this rule would imply that a cone is embeddable into a real vector space, thus leaving out some of the more interesting applications. A simple example for such a cone is the extended real line R, endowed with the usual algebraic operations of addition and multiplication by non-negative reals. Locally convex cones also carry a topological structure which is in some way compatible with the algebraic operations. Continuity of the scalar multiplication is not required in general, as it would exclude some of the most interesting and essential non-vector space examples. However, for invertible elements the topological requirements coincide locally with those for topological vector spaces. In this way, locally convex cones constitute canonical generalizations of locally convex topological vector spaces, still retaining many of their most important properties. In particular, locally convex cones yield a rich duality theory, where the study of the dual cone consisting of all continuous linear R-valued functionals oﬀers valuable insight into the given locally convex cone itself. There are powerful Hahn-Banach type extension and separation theorems using sublinear and superlinear functionals which allow inﬁnite values. Locally convex topologies can be generated using dual pairs of cones and a bilinear form. This leads to various notions of weak and polar topologies, including Mackey-Arens type theorems. There are the notions of continuous linear operators between locally convex cones and their adjoints as linear operators between the respective duals. There are analogues of the Uniform Boundedness and Open Mapping theorems. Although in instances considerably more delicate and complicated than their analogues for

Introduction

3

locally convex topological vector spaces, most of these concepts reduce to their full strength in those special cases. Locally convex cones carry an order relation which is only required to be reﬂexive and transitive and compatible with the algebraic operations. Equality is such an order, hence cones without an explicit order structure are also included. The topology of a locally convex cone is given through a convex semiuniform structure, but it can alternatively be expressed in terms of its order structure alone, using a subfamily of positive elements called abstract neighborhoods. This often allows for a concise and elegant formulation of topological properties. This latter formalization of locally convex cones shall therefore be used throughout this book. The relationship between topology and order was explored in detail in a seminal work by L. Nachbin [135]. Most importantly for the purposes of this book, and in contrast to vector spaces, locally convex cones allow for the presence of unbounded elements. These inﬁnity-type elements can be thoroughly investigated, classiﬁed and grouped into boundedness and topological connectedness components. Modiﬁed versions of the cancellation law apply within each of these components. The topological structure of locally convex cones is thus far richer, more complicated (and arguably more interesting) than that of topological vector spaces. Moreover, one shall investigate special types of locally convex cones called locally convex complete lattice cones, which allow suprema and inﬁma of subsets with respect to their order. These suprema and inﬁma are required to be compatible in a rather strong sense with the algebraic operations and with the topological neighborhoods, comparable to the requirements for Mtopologies in locally convex topological vector lattices. Cones of R-valued functions with their natural operations and pointwise order are the prime examples here. There is a canonical concept for order convergence of nets in a locally convex complete lattice cone, which is generally weaker than convergence in the given locally convex cone topology. This notion will turn out to be suitable for integration theory. Later on one shall establish and make use of the fact that every locally convex topological vector space can be canonically embedded into such a locally convex complete lattice cone. A normed space, for example, can be embedded into the locally convex complete lattice cone of all bounded-below R-valued functions on its dual unit ball. In developing this version of integration theory, locally convex cones shall be chosen for the sets P and Q from above, i.e. the ranges for the functions involved and their resulting integrals. Indeed, Q is required to be a locally convex complete lattice cone in order to allow the formation of suprema and inﬁma, which is essential for the evaluation of integrals of P-valued functions. Correspondingly, the values of the measures will be linear operators from P into Q and the convergence of a series of measures - essential for the explanation of countable additivity - will be deﬁned using a variation of strong operator convergence. The same notion of convergence will be used for the formulation of a variety of versions of the standard convergence theorems from integration theory, involving both sequences of functions and sequences of measures.

4

Introduction

The generality of this setting permits its application to a large variety of situations. In this way, one shapes a course for a uniﬁed approach to extended real-valued, vector-valued and operator-valued measures and the integration of corresponding classes of functions. Positive reals, for example, may be interpreted as linear operators from R into itself. Elements of R, on the other hand are linear operators on the positive reals. Vectors may be considered as linear operators from the scalar ﬁeld into their space, or alternatively, from their dual space into the scalars. Particular attention will be paid to the case in which Q, the range of the integrals, is some cone of linear operators on a locally convex cone or vector space. If this cone or vector space carries some additional structure, let us say a multiplication or an order with lattice properties, then additional conditions on the measures under consideration can guarantee certain desired properties for the resulting integrals. They may be multiplicative operators in the ﬁrst of the cases mentioned above, or lattice homomorphisms in the second. Among other applications, this type of condition will be used in the ﬁnal chapter in order to derive a generalized version of the Spectral representation theorem for linear operators on Banach spaces. In the ﬁrst chapter of this book, a summary of some known and previously established properties of locally convex cones will be provided, emphasizing essential attributes which turn out to be relevant to integration theory. Details are often referred to [100] and related works. The ﬁrst sections of Chapter I contain a review of the deﬁnitions, basic properties and a set of standard examples for locally convex cones. Many of these can be found in earlier texts. These standard examples serve in particular to illustrate the variety and the wide reach of the theory, and will be used and referred to throughout the book. In order to meet the demands of their utilization in integration theory, some speciﬁc properties of locally convex cones will be explored in detail. There is a review of the relative topologies and a thorough investigation of boundedness and connectedness components with respect to these topologies. These were introduced in earlier works by the author ([175], [176]), but some relevant details will be added. Some of the following investigations are new, in particular the parts about locally convex complete lattice cones, and are therefore provided in full detail and with all proofs. Not all of the concepts introduced in this chapter will be used later on, but appear to be of suﬃcient interest for the development of the general theory of locally convex cones to merit inclusion. Readers who are interested mainly in their application to integration theory may skip some of the details of these sections at a ﬁrst reading. Chapter II comprises the general theory of operator-valued measures and integrals of cone-valued functions. For this one considers two locally convex cones P and Q (the latter is supposed to be a complete lattice cone) and the cone L(P, Q) of linear operators from P into Q. L(P, Q)-valued measures are deﬁned on a σ-ﬁeld (or a σ-ring) of a set X. These measures are required to be bounded in some sense and countably additive with respect

Introduction

5

to order convergence in Q, which is generally weaker than the given convergence when a locally convex vector space is embedded into a locally convex complete lattice cone. This is followed by the deﬁnition of measurability and integrability for P-valued functions on X. The evaluation of their integrals are then elements of Q. The use of locally convex cones as the general setting allows the utilization of their order structure, and hence the concepts and techniques of classical real-valued measure theory. The integral of a function is deﬁned using an approximation in order from below by integrals over step functions. The general convergence theorems for integrals of sequences of integrable functions are also formulated (and valid only) in terms of order convergence, but may be strengthened in special situations. Most of the results in this chapter are new, as is the attempt to use locally convex cones in integration theory. Many of the deﬁnitions, statements and proofs turn out to be quite technical and often lengthy, but they are provided in full detail. Unfortunately, the inclusion of unbounded elements in locally convex cones tends to make arguments considerably more delicate and complicated compared to their counterparts for topological vector spaces. Chapter II concludes with an extensive section dealing with examples and special cases. These include the classical settings for extended real-, complex- and vector-valued measure theory that can be found in standard texts such as [25], [43], [55], [83], [178] or [179]. However, the generality of the approach allows a far wider range of applications. Of particular interest is the situation when both locally convex cones P and Q carry an additional algebraic or order theoretic structure. This structure then transfers naturally, that is pointwise, to the cone of all P-valued functions on X. In these circumstances, some canonical additional conditions on an L(P, Q)-valued measure can guarantee that integrals of P -valued functions with respect to such a measure will represent a structurepreserving linear operator from the cone of integrable P-valued functions on X into Q. Further applications include integrals for cone-valued functions with respect to real-valued measures, operator-valued functions with respect to cone-valued measures and positive, real- or complex-valued functions and cone- or operator-valued measures. Chapter III deals with topological measure theory, that is operator-valued measures on the σ-ﬁeld (or σ-ring) of all Borel subsets (or relatively compact Borel subsets) of a locally compact space X. As usual, some regularity properties are required for these measures. The presence of unbounded elements in a locally convex cone P suggests a slightly modiﬁed notion of continuity for P-valued functions in this case, with the result that one considers continuity with respect to the somewhat coarser relative topology of P. These functions yield interesting properties concerning the boundedness components of their range. Using suitable inductive limit-type topologies for these functions one can identify corresponding cones of continuous P-valued functions that vanish at inﬁnity with respect to these topologies. This generalizes the concept of weighted spaces of continuous real-valued functions on a locally compact space which was introduced by Nachbin in [136] and

6

Introduction

Prolla in [155]. The main result of this chapter, and perhaps in this book, is an integral representation theorem for continuous linear operators from function cones of this type into a locally convex lattice cone Q. It establishes that the given operator can be expressed as an integral with respect to some unique operator-valued measure. It is of course inspired by the classical Riesz representation theorem for continuous real-valued linear functionals on a function space. Unsurprisingly, the proof of this theorem requires considerable eﬀort and is probably the most technical and surely the longest in the entire book. The rewards are however plentiful. A comprehensive collection of special cases and applications is furnished at the end of this chapter. Some of these lead to known representation results for compact and weakly compact operators on Banach space-valued functions, but the more general cases are new. The results of Chapter II can be applied to derive additional properties for the representing measure if the given operator is of a special type. This applies in particular when the range cones P and Q carry an additional algebraic or order theoretic structure. The insertions of a special case yield the classical Spectral representation theorem for normal linear operators on a complex Hilbert space, for example. It might be interesting, although probably demanding, to try to extend the main theorem of Chapter III into a Choquet-type representation theory. In such an approach a continuous linear operator is given only on a subcone of continuous functions on X and an integral representation is sought using a measure with certain additional properties concerning its support. The techniques for proving Choquet’s theorem in the real-valued case rely heavily on the order properties of real-valued functions. Allowing locally convex cones with their rich order structure as the range of the functions involved might therefore present an avenue for the extension of this powerful classical result. Further investigations into cone-valued functions and operator valuedmeasures might also focus on generalizations of the Radon-Nikod´ ym theorem. This theorem probes the absolute continuity of a measure with respect to another one and the related fact that the ﬁrst measure can be expressed through the second one by the use of an integrable density function. Chapter II is concerned with some aspects of these questions, but further investigation would require more detailed studies of special properties of the locally convex cones involved. For vector-valued measures there are well-known relations between the Radon-Nikod´ ym and the Riesz representation theorems. Details about these can be found in the text by Diestel and Uhl [43], for example. Each chapter of this book concludes with a brief section containing notes and remarks. The bibliography at the end is far from complete. It contains the references and a somewhat arbitrary list of some of the better known publications on integration theory. A more exhaustive bibliography on the subject can be found in the books by Dunford and Schwartz [55], [56], [57] and Diestel and Uhl [43].

Introduction

7

The numbering of theorems, corollaries, deﬁnitions, examples, etc. is carried out consecutively and takes into account the sections, but not the chapters of the book. Cross-references without further qualiﬁcation are therefore meant within the same chapter. References to diﬀerent chapters are pointed out by the addition of the roman numeral for the relevant chapter.

Chapter I

Locally Convex Cones

The purpose of this chapter is twofold: Firstly, to provide the tools and the settings for the integration theory which will be developed in Chapters II and III, and secondly, to introduce the theory of locally convex cones to a wider audience. This theory generalizes locally convex topological vector spaces and has (in the author’s opinion, quite unsurprisingly) not yet received the attention that it deserves. Locally convex cones permit many more and substantially diﬀerent examples and applications than locally convex vector spaces. In the aspects of the theory that have been developed so far, the increase in generality leads only to minor, if any at all, compromises with respect to the depth of its results. While some of the methods and arguments employed may at times appear rather technical and indeed counterintuitive, this is largely the consequence of the inclusion of inﬁnity-type unbounded elements and the general non-availability of the cancellation law. So why is it worth the eﬀort? Endowed with suitable topologies, vector spaces yield rich and well-studied structures. Locally convex topological vector spaces permit an extensive duality theory whose study gives valuable insight into the spaces themselves. Some important mathematical settings, however, while close to the structure of vector spaces do not allow subtraction of their elements or multiplication by negative scalars. Examples are certain classes of functions that may take inﬁnite values or are characterized through inequalities rather than equalities. They arise naturally in integration theory, potential theory and in a variety of other settings. Likewise, families of convex subsets of vector spaces which are of interest in various contexts, do not form vector spaces. If the cancellation law fails, domains of this type can not be embedded into vector spaces in order to apply the results and techniques from classical functional analysis. The inclusion of these and similar examples into an analytical theory merits the investigation of a more general structure. Apart from being useful in this sense, the theory of locally convex cones allows for some interesting and occasionally insightful and elegant mathematics.

W. Roth, Operator-Valued Measures and Integrals for Cone-Valued Functions, Lecture Notes in Mathematics 1964, c Springer-Verlag Berlin Heidelberg 2009

9

10

I Locally Convex Cones

The ﬁrst three sections of this chapter present a review of some of the main concepts of this theory while often referring to [100] and other sources for details and proofs. A brief survey of the subject can also be found in [169]. Section 4 introduces the relative topologies of a locally convex cone and provides deﬁnitions and investigations of diﬀerent types of boundedness and connectedness components. Locally convex lattice cones, quasi-full locally convex cones and cones of linear operators, are studied in Sections 5, 6 and 7, respectively. These will be used extensively in the integration theory of Chapters II and III. Some of the more specialized parts of Sections 4 to 7 are included for reference in the later stages of Chapters II and III and may be skipped at ﬁrst reading.

1. Locally Convex Cones A cone is a set P endowed with an addition (a, b) → a + b and a scalar multiplication (α, a) → αa for real numbers α ≥ 0. The addition is supposed to be associative and commutative, and there is a neutral element 0 ∈ P. For the scalar multiplication the usual associative and distributive properties hold, that is α(βa) = (αβ)a, (α + β)a = αa + βa, α(a + b) = αa + αb, 1a = a and 0a = 0 for all a, b ∈ P and α, β ≥ 0. The cancellation law, stating that a + c = b + c implies a = b, however, is not required in general. It holds if and only if the cone P can be embedded into a real vector space. An ordered cone P carries a reﬂexive transitive relation ≤ such that a ≤ b implies a + c ≤ b + c and αa ≤ αb for all a, b, c ∈ P and α ≥ 0. Equality on P is obviously such an order. Note that anti-symmetry is not required for the relation ≤ . The theory of locally convex cones as developed in [100] uses order theoretical concepts to introduce a quasiuniform topological structure on an ordered cone. In a ﬁrst approach, the resulting topological neighborhoods themselves will be considered to be elements of the cone. In this vein, a full locally convex cone (P, V) is an ordered cone P that contains an abstract neighborhood system V, that is a subset of positive elements which is directed downward, closed for addition and multiplication by strictly positive scalars. The elements v of V deﬁne upper, resp. lower neighborhoods for the elements of P by v(a) = { b ∈ P | b ≤ a + v } resp. (a)v = { b ∈ P | a ≤ b + v }, Their intersection v s (a) = v(a) ∩ (a)v is the corresponding symmetric neighborhood of a. These neighborhoods create the upper, lower and symmetric topologies on P, respectively. All elements of P are supposed to be bounded below, that is for every a ∈ P and v ∈ V we have 0 ≤ a + λv for some λ ≥ 0.

1. Locally Convex Cones

11

Finally, a locally convex cone (P, V) is a subcone of a full locally convex cone not necessarily containing the abstract neighborhood system V. Every locally convex ordered topological vector space is a locally convex cone in this sense, as it can be canonically embedded into a full locally convex cone see Example 1.4(c) below, or Example I.2.7 in [100] . A subset V0 of the neighborhood system V is called a basis for V if for every v ∈ V there is v0 ∈ V0 and α > 0 such that αv0 ≤ v. An element a of a locally convex cone (P, V) is called bounded (above) if for every v ∈ V there is λ ≥ 0 such that a ≤ λv. All invertible elements of P are bounded. Indeed, if −a ∈ P for some a ∈ P, then given v ∈ V there is λ ≥ 0 such that 0 ≤ (−a) + λv since all elements of P are required to be bounded below. This yields a ≤ λv. For later reference we shall list a few basic properties of locally convex cone topologies. We shall use the following standard notations: A subset A of P is called decreasing if b ∈ A whenever b ≤ a for some a ∈ A, increasing if b ∈ A whenever b ≥ a for some a ∈ A, or order convex if b ∈ A whenever a ≤ b ≤ c for some a, c ∈ A. balanced if b ∈ A whenever b = λa or b + λa = 0 for some a ∈ A and 0 ≤ λ ≤ 1. The last of these deﬁnitions is of course derived from corresponding one for real vector spaces, that is the requirement that λa ∈ A whenever a ∈ A and −1 ≤ λ ≤ 1. Proposition 1.1. Let (P, V) be a locally convex cone. The upper (lower or symmetric) topology of P satisﬁes the following: (i) Every element of P admits a basis of convex and decreasing (increasing or order convex) neighborhoods. The symmetric neighborhoods in the basis for 0 ∈ P are also balanced. (ii) The mapping (a, b) → a + b : P × P → P is continuous. (iii) The mapping (α, a) → αa : [0, +∞)×P → P is continuous at all points (α, a) ∈ [0, +∞) × P such that a ∈ P is bounded. Proof. Clearly, for a ∈ P and v ∈ V the neighborhoods v(a), (a)v or v s (a) are convex and decreasing, increasing or order convex, respectively. The symmetric neighborhoods of 0 ∈ P are also balanced. Indeed, let v ∈ V and let a ∈ v s (0). Then a ≤ v and 0 ≤ a + v. Let 0 ≤ λ ≤ 1. Then λa ∈ v s (0) follows from the convexity of v s (0) since λa = λa + (1 − λ)0. If on the other hand b + λa = 0 for b ∈ P, then b ≤ b + λ(a + v) = (b + λa) + v = v

and

0 = b + λa ≤ b + v

Hence b ∈ v(0) holds in this case as well. For property (ii), let a, b ∈ P and v ∈ V and set u = (1/2)v ∈ V. Then for c ∈ u(a) and d ∈ u(b), that is c ≤ a + u and d ≤ b + u we

12

I Locally Convex Cones

have c + d ≤ (a + b) + v, hence c + d ∈ v(a + b). This shows continuity of the addition with respect to the upper topology. Likewise, c ∈ (a)u and d ∈ (b)u, that is a ≤ c + u and b ≤ d + u implies that a + b ≤ (c + d) + v, hence c + d ∈ (a + b)v. This yields continuity of the addition with respect to the lower topology. Combining the preceding arguments, we realize that c ∈ us (a) and d ∈ us (b) implies c + d ∈ v s (a + b), which proves continuity with respect to the symmetric topology. For Part (iii) let (α, a) ∈ [0, +∞) × P for a bounded element a ∈ P and let v ∈ V. There is λ > 0 such that both 0 ≤ a + λv and a ≤ λv. Set ε = min 1, 1/(2λ), 1/(2α + 2) > 0 and α0 = max{α − ε, 0} and α1 = α + ε. The interval [α0 , α1 ] then is a neighborhood for α in [0, +∞). For every α0 ≤ β ≤ α we observe that 1 αa = βa + (α − β)a ≤ βa + (α − β)λv ≤ βa + ελv ≤ βa + v 2 and 1 βa ≤ βa + (α − β)(a + λv) = αa + (α − β)λv ≤ αa + v. 2 Likewise, for α ≤ β ≤ α1 we have 1 αa ≤ αa + (β − α)(a + λv) = βa + (β − α)λv ≤ αa + v 2 and

1 βa = αa + (β − α)a ≤ αa + (β − α)λv ≤ βa + v. 2

Thus

1 1 αa ≤ βa + v and βa ≤ αa + v 2 2 holds for all β ∈ [α0 , α1 ]. Now let u = εv ∈ V. Then for every b ∈ u(a) and every β ∈ [α0 , α1 ] we have βb ≤ β(a + u) = βa + εβv 1 1 1 ≤ αa + v + ε(α + ε)v ≤ αa + v + v ≤ αa + v 2 2 2 by our construction of ε > 0. Thus βb ∈ v(αa). This shows continuity of the mapping (α, a) → αa : [0, +∞) × P → P at (α, a) with respect to the upper topology of P. Likewise, for b ∈ (a)u and β ∈ [α0 , α1 ] we infer using the above 1 1 1 αa ≤ βa + v ≤ β(b + u) + v ≤ βb + ε(α + ε)v + v ≤ βb + v. 2 2 2 This yields βb ∈ (αa)v and continuity of the mapping (α, a) → αa: [0, +∞) × P → P at (α, a) with respect to the lower topology of P. Combining the preceding arguments, we realize that b ∈ us (a) and β ∈ [α0 , α1 ]

1. Locally Convex Cones

13

implies βb ∈ v s (αa), which proves continuity with respect to the symmetric topology. On the subcone P0 of all invertible elements in a locally convex cone (P, V ) the scalar multiplication can be canonically extended to all real numbers if we set αa = (−α)(−a) for α < 0 and a ∈ P0 . Proposition 1.1 then yields Corollary 1.2. Let (P, V) be a locally convex cone and let P0 be the subcone of all invertible elements of P. The mapping (α, a) → αa : R × P0 → P0 is continuous with respect to the symmetric topology of P. Proof. First we observe that a ∈ v s (b) if and only if −a ∈ v s (−b) for a, b ∈ P0 and v ∈ V. Thus ai → a for a net (ai )i∈I in P0 implies that (−ai ) → (−a) . Next suppose that αi → α ∈ R for 0 ≤ αi ∈ R and ai → a for ai , a ∈ P0 . Then αi ai → αa by 1.1(iii) since every invertible element is bounded. Now ﬁnally, let αi → α in R and ai → a for ai , a ∈ P0 . Let βi = αi ∨ 0 and γi = −(αi ∧ 0). Then βi , γi ≥ 0 and αi = βi − γi . We have βi ai → βa and γi (−ai ) → γ(−a), where β = α ∨ 0 and γ = −(α ∧ 0), by the preceding. Thus αi ai = βi ai + γi (−ai ) → βa + γ(−a) = αa, again by 1.1(ii), as claimed.

1.3 Locally Convex Cones via Convex Quasiuniform Structures. As a subcone of a full locally convex cone, a locally convex cone (P, V) inherits both its order, algebraic structure and neighborhood system from the former. While this approach elegantly permits the use of the order structure of the full cone to describe the topologies of P, it is not always very practical, because for concrete examples such a full cone may be diﬃcult to access. Quite frequently, the topology of a locally convex cone is more visible as a convex quasiuniform structure as described in I.5 of [100]. This is a straightforward generalization of the uniform structures that deﬁne the topologies of locally convex topological vector spaces. In this vein, a neighborhood is a convex subset v of P 2 , where P is an ordered cone, satisfying the following conditions: (U1) If a ≤ b for a, b ∈ P, then (a, b) ∈ v. (U2) If (a, b) ∈ λv and (b, c) ∈ ρv for a, b, c ∈ P and λ, ρ > 0, then (a, c) ∈ (λ + ρ)v. (U3) For every a ∈ P there is λ ≥ 0 such that (0, a) ∈ λv. If a family V of such neighborhoods fulﬁlls the usual conditions for a quasiuniform structure (see [135]), that is (U4) For u, v ∈ V there is w ∈ V such that w ⊂ u ∩ v, (U5) If v ∈ V and λ > 0, then λv ∈ V,

14

I Locally Convex Cones

then a straightforward procedure (see I.5 in [100]) allows the embedding of V) whose neighborhood system P and V into a full locally convex cone (P, V is generated by the elements of V, and such that (a, b) ∈ v for a, b ∈ P Convex quasiuniform structures therefore and v ∈ V means a ≤ b + v in P. yield an equivalent approach to locally convex cones. Examples 1.4. (a) In the extended real number system R = R ∪ {+∞} we consider the usual order and algebraic operations, in particular a + ∞ = +∞ for all a ∈ R, α·(+∞) = +∞ for all α > 0 and 0·(+∞) = 0. Endowed with the neighborhood system V = {ε ∈ R | ε > 0}, R is a full locally convex cone. For a ∈ R the intervals (−∞, a + ε] are the upper and the intervals [a − ε, +∞] are the lower neighborhoods, while for a = +∞ the entire cone R is the only upper neighborhood, and {+∞} is open in the lower topology. The symmetric topology is the usual topology on R with +∞ as an isolated point. It is ﬁner than the usual topology of R, where the intervals [a, +∞] are the neighborhoods of +∞. (b) For the subcone R+ = {a ∈ R | a ≥ 0} of R we may also consider the singleton neighborhood system V = {0}. The elements of R+ are obviously bounded below even with respect to the neighborhood v = 0, hence R+ is a full locally convex cone. For a ∈ R the intervals (−∞, a] and [a, +∞] are the only upper and lower neighborhoods, respectively. The symmetric topology is the discrete topology on R+ . (c) Let (E, ≤) be a locally convex ordered topological vector space. Recall that equality is an order relation, hence this example will cover locally convex spaces in general. In order to interpret E as a locally convex cone we shall embed it into a larger full cone. This is done in a canonical way: Let P = Conv(E) be the cone of all non-empty convex subsets of E, endowed with the usual addition and multiplication of sets by non-negative scalars, that is αA = {αa | a ∈ A} and A + B = {a + b | a ∈ A and b ∈ B} for A, B ∈ P and α ≥ 0. We deﬁne the order on P by A≤B

if

A ⊂ ↓B,

where ↓B = {x ∈ E | x ≤ b for some b ∈ B} is the decreasing hull of the set B in E. Note that ↓B is again a convex subset of E. The requirements for an ordered cone are easily checked. The neighborhood system in P is given by a basis V ⊂ P of convex and balanced neighborhoods of the origin in E. That is A≤B+V if A ⊂ ↓(B + V ) for A, B ∈ P and V ∈ V. We observe that for every A ∈ P and V ∈ V there is ρ > 0 such that ρV ∩ A = ∅. This yields 0 ∈ A + ρV. Therefore {0} ≤ A + ρV, and every element A ∈ P is indeed bounded below. Thus (P, V) is a full locally convex cone. Via the embedding x → {x} : E → P of its elements onto singleton subsets, the locally convex ordered topological vector space E itself may be

1. Locally Convex Cones

15

considered as a subcone of P. This embedding preserves the order of E, and on its image in P, the upper or lower topologies of P reﬂect the order structure of E in the following sense: All upper or lower neighborhoods are decreasing or increasing, respectively, that is for elements a, b ∈ E and a neighborhood V ∈ V we have a≤b+V

if

a − b ∈ ↓V.

For a linear operator T : E → E in particular, continuity with respect to either the induced upper or lower topology requires that T is monotone (see Section 2 below). The symmetric topology of P, on the other hand induces a locally convex vector space topology on E in the usual sense. It coincides with the given topology of E if the neighborhoods V ∈ V are also order convex, that is if c ∈ V whenever a ≤ c ≤ b for a, b ∈ V and V ∈ V. If the given order on E is indeed the equality, then the upper, lower and symmetric topologies of P all coincide on E with the given topology since a ≤ b + V for a, b ∈ E and V ∈ V means that a − b ∈ V in this case, and since the neighborhoods in V were supposed to be balanced. In this way, every locally convex ordered topological vector space, endowed with a basis V of balanced, convex and order convex neighborhoods, is a locally convex cone, but not a full cone. Other subcones of P that merit further investigation are those of all closed, closed and bounded, or compact convex sets in P, respectively. Note that closed and bounded convex sets satisfy the cancellation law. Details on those and further related examples can be found in [100], I.1.7, I.2.7 and I.2.8. This example can be further generalized if we replace the vector space E by a locally convex cone. (d) If (P, V) is a locally convex cone and if P is indeed a vector space over R, that is the scalar multiplication in P is extended to all reals, then all elements of P are obviously bounded, as boundedness from above for the element a ∈ P follows from boundedness from below for the element −a ∈ P. We have a ∈ v(b) for a, b ∈ P and v ∈ V in this case if and only if a − b ∈ v(0). While the multiplication by negative scalars is in general not continuous with respect to the upper and lower topologies on P, the symmetric topology generated by the neighborhoods of the origin v s (0) = {a ∈ P | a ≤ v and − a ≤ v} is a locally convex vector space topology in the usual sense (see Corollary 1.2). If P is indeed a vector space over C, then we need to consider the modular symmetric topology instead (see Section 2 in [168]). It is generated by the neighborhoods of the origin v sm (0) = {a ∈ P | γa ≤ v for all γ ∈ Γ },

16

I Locally Convex Cones

where Γ = {γ ∈ C | |γ| = 1} denotes the unit circle of C. It is easy to verify that these sets are convex, balanced and absorbing. The modular symmetric topology is therefore a locally convex vector space topology in the usual sense and yields continuity for the multiplication by all scalars in C. Thus endowed with the modular neighborhoods Vm = {v sm | v ∈ V} and the equality as its order, (P, Vm ) is again a locally convex cone, and we have a ≤ b + v sm for a, b ∈ P and v ∈ V if γ(a − b) ≤ v for all γ ∈ Γ. In the sequel, we shall say that a locally convex cone (P, V) is a locally convex topological vector space over R or C if P is a vector space over R or C, endowed with the equality as order and a system V of neighborhoods such that v(0) = v sm (0) holds for all v ∈ V. The subsets v(0) of P then are convex, balanced and absorbing, and P carries its modular symmetric topology. (e) Let (P, V) be a locally convex cone, X a set and let F(X, P) be the cone of all P-valued functions on X, endowed with the pointwise op is a full cone containing both P and V, then we erations and order. If P may identify the elements v ∈ V with the constant functions vˆ on X, that = {ˆ is x → v for all x ∈ X. Hence V v | v ∈ V} is a subset and a neighbor is uniformly bounded hood system for F(X, P). A function f ∈ F(X, P) below, if for every vˆ ∈ V there is ρ ≥ v . These func 0 ≤ f + ρˆ 0 such that V , carrying the topology of tions form a full locally convex cone Fb (X, P), is a locally convex cone. uniform convergence. As a subcone, Fb (X, P), V for F(X, P) may be Alternatively, a more general neighborhood system V created using a family of V-valued functions on X, where V = V ∪ {0, ∞} consists of the neighborhood system V for P augmented by 0 ∈ P and a maximal element ∞. (We use a + ∞ = v + ∞ = α · ∞ = ∞ and a ≤ ∞ are deﬁned for for all a ∈ P, v ∈ V and α > 0.) The neighborhoods vˆ ∈ V functions f, g ∈ F(X, P) as f ≤ g + vˆ

if

f (x) ≤ g(x) + vˆ(x) for all x ∈ X.

In this case we consider the subcone FVb (X, P) of all functions in F(X, P) that is f ∈ F (X, P) that are bounded below relative to the functions in V, Vb there is λ ≥ 0 such that 0 ≤ f + λˆ if for every v ˆ ∈ V v . In this way forms a locally convex cone. Of particular interest is the case FVb (X, P), V when V is generated by a suitable family Y of subsets Y of X and the V-valued functions vˆY (x) = v for x ∈ Y and vˆY (x) = ∞, else, correspond carries the ing to some v ∈ V and Y ∈ Y. In this case FVb (X, P), V topology of uniform convergence on the sets in Y. If X is a topological space, then suitable subcones for further investigation are those of continuous functions with respect to any of the given (upper, lower or symmetric) topologies on P. We shall explore diﬀerent notions of continuity for cone-valued functions and discuss an even wider range of suitable locally convex cone topologies in Chapter III.

2. Continuous Linear Operators and Functionals

17

Occasionally in applications of this type (see for example the proof of Proposition 5.37 and the construction of the standard lattice completion in of V-valued functions under consideration is not 5.57 below) the family V naturally closed for addition, and including all pointwise sums of the functions might not be desirable. This situation can often be remedied if we in V as a system of abstract neighborhoods instead, with a suitably consider V is closed and which is compatible with the modiﬁed addition ⊕ for which V scalar multiplication. The neighborhoods vˆ ∈ F(X, P) are deﬁned as above using associated V-valued functions which for simplicity we also denote by vˆ. The latter amounts to a slight abuse of notation, since for this concept to work we need to allow that the association between neighborhoods and V-valued functions is not one-to-one. In order to create a convex quasiuniform , we structure in the sense of 1.3, hence a locally convex cone FVb (X, P), V where + stands for require that u ˆ ⊕ vˆ ≥ u ˆ + vˆ holds for all u ˆ, vˆ ∈ V, the pointwise sum of the associated V-valued functions. We shall use this approach in 5.37 and 5.57 below. (f) For x ∈ R denote x+ = max{x, 0} and x− = − min{x, 0}. For 1 ≤ p ≤ +∞ and a sequence (xi )i∈N in R let (xi )p denote the usual lp ∞ p (1/p) norm, that is (xi )p = ∈ R for p < +∞ and (xi )∞ = i=1 |xi | sup{|xi | | i ∈ N} ∈ R. Now let lp be the cone of all sequences (xi )i∈N in p and the R such that (x− i )p < +∞. We use the pointwise order in l neighborhood system Vp = {ρvp | ρ > 0}, where (xi )i∈N ≤ (yi )i∈N + ρvp means that (xi − yi )+ p ≤ ρ. (In this expression the lp norm is evaluated only over the indexes i ∈ N for which yi < +∞.) It can be easily veriﬁed that (lp , Vp ) is a locally convex cone. In fact (lp , Vp ) can be embedded into a full cone following a procedure analogous to that in 1.4(c). The case for p = +∞ is of course already covered by Example 1.4(e).

2. Continuous Linear Operators and Functionals For cones P and Q a mapping T : P → Q is called a linear operator if T (a + b) = T (a) + T (b)

and

T (αa) = αT (a)

holds for all a, b ∈ P and α ≥ 0. If both P and Q are indeed vector spaces over R, then 0 = T (a − a) = T (a) + T (−a) implies that such an operator is linear over R. If both P and Q are ordered, then T is called monotone, if a ≤ b implies T (a) ≤ T (b). If both (P, V) and (Q, W) are locally convex cones, the operator T is called (uniformly) continuous if for every w ∈ W one can ﬁnd v ∈ V such that T (a) ≤ T (b) + w whenever a ≤ b + v for a, b ∈ P. A family T of linear operators is called equicontinuous if the above condition holds for every w ∈ W with the same v ∈ V for all T ∈ T.

18

I Locally Convex Cones

Uniform continuity is not just continuity. It is immediate from the deﬁnition that it implies and combines continuity for the operator T : P → Q with respect to the upper, lower and symmetric topologies on P and Q, respectively. A linear functional on P is a linear operator μ : P → R. The dual cone P ∗ of a locally convex cone (P, V) consists of all continuous linear functionals on P and is the union of all polars v ◦ of neighborhoods v ∈ V, where μ ∈ v ◦ means that μ(a) ≤ μ(b) + 1, whenever a ≤ b + v for a, b ∈ P. Continuity implies that a linear functional μ is monotone, and for a full cone P it requires just that μ(v) ≤ 1 holds for some v ∈ V in addition. Continuous linear functionals can take only ﬁnite values on bounded elements. Indeed, let μ ∈ v ◦ for some v ∈ V and let a ∈ P be a bounded element. Then a ≤ λv for some λ ≥ 0, hence μ(a) ≤ λ as claimed. We endow P ∗ with the canonical algebraic operations and the topology w(P ∗ , P) of pointwise convergence on the elements of P, considered as functions on P ∗ with values in R with its usual topology. As in locally convex topological vector spaces, the polar v ◦ of a neighborhood v ∈ V is seen to be w(P ∗ , P)-compact and convex ( [100], Theorem II.2.4). Examples 2.1. Revisiting the preceding Examples 1.4 we observe the following: ∗ (a) The dual cone R of R see 1.4(a) consists of all positive reals (via the usual multiplication), and the singular functional ¯0 such that ¯0(a) = 0 for all a ∈ R and ¯ 0(+∞) = +∞. (b) Likewise, in 1.4(b), the continuous linear functionals on R+ , endowed with the neighborhood system V = {0}, are the positive reals together with ¯ 0, but further include the element +∞, acting as +∞(0) = 0 and +∞(a) = +∞ for all 0 = a ∈ R+ . This functional is obviously contained in the polar of the neighborhood 0 ∈ V. (c) If both (P, V) and (Q, W) are locally convex cones and ordered vector spaces over K = R or K = C, let us also consider the modular symmetric topologies on P and Q which are deﬁned by the modular symmetric neighborhoods v sm and wsm corresponding to the given neighborhoods v ∈ V and w ∈ W, respectively. Recall from 1.4(d) that a ≤ b + v sm for a, b ∈ P and v ∈ V means that γ(a − b) ≤ v for all γ ∈ K such that |γ| = 1. The modular topologies were seen to be locally convex vector space topologies. If a linear operator T : P → Q is continuous with respect to the given locally convex cone topologies and indeed linear over K, then it is straightforward to verify that T is also continuous with respect to the respective modular symmetric topologies of P and Q. The converse does not hold true in general. For Q = R, however, that is for linear functionals, we have the following: ∗ denotes the dual of P if If P ∗ denotes the given dual of P, and if Pm ∗ since endowed with the modular symmetric neighborhoods, then P ∗ ⊂ Pm the latter topology is ﬁner that the given one. According to Theorem 3.3 ∗ there are μi ∈ P ∗ for i = 1, 2 in [168], for every linear functional μ ∈ Pm in the real or i = 1, 2, 3, 4 in the complex case such that

2. Continuous Linear Operators and Functionals

μ(a) = μ1 (a) + μ2 (−a)

or

19

μ(a) = μ1 (a) + μ2 (−a) + μ3 (ia) + μ4 (−ia)

for all a ∈ P, respectively. (d) Let (P, V) be a locally convex vector space over K, that is a locally convex cone which is a vector space over K and carries the modular symmetric topology. The functionals in the dual cone P ∗ of P are real-valued, but there exists a canonical correspondence between the dual cone P ∗ and the usual dual space PK∗ of P as a locally convex topological vector space. PK∗ consists of all K-valued continuous K-linear functionals on P. In the real case this correspondence is obvious, as P ∗ and PK∗ coincide. (If both a, −a ∈ P, then μ(a) + μ(−a) = 0 for every μ ∈ P ∗ , hence μ is linear over R.) In the complex case there is an established correspondence between P ∗ and PK∗ : The real part μ of every continuous complex linear functional μK on P is in P ∗ and, conversely, for every μ ∈ P ∗ , the mapping a → μ(a) − i μ(ia) deﬁnes a continuous complex linear functional μK ∈ PK∗ . PK∗ is again a vector space over K, and for μK and α ∈ K the respective projections μ and (αμ) into P ∗ of the functionals μK and αμK relate as αμ (a) = e (αμK )(a) = e μK (αa) = μ(αa). The above formula eﬀectively extends the multiplication by non-negative reals in P ∗ to all scalars in K in such a way that the mapping μK → μ : PK∗ → P ∗ becomes a vector space isomorphism. Similarly, every element ϕK of the (algebraic) second vector space dual PK∗∗ of PK corresponds to a real-linear functional ϕ on the dual cone P ∗ by ϕ(μ) = e ϕK (μK ) for μ ∈ P ∗ . On the other hand, every functional ϕ on P ∗ that is linear with respect to the non-negative reals corresponds to a K-valued linear functional ϕK ∈ PK∗∗ on PK∗ by ϕK (μK ) = ϕ(μ)

or

ϕK (μK ) = ϕ(μ) − iϕ(iμ)

for μK ∈ PK∗ in the real or complex case, respectively. Here we use the above deﬁned extension of the scalar multiplication in P ∗ . K-linearity for ϕK is easily checked. Indeed, additivity is obvious for ϕK . For compatibility with the scalar multiplication, the real case follows from ϕ(μ) + ϕ (−1)μ = ϕ(0) = 0, hence ϕ (−1)μ = −ϕ(μ). In the complex case we calculate for μK ∈ PK∗ and α = x + iy ∈ C ϕK (x + iy)μK = ϕ (x + iy)μ − iϕ (x + iy)iμ = (x + iy)ϕ(μ) + (y − ix)ϕ(iμ) = (x + iy) ϕ(μ) − iϕ(iμ) = (x + iy)ϕK (μK ).

20

I Locally Convex Cones

∗∗ Thus there is also a canonical correspondence between P , the cone of all real-valued linear functionals on P ∗ see 7.3(i) below and the second vector space dual PK∗∗ of PK . (e) In 1.4(c) and (e) on the other hand, due to the generality of the settings, a complete description for the respective dual cones is not immediately available. We may, however, identify some of their elements: In 1.4(c), let μ be a continuous monotone linear function on the locally convex ordered topological vector space (E, ≤). Then the mapping

A → sup{μ(a) | a ∈ A} : Conv(E) → R is seen to be an element of Conv(E)∗ . (f) In 1.4(e), if μ ∈ v ◦ ⊂ P ∗ for some v ∈ V, and if vˆ(x) ≤ v for some and x ∈ X, then the mapping μx : F (X, P) → R such that v∈V Vb μx (f ) = μ f (x)

for all f ∈ FVb (X, P)

is a continuous linear functional on FVb (X, P); more precisely μx ∈ vˆ◦ . (g) In 1.4(g), for p < +∞ the dual cone of lp consists of all sequences (yi )i∈N such that yi ≥ 0 for all i ∈ N and (yi )q < +∞, where q is the conjugate index of p. 2.2 Embeddings. We have intuitively used the term embedding before. Let us now establish a precise deﬁnition: Let (P, V) and (Q, W) be locally convex cones. A linear operator Φ : P → Q is called an embedding of (P, V) into (Q, W) if it can be extended to a mapping Φ : (P ∪ V) → (Q ∪ W) such that Φ(V) = W and a≤b+v

holds if and only if

Φ(a) ≤ Φ(b) + Φ(w)

for all a, b ∈ P and v ∈ V. This condition implies that Φ is continuous, and in case that Φ is one-toone, that the inverse operator Φ−1 : Φ(P) → P is also continuous. It is easily veriﬁed that the composition of two embeddings is again an embedding in this sense. Embeddings are meant to preserve not just the topological structure, but also the particular neighborhood system of a locally convex cone. Lemma 2.3. Let (P, V) and (Q, W) be locally convex cones and let Φ : P → Q be an embedding of (P, V) into (Q, W). If the symmetric topology of P is Hausdorﬀ, then Φ is one-to-one. Proof. Under the assumptions of the Lemma, suppose that Φ(a) = Φ(b) holds for a, b ∈ P. Then a ≤ b + v and b ≤ a + v, hence a ∈ v s (b) for all v ∈ V follows from 2.2. This yields a = b since the symmetric topology of P is supposed to be Hausdorﬀ.

2. Continuous Linear Operators and Functionals

21

An embedding Φ of (P, V) into (Q, W) is called an isomorphism if the mapping Φ : (P ∪ V) → (Q ∪ W) is invertible. Then Φ−1 is an embedding of (Q, W) into (P, W). Hahn-Banach type extension and separation theorems for linear functionals are most important for the development of a powerful duality theory for locally convex cones. We shall mention a few results from [100] and [172]. A sublinear functional on a cone P is a mapping p : P → R such that p(αa) = αp(a)

and

p(a + b) ≤ p(a) + p(b)

holds for all a, b ∈ P and α ≥ 0. Likewise, an extended superlinear functional on P is a mapping q : P → R = R ∪ {+∞, −∞} such that q(αa) = αq(a)

and

q(a + b) ≥ q(a) + q(b)

holds for all a, b ∈ P and α ≥ 0. (We set α + (−∞) = −∞ for all α ∈ R, α · (−∞) = −∞ for all α > 0 and 0 · (−∞) = 0 in this context.) We cite Theorem 3.1 from [172]: Sandwich Theorem 2.4. Let (P, V) be a locally convex cone, and let v ∈ V. For a sublinear functional p : P → R and an extended superlinear functional q : P → R there exists a linear functional μ ∈ v ◦ such that q ≤ μ ≤ p if and only if q(a) ≤ p(b) + 1 holds whenever a ≤ b + v for a, b ∈ P. This theorem is the basic tool for the development of a duality theory for locally convex cones. It leads to a variety of Hahn-Banach type extension and separation results, the most general ones being Theorems 4.1 and 4.4 in [172]. For future use we shall quote both of these: Extension Theorem 2.5. Let (P, V) be a locally convex cone, C and D non-empty convex subsets of P, and let v ∈ V. Let p : P → R be a sublinear and q : P → R an extended superlinear functional. For a convex function f : C → R and a concave function g : D → R there exists a monotone linear functional μ ∈ v ◦ such that q ≤ μ ≤ p,

g≤μ

on D

and

μ≤f

on C

if and only if q(a) + ρg(d) ≤ p(b) + σf (c) + 1

whenever

a + ρd ≤ b + σc + v

for a, b ∈ P, c ∈ C, d ∈ D and ρ, σ ≥ 0. In the context of this theorem (Theorem 4.1 in [172]), an R-valued function f deﬁned on a convex subset C of an ordered cone P is called convex if f λc1 + (1 − λ)c2 ≤ λf (c1 ) + (1 − λ)f (c2 )

22

I Locally Convex Cones

holds for all c1 , c2 ∈ C and λ ∈ [0, 1]. Likewise, f : C → R is called concave if f λc1 + (1 − λ)c2 ≥ λf (c1 ) + (1 − λ)f (c2 ) holds for all c1 , c2 ∈ C and λ ∈ [0, 1]. An aﬃne function f : C → R is both convex and concave. The generality of Theorem 2.5 leads to a wide variety of applications and special cases. An extension theorem in the true meaning of the words can be obtained by identifying the convex sets C and D and the functions f and g. For the following (still very general) corollary we shall also leave out (by setting them equal to +∞ and −∞ outside 0 ∈ P, respectively) the functionals p and q. Corollary 2.6. Let (P, V) be a locally convex cone, C a non-empty convex subsets of P, and let v ∈ V. For an aﬃne function f : C → R there exists a monotone linear functional μ ∈ v ◦ such that μ = f on C if and only if ρf (d) ≤ σf (c) + 1

whenever

ρd ≤ σc + v

for c, d ∈ C, and ρ, σ ≥ 0. If C is indeed a subcone of P, that is (C, V) is a locally convex subcone of (P, V), then the condition of Corollary 2.6 reduces to: f (0) = 0 and f (d) ≤ f (c) + 1 holds whenever d ≤ c + v for c, d ∈ C. But this means that the aﬃne function f is indeed a linear functional on C and contained in the polar of the neighborhood v ∈ V. This observation leads to the following most frequently used consequence of Theorem 2.5 (see also Theorem II.2.9 in [100]). Corollary 2.7. Let (N , V) be a subcone of the locally convex cone (P, V). Every continuous linear functional on N can be extended to a continuous ◦ there is μ ˆ ∈ vP◦ linear functional on P; more precisely: For every μ ∈ vN such that μ ˆ coincides with μ on N . Theorem 4.4 in [172] deals with the separation of convex sets by continuous linear functionals, a result that can also be obtained by special insertions in Theorem 2.5. Separation Theorem 2.8. Let (P, V) be a locally convex cone, C and D non-empty convex subsets of P, and let v ∈ V. For α ∈ R there exists a monotone linear functional μ ∈ v ◦ such that μ(c) ≤ α ≤ μ(d)

for all c ∈ C and d ∈ D

if and only if αρ ≤ ασ + 1 for c ∈ C, d ∈ D and ρ, σ ≥ 0.

whenever

ρd ≤ σc + v

3. Weak Local and Global Preorders

23

In a special result for points and closed case, this leads to a separation convex sets see Corollary 4.6 in [172] : Corollary 2.9. Let A be a non-empty convex subset of a locally convex cone (P, V) such that 0 ∈ A. (i) If A is closed with respect to the lower topology on P, then for every element b ∈ / A in P there exists a monotone linear functional μ ∈ P ∗ such that μ(a) ≤ 1 ≤ μ(b) for all a ∈ A and indeed 1 < μ(b) if b is bounded above. (ii) If A is closed with respect to the upper topology on P, then for every element b ∈ / A in P there exists a monotone linear functional μ ∈ P ∗ such that μ(b) < −1 ≤ μ(a) for all a ∈ A. In view of the corresponding separation results for locally convex topological vector spaces, Corollary 2.9 is not entirely satisfying, in particular since it requires that 0 ∈ A. A stronger and more suitable separation statement will be derived in Section 4 (Theorem 4.30). It will make use of the relative topologies of a locally convex cone which are to be introduced below. We shall quote and make use of another result from [172]. The Range Theorem (Theorem 5.1 in [172]) describes the scope of all linear functionals whose existence is guaranteed by the Sandwich Theorem. It is a powerful and indeed non-trivial consequence even in the special case of vector spaces, where its formulation can however be considerably simpliﬁed. Range Theorem 2.10. Let (P, V) be a locally convex cone. Let p and q be sublinear and extended superlinear functionals on P and suppose that there is at least one linear functional μ ∈ P ∗ satisfying q ≤ μ ≤ p. Then for all a ∈ P sup μ(a) = sup inf {p(b) − q(c) | b, c ∈ P, q(c) ∈ R, a + c ≤ b + v},

μ∈P ∗ q≤μ≤p

v∈V

and for all a ∈ P such that μ(a) is ﬁnite for at least one μ ∈ P ∗ satisfying q ≤ μ ≤ p sup μ(a) = inf sup {q(c) − p(b) | b, c ∈ P, p(b) ∈ R, c ≤ a + b + v}.

μ∈P ∗ q≤μ≤p

v∈V

3. Weak Local and Global Preorders In addition to the given order ≤ on a locally convex cone, we shall frequently use the weak (global) preorder (for details, see [175] and Section 4 below)

24

I Locally Convex Cones

which is slightly weaker then the given order and deﬁned for a, b ∈ P by ab

if

a ≤ γb + εv

for all v ∈ V and ε > 0 with some 1 ≤ γ ≤ 1 + ε. This order represents a closure of the given order with respect to the linear and topological structures of P. It is obviously coarser than the given order, that is a ≤ b implies a b for a, b ∈ P. In the preceding Examples 1.4(a) and (b), however, both orders coincide. In 1.4(e) this depends on the order in P and the neighborhood such ˆ If P = R and if for every x ∈ X there is vˆ ∈ V valued functions in V. that vˆ(x) < +∞, then the given and the weak preorder coincide. In 1.4(c), on the other hand, we have A B if A ⊂ ↓B , where ↓B denotes the topological closure in E of the decreasing hull ↓B of B. Note that ↓B is again a convex subset of E. In 1.4(d), that is the case of a vector space P over R or C, the weak preorder is given by a b if a − b ∈ v(0) for all v ∈ V. In this way (P, ) becomes a locally convex ordered topological vector space in the usual sense if endowed with the (modular) symmetric topology resulting from the neighborhood system. The weak preorder on P is again compatible with the algebraic operations, as Lemma 4.1 below will imply. In Corollary 4.31 below (see also Theorem 3.1 in [175]) we shall establish that the weak preorder on a locally convex cone P is entirely determined by its dual cone P ∗ , that is a b holds for a, b ∈ P if and only if μ(a) ≤ μ(b) for all μ ∈ P ∗ . The weak preorder may also be used in a full cone containing P and V. Consequently, the respective relation involving the neighborhoods in V is deﬁned for elements a, b ∈ P and v ∈ V as ab+v if a ≤ γ(b + v) + εu for all u ∈ V and ε > 0 with some 1 ≤ γ ≤ 1 + ε. This condition can be slightly simpliﬁed: Lemma 3.1. Let a, b ∈ P and v ∈ V. We have a b + v if and only if for every ε > 0 there is 1 ≤ γ ≤ 1 + ε such that a ≤ γb + (1 + ε)v. Proof. Let a, b ∈ P and v ∈ V. Suppose that a b + v and let ε > 0. According to the preceding deﬁnition of the weak preorder involving neighborhoods, for u = v and ε/2 in place of ε, there is 1 ≤ γ ≤ 1 + ε/2 such that a ≤ γ(b+v)+(ε/2)v ≤ γb+εv. For the reverse implication suppose that the condition of the Lemma holds, and let u ∈ V and ε > 0. There is λ ≥ 0 such that 0 ≤ b + λu. Choose 0 < δ ≤ ε/λ. Then there is 1 ≤ γ ≤ 1 + δ such that a ≤ γb + (1 + δ)v ≤ γb + (1 + δ)v + (1 + δ − γ)(b + λu) ≤ (1 + δ)(b + v) + δλ u ≤ (1 + δ)(b + v) + εu. This shows a b + v.

3. Weak Local and Global Preorders

25

Endowed with the weak preorder (P, V) forms again a locally convex cone. For details we refer to [175]. In Corollary 4.34 below (see also Theorem 3.2 in [175]) we shall demonstrate that for a, b ∈ P and a neighborhood v ∈ V, we have a b + v if and only if μ(a) ≤ μ(b) + 1 holds for all μ ∈ v ◦ . The neighborhoods with respect to the weak preorder in P are therefore entirely determined by their polars. Given a neighborhood v ∈ V the weak local preorder (see [175]) v on P is the weak (global) preorder with respect to the neighborhood subsystem Vv = {αv | α > 0}. That is, for a, b ∈ P we have a v b

if

a ≤ γb + εv

for all ε > 0 with some 1 ≤ γ ≤ 1 + ε. Corollary 4.31 below (see also Theorem 3.1 in [175]) states that a v b if and only if μ(a) ≤ μ(b) holds for all μ ∈ v ◦ . Lemma 3.2. Let a, b ∈ P. (a) a b if and only if for every v ∈ V and ε > 0 there is 1 ≤ γ ≤ 1 + ε such that a γb + εv. (b) a v b for v ∈ V if and only if for every ε > 0 there is 1 ≤ γ ≤ 1 + ε such that a γb + εv. Proof. Part (a) follows from Part (b) as a b holds if and only if a v b for all v ∈ V. For Part (b) let a, b ∈ P and v ∈ V such that the second condition in (b) holds. Given ε > 0 set δ = min{ε/3, 1}. Then a γb + δv holds with some 1 ≤ γ ≤ 1 + δ. We infer from Lemma 3.1 that there is 1 ≤ γ ≤ 1 + δ such that a ≤ (γ γ)b + (1 + δ)δv. Since (1 + δ)δ ≤ ε and 1 ≤ γ γ ≤ (1 + δ)2 ≤ 1 + ε, and since ε > 0 was arbitrarily chosen, we conclude that a v b. The reverse implication is trivial since a ≤ γb + εv implies that a γb + εv. Lemma 3.2 shows in particular that the second iteration of the weak preorder, that is the second weak preorder generated by the ﬁrst one does indeed coincide with the ﬁrst one. We observe that for a linear operator T between locally convex cones (P, V) and (Q, W), continuity with respect to the given orders implies continuity and monotonicity with respect to the respective weak preorders on P and Q. Indeed, suppose that for v ∈ V and w ∈ W we have T (a) ≤ T (b)+w whenever a ≤ b + v for a, b ∈ P. Let a b + v and let ε > 0. According to Lemma 3.1 there is 1 ≤ γ ≤ 1 + ε such that a ≤ γb + (1 + ε)v. Thus T (a) ≤ γT (b) + (1 + ε)w. Since ε > 0 was arbitrarily chosen, we conclude that T (a) T (b) + w, thus establishing our claim. The weak preorder may also be used to establish a representation for a locally convex cone (P, V) as a cone of continuous R-valued functions on some topological space and as a cone of convex subsets of some locally convex topological vector space, respectively. We shall cite Theorem 4.1 from [175]. Recall the deﬁnition of an embedding from 2.2.

26

I Locally Convex Cones

Theorem 3.3. Every locally convex cone (P, V) can be embedded with respect to its weak preorder into (i) a locally convex cone of continuous R-valued functions on some topological space X, endowed with the pointwise order and operations and the topology of uniform convergence on a family of compact subsets of X. (ii) a locally convex cone of convex subsets of a locally convex topological vector space, endowed with the usual addition and multiplication by scalars, the set inclusion as order and the neighborhoods inherited from the vector space.

4. Boundedness and the Relative Topologies While all elements of a locally convex cone are bounded below by deﬁnition, they need not to be bounded above. Given a neighborhood v ∈ V, an element a of a locally convex cone (P, V) is called v-bounded (above) (see [100], I.2.3) if there is λ ≥ 0 such that a ≤ λv. The subset Bv ⊂ P of all v-bounded elements is a subcone and even a face of P. Correspondingly, by

B = v∈V Bv we denote the subcone (and face) of all bounded elements of P (see Section 1 and Proposition 4.11 below). All invertible elements of P were seen to be bounded, and continuous linear functionals take only ﬁnite values on bounded elements (see Section 2). The presence of unbounded elements constitutes a signiﬁcant diﬀerence between locally convex cones and locally convex topological vector spaces. It tends to make matters more interesting, but also considerably more complicated. If, for example, the element a ∈ P is not bounded, then the mapping α → αa : [0, +∞) → P, is not necessarily continuous if we consider the usual topology of [0, +∞) and any of the given (upper, lower or symmetric) topologies on P see Proposition 1.1(iii) . Hence these topologies appear to be rather restrictive. For similar reasons, our upcoming deﬁnition of measurability for P-valued functions in Chapter II would turn out to be very limiting if applied to the given topologies of a locally convex cone. We shall therefore introduce slightly coarser neighborhoods on P which take unbounded elements suitably into account. Given a neighborhood v ∈ V and ε > 0, we deﬁne the corresponding upper and lower relative neighborhoods vε (a) and (a)vε for an element a ∈ P by vε (a) = { b ∈ P | b ≤ γa + εv for some 1 ≤ γ ≤ 1 + ε } (a)vε = { b ∈ P | a ≤ γb + εv for some 1 ≤ γ ≤ 1 + ε }. Their intersection vεs (a) = vε (a) ∩ (a)vε is the corresponding symmetric relative neighborhood. These are of course convex subsets of P. Note that for a positive element a ∈ P the above expressions somewhat simplify. Since γa ≤ (1 + ε)a in this case, we have vε (a) = { b ∈ P | b ≤ (1 + ε)a + εv} and

4. Boundedness and the Relative Topologies

27

(a)vε = { b ∈ P | a ≤ (1 + ε)b + εv}. We shall frequently use the following observations: Lemma 4.1. Let a, b, c, ai , bi ∈ P, v ∈ V, λ ≥ 0 and ε, δ > 0. (a) If a ∈ vε (b) and b ∈ vδ (c), then a ∈ v(ε+δ+εδ) (c). (b) If a ∈ vε (b) and 0 ≤ b + λv, then a ≤ (1 + ε)b + ε(1 + λ)v. (c) If a ∈ vε (b) and 0 ≤ a + λv, then a ≤ (1 + ε)b + ε(1 + λ + ε)v and 0 ≤ b + (λ + ε)v. (d) If ai ∈ vε (bi ) and if 0 ≤ bi + λv for i = 1, . . . , n, then (a1 + . . . + an ) ∈ vεn(1+λ) (b1 + . . . + bn ). Proof. For (a), let a ∈ vε (b) and b ∈ vδ (c), that is a ≤ γb + εv and b ≤ λc+δv for some 1 ≤ γ ≤ 1+ε and 1 ≤ λ ≤ 1+δ. Then a ≤ γλc+(γδ +ε)v. As γδ + ε ≤ (1 + ε)δ + ε = ε + δ + εδ and 1 ≤ γλ ≤ (1 + ε)(1 + δ) = 1 + ε + δ + εδ, we have a ∈ v(ε+δ+εδ) (c). For (b), let a ∈ vε (b), that is a ≤ γb + εv for some 1 ≤ γ ≤ 1 + ε. If 0 ≤ b + λv, then a ≤ γb + εv + (1 + ε − γ)(b + λv) ≤ (1 + ε)b + (ε + ελ)v. For (c), let a ∈ vε (b) and λ ≥ 0 such that 0 ≤ a + λv. Then a ≤ γb + εv with some 1 ≤ γ ≤ 1+ε, hence 0 ≤ γb+(ε+λ)v, and indeed 0 ≤ b+ ε+λ γ v ≤ b+(ε+λ)v. Part (b) yields a ≤ (1+ε)b+ε(1+λ+ε)v. For (d), let ai ∈ vε (bi ) and 0 ≤ bi + λv. Then ai ≤ (1 + ε)bi + ε(1 + λ)v by Part (b). This yields a1 + . . . + an ≤ (1 + ε)(b1 + . . . + bn ) + nε(1 + λ)v, hence our claim.

Property 4.1(a) implies in particular that vε (a) ⊂ v3ε (c) whenever a ∈ vε (b) and b ∈ vε (c) for a, b, c ∈ P and 0 < ε ≤ 1. Similar statements as in Lemma 4.1 hold for the lower and for the symmetric relative neighborhoods. For elements a, b ∈ P the weak local and global preorders on P as deﬁned in Section 3 can be recovered as a v b

if

a ∈ vε (b)

if

a ∈ vε (b)

for some v ∈ V and all ε > 0, and ab

for all ε > 0 and v ∈ V. Lemma 4.1(d) implies that these orders are compatible with the algebraic operations in P.

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For varying v ∈ V and ε > 0 the neighborhoods vε (·), (·)vε and vεs (·) create the upper, lower and symmetric relative topologies on P, respectively. We notice that a ≤ b + v for a, b ∈ P and v ∈ V implies that a b + v, and for a given ε > 0, with δ = min{1, ε/2}, we notice that a b + δv implies a ≤ γb + (1 + δ)δv ≤ γb + εv for some 1 ≤ γ ≤ 1 + δ see Lemma 3.1 , hence b ∈ vε (a). This observation demonstrates that the given upper, lower and symmetric topologies on P are ﬁner than those induced by the same neighborhood system using the weak preorder, and that in turn these topologies are ﬁner than the above deﬁned relative topologies. However, while the relative neighborhoods form convex subsets of P, they do Indeed, the sets in general not create a locally convex cone topology. (a, b) | a ∈ vε (b) are not necessarily convex in P 2 , hence do not establish a convex semiuniform structure on P in the sense of 1.3. For later reference we shall list some further properties of the relative topologies and use the earlier introduced standard notations for subsets of P (see 1.1). Proposition 4.2. Let (P, V) be a locally convex cone. The upper (lower or symmetric) relative topology of P is coarser than the given upper (lower or symmetric) topology and satisﬁes the following: (i) Every element of P admits a basis of convex and decreasing (increasing or order convex) neighborhoods. The symmetric relative neighborhoods in the basis for 0 ∈ P are also balanced. (ii) The mapping (a, b) → a + b : P × P → P is continuous. (iii) The mapping (α, a) → αa : [0, +∞)×P → P is continuous at all points (α, a) ∈ [0, +∞) × P such that either α > 0 or a ∈ P is bounded. (iv) For bounded elements of P the neighborhoods in the upper (lower or symmetric) relative topology are equivalent to the neighborhoods in the given upper (lower or symmetric) topology. Proof. We observed before that the relative topologies are coarser than the given topologies on P. Clearly, for a ∈ P, v ∈ V and ε > 0 the relative neighborhoods vε (a), (a)vε or vεs (a) are convex and decreasing, increasing or order convex, respectively. The symmetric relative neighborhoods of 0 ∈ P are also balanced. Indeed, let v ∈ V and ε > 0 and let a ∈ vεs (0). Then a ≤ εv and 0 ≤ γa + εv for some 1 ≤ γ ≤ 1 + ε. Thus 0 ≤ a + (ε/γ)v ≤ a + εv. Let 0 ≤ λ ≤ 1. Then λa ∈ vεs (0) follows from the convexity of vεs (0) since λa = λa + (1 − λ)0. If on the other hand b + λa = 0 for b ∈ P, then b ≤ b + λ(a + εv) ≤ εv

and

0 = b + λa ≤ b + εv

Hence b ∈ vε (0) holds in this case as well. For property (ii), let a, b ∈ P, v ∈ V and ε > 0. There is λ ≥ 0 such that 0 ≤ a + λv and 0 ≤ b + λv. Choose δ = ε/(2λ + 4). Then for c ∈ vδ (a) and d ∈ vδ (b) we have c + d ∈ v2δ(1+λ) (a + b) = vε (a + b) by Lemma 4.1(d). This shows continuity of the addition with respect to the

4. Boundedness and the Relative Topologies

29

upper relative topology. Next with the same choice for δ, let c ∈ (a)vδ and d ∈ (b)vδ , that is a ∈ vδ (c) and b ∈ vδ (d). Then we have a ≤ γc + δv for some 1 ≤ γ ≤ 1 + δ, hence 0 ≤ γc + (λ + δ)v and 0 ≤ c + (λ + 1)v. Likewise, 0 ≤ d + (λ + 1)v. Now 4.1(d) yields a + b ∈ v2δ(2+λ) (c + d) ⊂ vε (c + d). Thus c + d ∈ (a + b)vε . This shows continuity of the addition with respect to the lower relative topology. Combining the preceding arguments, we realize that c ∈ vδs (a) and d ∈ vδs (b) yields c + d ∈ vεs (a + b), which proves continuity with respect to the symmetric relative topology. Next we shall argue Part (iv): Let a ∈ P be a bounded element, let v ∈ V and ε > 0. There is λ ≥ 0 such that a ≤ λv. We shall verify that (εv)(a) ⊂ vε (a) ⊂ (ρv)(a)

and

(a)(εv) ⊂ (a)vε (a) ⊂ (a)(ρv).

with ρ = ε(1 + λ). Indeed, the inclusions (εv)(a) ⊂ vε (a) and (εv)(a) ⊂ vε (a) are obvious. Moreover, for b ∈ vε (a) we have b ≤ γa + εv with some 1 ≤ γ ≤ 1 + ε. Then γa = a + (γ − 1)a ≤ a + ελv implies that b ≤ a + ε(1 + λ)v = a + ρv, hence b ∈ (ρv)(a). For b ∈ (a)vε on the other hand, we have a ≤ γb + εv with 1 ≤ γ ≤ 1 + ε. Then γa ≤ a + ελv implies γa ≤ γb + ε(1 + λ)v = γb + ρv, hence a ≤ b + (ρ/γ)v ≤ b + ρv, and therefore b ∈ (a)(ρv). For the ﬁrst case in Part (iii) let (α, a) ∈ (0, +∞)×P. For v ∈ V andε > 0 let λ ≥ 0 such that 0 ≤ a+λv. For 0 < δ < min 1, ε/3, ε/ 2α(1+λ) we consider the neighborhoods uδ (α) = α/(1 + δ) , α(1 + δ) of α in [0, +∞) and vδ (a) of a in P. For every b ∈ vδ (a) we have b ≤ (1 + δ)a + δ(1 + λ)v by 4.1(b). For β ∈ uδ (α) we set γ = β(1 + δ)/α and estimate βb ≤ β(1 + δ)a + βδ(1 + λ)v = γ(αa) + βδ(1 + λ)v. Now α/(1+δ) ≤ β ≤ α(1+δ) and our choice for δ implies 1 ≤ γ ≤ (1+δ)2 ≤ 1 + ε as well as βδ(1 + λ) ≤ α(1 + δ)δ(1 + λ) ≤ 2αδ(1 + λ) ≤ ε. Thus βb ∈ vε (αa). This shows continuity for the scalar multiplication at (α, a) with respect to the upper relative topology. For the lower topology, with the same choice for δ, let b ∈ (a)vδ and β ∈ uδ (α). Then a ≤ (1+δ)b+δ(2+λ)v by 4.1(c). We set γ = α(1 + δ)/β and obtain αa ≤ α(1 + δ)b + αδ(2 + λ)v = γ(βb) + αδ(2 + λ)v. We verify 1 ≤ γ ≤ 1 + ε and αδ(2 + λ) ≤ ε and infer that αa ∈ (βb)vε , hence βb ∈ (αa)vε . This shows continuity with respect to the lower relative topology. The combination of both arguments yields continuity with respect to the symmetric relative topology. The second case of Part (iii), that is the continuity of the scalar multiplication at (α, a) ∈ [0, +∞) × P for a bounded element a ∈ P, follows directly from Part (iv) and from Part (iii) of Proposition 1.1. Indeed, the given and the relative upper (lower or symmetric) topologies coincide locally at a ∈ P by (iv), thus continuity with respect to any of the given topologies

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which was established in Proposition 1.1(iii) implies continuity with respect to the corresponding relative topology. For P = R, in particular, Part (iv) of the preceding proposition implies that the given and the relative topologies coincide on all reals. They also coincide on the element +∞, thus everywhere, as can be easily veriﬁed (for details on this, see Example 4.37(a) below). Part (iv) together with Corollary 1.2 also yields: Corollary 4.3. Let (P, V) be a locally convex cone and let P0 be the subcone of all invertible elements of P. The mapping (α, a) → αa : R × P0 → P0 is continuous with respect to the symmetric relative topology of P. We observe that the given upper (lower or symmetric) topologies do of course satisfy the properties listed in Proposition 4.2 with the exception of 4.2(iii). More precisely, we take note: Proposition 4.4. Let (P, V) be a locally convex cone. The upper (or lower) relative topology is the ﬁnest topology on P which is coarser than the given upper (or lower) topology and satisﬁes property (iii) from Proposition 4.2. Proof. Let τ be any topology on P which is ﬁner than the upper (or lower) topology and satisﬁes property (iii) from Proposition 4.2. Let a ∈ P and let U (a) be a neighborhood in τ for a. We shall show that U (a) contains some upper (or lower) relative neighborhood of a. The mapping (α, b) → αb : [0, +∞) × P → P is continuous with respect to τ at the point (1, a). Thus there is a neighborhood V (a) in τ and 0 < ε ≤ 1 such that βb ∈ U (a) for all b ∈ V (a) and β ∈ [1 − ε, 1 + ε]. Moreover, since τ is coarser than the upper (or lower) topology of P there is v ∈ V such that v(a) ⊂ V (a) or (a)v ⊂ V (a) . In the case of the upper topology, then for every c ∈ vε (a) we have c ≤ γa + εv for some 1 ≤ γ ≤ 1 + ε. Thus d ≤ a + (ε/γ)v ≤ a + v for d = (1/γ)c. We infer that d ∈ v(a) ⊂ V (a), hence c = γd ∈ U (a) since γ ∈ [1 − ε, 1 + ε]. This shows vε (a) ⊂ U (a). Likewise, in the case of the lower topology, for c ∈ (a)vε we have a ≤ γc + εv for some 1 ≤ γ ≤ 1 + ε, hence d = γc ∈ (a)(v) ⊂ V (a). This yields c = (1/γ)d ∈ U (a) since (1/γ) ∈ [1 − ε, 1 + ε]. We conclude that (a)vε ⊂ U (a) in this case. Proposition 4.5. Let (P, V) and (Q, W) be locally convex cones. A continuous linear operator T : P → Q is also continuous if both P and Q are endowed with either their respective upper, lower or symmetric relative topologies. Proof. Let T : P → Q be a continuous linear operator. Given w ∈ W , there is v ∈ V such that a ≤ b + v implies T (a) ≤ T (b) + w for elements a, b ∈ P. some 1 ≤ γ ≤ 1 + ε, implies Thus a ∈ vε (b), that is a ≤ γb + εv with T (a) ≤ γT (b) + εw, hence T (a) ∈ wε T (b) . A similar argument shows continuity with respect to either the lower or symmetric relative topologies of P and Q.

4. Boundedness and the Relative Topologies

31

For Q = R, in particular, we remarked earlier (see also Example 4.37(a) below) that the given and the relative topologies coincide. A linear functional μ ∈ P ∗ is therefore also continuous if we endow P with either of its relative and R with the corresponding given topology. We shall also use the (upper, lower, symmetric) relative v-topologies on P, generated by the relative neighborhoods for a ﬁxed v ∈ V. The symmetric relative v-topology, in particular, is induced by the pseudometric √ dv (a, b) = inf 1, ε | a ∈ vεs (b) . The properties of a pseudometric (see Section 2.1 in [198]) are readily checked for this expression: We obviously have dv (a, b) ≥ 0, dv (a, a) = 0 and dv (a, b) = dv (b, a) for a, b ∈ P. The triangular inequality, namely dv (a, c) ≤ dv (a, b) + dv (b, c) for a, b, c ∈ P, holds trivially true if either dv (a, b) = 1 or dv (b, c) = 1. Otherwise, if dv (a, b) < ε < 1 and dv (b, c) < δ < 1, then a ∈ vεs2 (b) and b ∈ vδs2 (c) implies by Lemma 4.1(a) that a ∈ vρs (c), where ρ = ε2 + δ 2 + ε2 δ 2 ≤ (ε + δ)2 . Thus dv (a, c) ≤ ε + δ, hence the triangular inequality holds. As a consequence of the availability of a pseudometric for the symmetric relative v-topology, arbitrary subsets of separable subsets of P remain separable (see 16G in [198]). We shall use this fact in Chapter II. The (upper, lower, symmetric) relative topologies on P are the common reﬁnements of all (upper, lower, symmetric) relative v-topologies. 4.6 The Weak Topology σ(P, P ∗ ). The weak topology σ(P, P ∗ ) on a locally convex cone (P, V) is generated by its dual cone in the following way: For an element a ∈ P an upper neighborhood VΥ (a), corresponding to a ﬁnite subset Υ = {μ1 , . . . , μn } of P ∗ , is given by VΥ (a) = b ∈ P | μi (b) ≤ μi (a) + 1 for all μi ∈ Υ . Endowed with these neighborhoods, P forms again a locally convex cone (see Section II.3 in [100]). We are mostly interested in the resulting symmetric topology σ(P, P ∗ ) which is generated by the symmetric neighborhoods

1 , if μi (a) < +∞ |μi (b) − μi (a)|≤ s VΥ (a) = b ∈ P μi (b)= +∞ , if μi (a) = +∞ In this way weak convergence for a net (ai )i∈I in (P, V) means that μ(ai ) i∈I converges towards μ(a) in R (with respect to the symmetric locally convex cone topology of R) for every continuous linear functional μ ∈ P ∗. While the relative topologies of a locally convex cone are generally coarser than the given ones, we observe from the preceding deﬁnition that the relative upper, lower and symmetric weak topologies do indeed coincide with the given upper, lower and symmetric weak topologies on P.

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Lemma 4.7. The weak topology σ(P, P ∗ ) on a locally convex cone (P, V) is coarser than the symmetric relative topology. Proof. For this, let a ∈ P, let Υ be a ﬁnite subset of P ∗ and consider the weak neighborhood VΥs (a) from above. Choose v ∈ V such that μi ∈ v ◦ for all i = 1, . . . , n. We shall show that for a suitable ε > 0 the symmetric neighborhood vεs (a) is contained in VΥs (a). Indeed, let b ∈ vεs (a), that is b ≤ γa + εv

and

a ≤ γ b + εv

and

μ(a) ≤ γ μ(b) + ε

for some 1 ≤ γ, γ ≤ 1 + ε. Thus μ(b) ≤ γμ(a) + ε

for all μ ∈ Υ. If μ(a) = +∞, then μ(a) = +∞. Moreover, ε > 0 may be chosen such that the above implies |μ(b) − μ(a)| ≤ 1 for all μ ∈ Υ such that μ(a) < +∞. This shows b ∈ VΥs (a). Proposition 4.8. Let (P, V) be a locally convex cone. The following statements are equivalent: (i) The symmetric relative topology on P is Hausdorﬀ. (ii) The weak topology on P is Hausdorﬀ. (iii) The weak preorder on P is antisymmetric. Proof. Clearly, (ii) implies (i), since the symmetric relative topology is ﬁner than σ(P, P ∗ ). If a b and b a for a, b ∈ P, then a ∈ vεs (b) for all v ∈ V and ε > 0. If the symmetric relative topology is Hausdorﬀ, then this implies a = b. Thus (i) implies (iii). If the weak preorder is antisymmetric, then for distinct elements a, b ∈ P we have either a b or b a, thus a b + v or b a+v for some v ∈ V by Lemma 3.2. Then there exists a linear functional μ ∈ v ◦ such that μ(a) > μ(b) + 1 or μ(b) > μ(a) + 1, respectively (see s (a) Section 3 and Corollary 4.34 below). The weak neighborhoods V{(1/3)μ} s and V{(1/3)μ} (b) are therefore seen to be disjoint. Thus (iii) implies (ii) as well. 4.9 Boundedness Components. For an element a ∈ P we deﬁne the upper and lower boundedness components of a as B(a) = vε (a) and (a)B = (a)vε , v∈V ε>0

v∈V ε>0

respectively. The elements of B(a) are called bounded above relative to a. Correspondingly, the elements of (a)B are called bounded below relative to a. By the deﬁnition of a locally convex cone we have 0 ∈ B(a) for all a ∈ P, and B(0) = B consists of all bounded elements of P. We shall ﬁrst list a few basic properties of the upper boundedness components.

4. Boundedness and the Relative Topologies

33

Proposition 4.10. Let a, b, ∈ P. The following are equivalent: (i) b ∈ B(a). (ii) B(b) ⊂ B(a). (iii) For every v ∈ V there are α, β ≥ 0 such that b ≤ αa + βv. (iv) The mapping α → a + αb : [0, +∞) → P is continuous with respect to the symmetric relative topology of P. (v) For all μ ∈ P ∗ , μ(a) < +∞ implies μ(b) < +∞. Proof. Let a, b ∈ P. We shall ﬁrst establish the equivalence of (i), (ii) and (iii): Suppose that b ∈ B(a) and let c ∈ B(b). Then for every v ∈ V there are ε, δ > 0 such that c ∈ vε (b) and b ∈ vδ (a). Following Lemma 4.1(a), this implies c ∈ v(ε+δ+εδ) (a). We conclude that c ∈ B(a), hence B(b) ⊂ B(a), and (i) implies (ii). If B(b) ⊂ B(a), then b ∈ B(a) since b ∈ B(b) trivially holds. Thus for every v ∈ V there is ε > 0 such that b ∈ vε (a), that is b ≤ αa + βv for some α, β ≥ 0. Therefore (ii) implies (iii). If, on the other hand, for some v ∈ V we have b ≤ αa + βv for α, β ≥ 0, we choose λ ≥ 0 such that 0 ≤ a + λv. Then b ≤ (αa + βv) + (a + λv) = (1 + α)a + (β + λ)v, hence b ∈ vε (a) for every ε > max{α, β + λ}. If this argument can be made for all v ∈ V, then we have b ∈ B(a), hence (iii) implies (i) as well, and the Conditions (i), (ii) and (iii) are seen to be equivalent. Next we shall verify that (iii) implies (iv): Following Proposition 4.2(iii), for any choice of b ∈ P the mapping α → αb is continuous with respect to the symmetric relative topology of P on the open interval (0, +∞). Likewise, of course, is the constant mapping α → a. Thus by the continuity of the addition in P see Proposition 4.2(ii) , the mapping f : [0, +∞) → P such that f (α) = a + αb is also continuous on (0, +∞). In case that (iii) holds, we shall verify continuity at α = 0 as well: Given v ∈ V and ε > 0 there is λ > 0 such that 0 ≤ b + λv, and by (iii) there are γ, ρ ≥ 0 such that b ≤ γa + ρv. Then for δ = min{ ε/γ, ε/ρ, ε/λ} and all α ∈ [0, δ) we have a + αb ≤ a + α(γa + ρv) ≤ (1 + αγ)a + αρv. Since our choice of δ guarantees that both αγ ≤ ε and αρ ≤ ε, we infer that f (α) ∈ vε f (0) . Similarly, one observes that a ≤ a + α(b + λv) ≤ (a + αb) + αλv holds for all α ≥ 0. If indeed α ∈ [0, δ), then our choice for δ guarantees that αλ ≤ ε. This shows f (0) ∈ vε f (α) , that is f (α) ∈ f (0) vε , and

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together with the above, f (α) ∈ vεs f (0) for all α ∈ [0, δ). We infer continuity for the function f at α = 0 with respect to the symmetric relative topology of P. Next suppose that (iv) holds. Then for every linear functional μ ∈ P ∗ the mapping ϕ : [0, +∞) → R such that ϕ(α) = μ(a + αb) = μ(a) + αμ(b) is also continuous at α = 0 (see the remark after Proposition 4.5) if we consider R in its symmetric topology, for which = ∞ is an isolated point (see Example 4.37(a) below). Therefore μ(b) is ﬁnite whenever ϕ(0) = μ(a) is ﬁnite, and we infer that (iv) implies (v). Finally, suppose that Condition (iii) fails for the element b. Given a neighborhood v ∈ V, we deﬁne a corresponding functional μv on P setting μv (c) = 0 for all c ∈ P such that c ≤ αa + βv for some α, β ≥ 0, and μv (c) = +∞, else. It is straightforward to check that μv is linear. Indeed, if μv (c) = μv (d) = 0, that is c ≤ αa + βv and c ≤ γ + δv for some α, β, γ, δ ≥ 0, then c + d ≤ (α + γ)a + (β + δ)v, hence μv (c + d) = 0 as well. If, on the other hand, μv (c+d) = 0, that is c+d ≤ αa+βv for some α, β ≥ 0, we choose λ ≥ 0 such that 0 ≤ d+λv and have c ≤ c+d+λv ≤ αa+(β+λ)v. This shows μv (c) = 0. Similarly, one veriﬁes that μv (d) = 0. Moreover, we realize that μv is an element of the polar v ◦ of v, as for c ≤ d + v, we have μv (c) = 0 whenever μv (d) = 0, hence μv (c) ≤ μv (d) + 1 holds in any case. Using this construction, we proceed with our argument: If (iii) fails for b, then there is a neighborhood v ∈ V such that b αa + βv for all choices of α, β ≥ 0, hence μv (b) = +∞, while μv (a) = 0. Thus Condition (v) does not hold either. This in turn shows that (v) implies (iii) and completes our argument. Proposition 4.11. Let a, b, c ∈ P. Then (a) B(a) is a subcone of P, and B ⊂ B(a). (b) B(a) is a face in P, that is b + c ∈ B(a) implies both b, c ∈ B(a). (c) B(αa) = B(a) for α > 0, and B(a) + B(b) ⊂ B(a + b). (d) B(a) is closed in P with respect to the lower relative topology of P. Proof. Part (a) is obvious from Proposition 4.10(iii), since b ≤ αa + βv and c ≤ γa + δv for v ∈ V and α, β, γ, δ ≥ 0 implies that b + c ≤ (α + γ)a + (β + δ)v and λb ≤ λαa + λβv for λ ≥ 0. Moreover, since 0 ∈ B(a), Proposition 4.10(ii) yields that B = B(0) ⊂ B(a). For (b), let b+c ∈ B(a), that is, given v ∈ V, we have b+c ≤ αa+βv for some α, β ≥ 0. Because all elements of a locally convex cone are bounded below, there is λ ≥ 0 such that 0 ≤ c + λv. Thus b ≤ b + c + λv ≤ αa + (β + λ)v. Hence b ∈ B(a). Similarly, one veriﬁes that c ∈ B(a). The ﬁrst statement of (c) is obvious from 4.10(iii). For the second statement, let c ∈ B(a), d ∈ B(b) and v ∈ V. Then c ≤ αa+βv and d ≤ γb+δv for some α, β, γ, δ ≥ 0. Let λ ≥ 0 such that both 0 ≤ a+λv and 0 ≤ b+λv.

4. Boundedness and the Relative Topologies

35

In case that α ≤ γ, this yields c ≤ αa + βv + (γ − α)(a + λv) = γa + ρv, where ρ = β + (γ − α)λ. Thus c + d ≤ γ(a + b) + (ρ + δ)v. In case that α > γ, a similar argument leads to c + d ≤ α(a + b) + (ρ + β)v, where ρ = δ + (α − γ)λ. This veriﬁes c + d ∈ B(a + b). Finally, for Part (c), we remarked before that a linear functional μ ∈ P ∗ is a continuous mapping from P into R if we endow P with either its upper, lower or symmetric relative topology, and R with either its given upper, lower or symmetric topology, respectively. We shall use this observation for the functionals μv ∈ P ∗ for v ∈ V, that we constructed in the argument for the implication (v) ⇒ (iii) in the proof of Proposition 4.10, that is μv (c) = 0 if c ≤ αa + βv for some α, β ≥ 0, and μ(c) = +∞, else. Because R is a closed subset of R in the lower topology of R (see Example 1.4(a)), its (R) under μv is closed in the lower relative topology of P. inverse image μ−1 v

We have B(a) = v∈V μ−1 v (R) by Proposition 4.10(v). Thus B(a) is indeed closed in the lower relative topology of P. We proceed to identify the corresponding properties of the lower boundedness components. Proposition 4.12. Let a, b, ∈ P. The following are equivalent: (i) b ∈ (a)B. (ii) a ∈ B(b). (iii) B(a) ⊂ B(b). (iv) (b)B ⊂ (a)B. (v) For every v ∈ V there are α, β > 0 such that αa ≤ b + βv. (vi) The mapping α → αa + b : [0, +∞) → P is continuous with respect to the symmetric relative topology of P. (vii) For all μ ∈ P ∗ , μ(a) = +∞ implies μ(b) = +∞. Proof. Let a, b ∈ P. First we observe that b ∈ (a)B holds for a, b ∈ P if and only if for every v ∈ V there is ε > 0 such that b ∈ (a)vε , that is a ∈ vε (b). The latter means that a ∈ B(b). Hence (i) and (ii) are indeed equivalent. The equivalence of (ii) and (iii) follows from the corresponding one in Proposition 4.10: We have b ∈ (a)B if and only if a ∈ B(b) by the preceding argument, and the latter holds if and only if B(a) ⊂ B(b) by 4.10. Now suppose that B(a) ⊂ B(b) holds and let c ∈ (b)B. Then b ∈ B(c), hence B(a) ⊂ B(b) ⊂ B(c) by 4.10(ii). Thus a ∈ B(c), hence c ∈ (a)B. This shows (b)B ⊂ (a)B. For the converse suppose that (b)B ⊂ (a)B. This implies b ∈ (a)B, hence a ∈ B(b) and B(a) ⊂ B(b) by 4.10(ii). Therefore (iii) and (iv) are also equivalent. Next suppose that for every v ∈ V there are α, β > 0 such that αa ≤ b + βv. Then a ≤ (1/α)b+(β/α)v, hence a ∈ B(b) by 4.10(iii). For the converse, let a ∈ B(b), that is a ∈ vε (b) for every v ∈ V with some ε > 0. This yields

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a ≤ γb + εv for some 1 ≤ γ ≤ 1 + ε, hence (1/γ)a ≤ b + (ε/γ)v, as claimed. We infer that (ii) and (v) are also equivalent. The remaining parts of this proof require only little eﬀort if we use the already established equivalence of (i) and (ii) and the corresponding results for the upper boundedness components in Proposition 4.10: The equivalence of (ii) and (vi) follows from the equivalence of (i) and (iv) in Proposition 4.10. The equivalence of Conditions (i) and (v) from 4.10, on the other hand, yields that a ∈ B(b) if and only if μ(b) < +∞ implies μ(a) < +∞ for every μ ∈ P ∗ . But the latter is equivalent to the formulation of Condition (vii) in the present proposition. Proposition 4.13. Let a, b, c ∈ P. Then (a) If b ∈ (a)B and c ∈ P, then βb + c ∈ (a)B for all β > 0. (b) (αa)B = (a)B for α > 0, and (a + b)B = (a)B ∩ (b)B. (c) (a)B is closed in P with respect to the upper relative topology of P. Proof. For Part (a), let b ∈ (a)B, that is a ∈ B(b), let c ∈ P and β > 0. 4.11(c) shows that a ∈ B(βb), hence a ∈ B(βb) + B(c) ⊂ B(βb + c). Thus βb + c ∈ (a)B. The ﬁrst part of (b) is obvious from 4.12(v). For the second part let c ∈ (a + b)B. Then a + b ∈ B(c), hence both a ∈ B(c) and b ∈ B(c), since B(c) is a face in P by Proposition 4.11(b). Thus c ∈ (a)B ∩ (b)B. This argument is indeed reversible: If c ∈ (a)B ∩ (b)B, then both a ∈ B(c) and b ∈ B(c). This implies a + b ∈ B(c), since B(c) is a subcone of P see 4.11(a) . Thus c ∈ (a + b)B. For Part (c) we recall that the singleton set {+∞} is closed in the upper topology of R, hence its inverse image μ−1 ({+∞}) under any linear functional μ ∈ P ∗ is closed with respect to the upper relative topology of P. Following Proposition 4.12(vii), (a)B is the intersection of the sets μ−1 ({+∞}) for all μ ∈ P ∗ such that μ(a) = +∞, hence (a)B is indeed closed for the upper relative topology. The sets B s (a) = B(a) ∩ (a)B are called the symmetric boundedness components of P. The elements of B s (a) are called bounded relative to a. The symmetric boundedness components are of particular interest, since they will provide a natural partition of a locally convex cone into boundedness equivalence classes. Before establishing this feature, we shall list a few properties of the symmetric boundedness components: Proposition 4.14. Let a, b, ∈ P. The following are equivalent: (i) b ∈ B s (a). (ii) a ∈ Bs (b). (iii) B(b) = B(a).

4. Boundedness and the Relative Topologies

37

(iv) (b)B = (a)B. (v) B s (b) = B s (a). (vi) For every v ∈ V there are α, β ≥ 0 such that both b ≤ αa + βv

and

a ≤ αb + βv.

(vii) The mapping α → αa + (1 − α)b : [0, 1] → P is continuous with respect to the symmetric relative topology of P. (viii) For all μ ∈ P ∗ , μ(a) = +∞ if and only if μ(b) = +∞. Proof. Let a, b ∈ P. If b ∈ B s (a) = B(a) ∩ (a)B, then b ∈ B(a) and a ∈ B(b). This implies B(a) = B(b) by 4.10(ii). On the other hand, if B(a) = B(b), then b ∈ B(a) and a ∈ B(b), hence b ∈ B(a) ∩ (a)B. This yields the equivalence of (i) and (iii). The equivalence of (iii) and (iv) follows immediately from 4.12(iii) and (iv). Conditions (iii) and (iv) are symmetric in a and b and therefore also equivalent to (ii). Conditions (iii) and (iv) imply (v), which in turn obviously renders (i), since Bs (b) = B s (a) implies b ∈ B s (b) = B s (a). Clearly, (vi) implies (i), since by Proposition 4.10(iii) it yields b ∈ B(a) and a ∈ B(b), hence b ∈ Bs (a). On the other hand, if b ∈ Bs (a), then b ∈ B(a) and a ∈ B(b), and by 4.10(iii), given v ∈ V, there are α , α , β , β ≥ 0 such that and a ≤ α b + β v. b ≤ α a + β v There is λ ≥ 0 such that both 0 ≤ a + λv and 0 ≤ b + λv. Set α = max{α , α } and β = max{ β + (α − α )λ, β + (α − α )λ }. Then b ≤ (α a + β v) + (α − α )(a + λv) ≤ αa + β + (α − α )λ v ≤ αa + βv, and, likewise, a ≤ (α a + β v) + (α − α )(a + λv) ≤ αa + β + (α − α )λ v ≤ αa + βv. Therefore (i) implies (vi) as well. Condition (viii) of this proposition is the combination of the corresponding conditions in Propositions 4.10 and 4.12 and therefore also equivalent to Conditions (i) to (vi). All left to show is that (vii) is equivalent to the rest. First let us verify that (vii) implies (viii). If the mapping α → αa + (1 − α)b : [0, 1] → P

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I Locally Convex Cones

is continuous with respect to the symmetric relative topology of P, then for every linear functional μ ∈ P ∗ the mapping ϕ : [0, 1] → R such that ϕ(α) = μ αa + (1 − α)b = αμ(a) + (1 − α)μ(b) is also continuous (see the remark after Proposition 4.5) if we consider R in its symmetric topology, for which = ∞ is an isolated point. Therefore ϕ(0) = μ(b) is ﬁnite if and only if ϕ(1) = μ(a) is ﬁnite. Hence (vii) implies (viii). Finally, we shall demonstrate how the other conditions imply (vii). Following Proposition 4.2(iii), for any choice of a, b ∈ P the mappings α → αa and α → (1 − α)b are continuous with respect to the symmetric relative topology of P on the open intervals (0, +∞) and (−∞, 1), respectively. Thus by Proposition 4.2(ii), that is the continuity of the addition in P, the mapping f : [0, 1] → P such that f (α) = αa + (1 − α)b is continuous on the interval (0, 1). In case that b ∈ Bs (a), we shall verify continuity at the endpoints α = 0 and α = 1 as well: Proposition 4.12(vi), if applied to the element (1/2)b ∈ (a)B, states that the mapping 1 α → αa + b : 0, +∞ → P 2 is continuous at α = 0. The mapping 1 1 − α b : −∞, α → → P, 2 2 on the other hand, is continuous at 0 by 4.2(iii). Thus the sum of these mappings, that is the function f, is also continuous at 0. A similar argument holds for α = 1. Following Propositions 4.10(iv) and 4.2(iii), respectively, the mappings 1 α → a + (1 − α)b : − ∞, 1 2 and

α →

1 α− 2

a :

1 , +∞ 2

→P

are continuous at α = 1. So is their sum, the function f. This concludes our argument. Proposition 4.15. Let a, b, c ∈ P. Then (a) If b, c ∈ B s (a), then βb + γc ∈ Bs (a) for all β, γ > 0. (b) B s (αa) = B s (a) for α > 0, and B s (a + b) ⊃ B s (a) ∩ B s (b). (c) B s (a) is closed in P with respect to the symmetric relative topology of P.

4. Boundedness and the Relative Topologies

39

Proof. Part (a) follows directly from Propositions 4.11(a) and 4.13(a). The ﬁrst part of (b) follows from the ﬁrst parts of 4.11(c) and 4.13(b). The same sources yield the second part of (b) as well, since the relations (a + b)B = (a)B ∩ (b)B and B(a + b) ⊃ B(a) + B(b) ⊃ B(a) ∩ B(b) imply that B s (a + b) = B(a + b) ∩ (a + b)B ⊃ B(a) ∩ B(b) ∩ (a)B ∩ (b)B = B s (a) ∩ B s (b). Finally, by Propositions 4.11(d) and 4.13(c), the sets B(a) and (a)B are closed in the lower and upper relative topologies of P, respectively. Consequently, both of these sets as well as their intersection, that is Bs (a), are also closed in the symmetric relative topology of P, which is ﬁner than both the upper and the lower relative topologies. Proposition 4.16. The symmetric boundedness components satisfy a version of the cancellation law, that is a + c b + c for elements a, b and c of the same boundedness component implies that a b. Proof. Suppose that the elements a, b, c ∈ P are bounded relative to each other and that a + c b + c. Given v ∈ V there is λ ≥ 0 such that 0 ≤ c + λv. Thus a + (c + λv) b + (c + λv). As we observed before, (P, V) endowed with the weak preorder forms again a locally convex cone. Following Lemma I.4.2 in [100], if applied to this order and the positive element (a + λv) of a full cone containing P, the above implies a b + ε(c + λv) for all ε > 0. By our assumption, there are α, β ≥ 0 such that c ≤ αb + βv. Now combining the above yields a b + ε αb + (β + λ)v = (1 + εα)b + ε(β + λ)v for all ε > 0. This shows a v b by our deﬁnition of the weak local preorder in Section 3. Finally, because a v b holds for all v ∈ V, we infer that a b. Proposition 4.17. The symmetric boundedness components furnish a partition of P into disjoint convex subsets that are closed and connected in the symmetric relative topology. Proof. Proposition 4.15(a) implies that the symmetric boundedness components are convex subset of P. They are closed in the symmetric relative topology by 4.15(c). Moreover, the equivalence of (i) and (v) in Proposition 4.14 shows that any two symmetric boundedness components of P either coincide or are disjoint. For connectedness, let a ∈ P, and let b, c ∈ B s (a). Then B s (a) = B s (b) = B s (b) by Proposition 4.14(v), and by the equivalent condition in 4.14(vii), the mapping f : [0, 1] → B s (a) such that f (α) = αb + (1 − α)c is continuous with respect to the symmetric relative topology of P. As f (0) = c and f (1) = b, this shows that B s (a) is pathwise connected,

40

I Locally Convex Cones

hence connected in the symmetric relative topology of P (see Theorem 27.2 in [198]). We shall also consider the local boundedness components of a locally convex cone P that arise if we endow P with the neighborhood subsystem Vv = {αv | α > 0} consisting of the multiples of a single neighborhood v ∈ V. For an element a ∈ P and a neighborhood v ∈ V, we deﬁne the (local) upper, lower and symmetric v-boundedness components of a as vε (a), (a)Bv = (a)vε , and Bvs (a) = Bv (a) ∩ (a)Bv , Bv (a) = ε>0

ε>0

respectively. The elements of Bv (a) are called v-bounded above relative to a. Bv (0) = Bv consists of all v-bounded elements of P. The global boundedness components may be recovered as Bv (a), (a)B = (a)Bv and B s (a) = Bvs (a), B(a) = v∈V

v∈V

v∈V

respectively. Obviously, the statements of Propositions 4.10 to 4.17 apply also to the local boundedness components, since we may replace the given neighborhood system V by the subsystem Vv and consider the locally convex cone (P, Vv ) for this purpose. The cancellation law in Proposition 4.16 holds with the weak local preorder v in this case. The dual cone P ∗ of (P, Vv ) consists only of the multiples of the functionals in v ◦ , and the relative topologies of P are the relative v-topologies. The main beneﬁt in considering the local boundedness components as compared to the global ones, is the following: We shall proceed to verify that the disjoint partition of P into symmetric local boundedness components provides indeed a topological partition as well. Proposition 4.18. Let a ∈ P and v ∈ V. (a) Bv (a) is open in P with respect to the upper, closed with respect to the lower and both open and closed with respect to the symmetric relative v-topology of P. (b) (a)Bv is closed in P with respect to the upper, open with respect to the lower and both open and closed with respect to the symmetric relative v-topology of P. Proof. Let a ∈ P and v ∈ V Proposition 4.11(d) states that Bv (a) is closed in the lower relative v-topology of P. Let b ∈ Bv (a), that is b ≤ αb + βv for some α, β ≥ 0, and let vε (b) be a lower neighborhood of b. Then for c ∈ vε (b) we have c ≤ γb + εv with some 1 ≤ γ ≤ 1 + ε, and therefore c ≤ (αγ)a + (βγ + ε)v. This shows c ∈ Bv (a), hence vε (b) ⊂ Bv (a), and Bv (a) is seen to be open in the lower relative v-topology of P. Moreover, because the symmetric relative v-topology is the common reﬁnement of the

4. Boundedness and the Relative Topologies

41

upper and lower topologies, Bv (a) is indeed both open and closed in this topology. This completes Part (a). The argument for Part (b) is similar: Proposition 4.13(c) states that (a)Bv is closed in the upper relative v-topology of P. Let b ∈ (a)Bv , that is αa ≤ b + βv for some α, β > 0, and let (b)vε be a lower neighborhood of b. Then for c ∈ (b)vε we have b ≤ γc + εv with some 1 ≤ γ ≤ 1 + ε, and therefore αa ≤ γc + (ε + δ)v, hence (α/γ)a ≤ c + (ε + δ)/γv. This shows c ∈ (a)Bv , hence (b)vε ⊂ (a)Bv , and (a)Bv is seen to be open in the lower relative v-topology of P. Hence (a)Bv is both open and closed in the symmetric relative v-topology. Propositions 4.18 and 4.17 now yield a topological and algebraic partition of a locally convex cone into local boundedness components. Proposition 4.19. For every neighborhood v ∈ V, the symmetric v-boundedness components furnish a partition of P into disjoint convex subsets that are open, closed and connected in the symmetric relative v-topology. A subset of P that is open or closed in any of the relative v-topologies is of course also open or closed in the corresponding (global) relative topology of P. The same statement does however not hold for connectedness. 4.20 Connectedness. Topological vector spaces are connected and all of their elements are bounded. This does not hold for locally convex cones in general. However, Propositions 4.17 and 4.19 suggest relations between the boundedness and the connectedness components of a locally convex cone. Let us recall some of the relevant concepts from topology: The quasi-component of a point x in a topological space X is the intersection of all closed and open subsets of X which contain x. The quasi-components constitute a decomposition of X into pairwise disjoint and closed subsets (see VIII.26 in [198] or VI.1 in [59]). The component of a point x ∈ X, on the other hand is the largest connected subset of X which contains the point x. The components are subsets of the quasi-components and constitute a decomposition of X into pairwise disjoint, connected and closed subsets. A topological space is locally connected, if each of its points has a basis of connected neighborhoods. In locally connected spaces the quasi-components and components coincide and are both open and closed (see Corollary 27.10 in [198]). Proposition 4.21. Let (P, V) be a locally convex cone. (a) In the symmetric relative topology of P the components, quasicomponents and the symmetric boundedness components coincide. (b) For every neighborhood v ∈ V and the symmetric relative v-topology, P is locally connected and the components, quasi-components and the symmetric v-boundedness components coincide. Proof. (a) For an element a ∈ P Proposition 4.17 implies that B s (a) is contained in its (connectedness) component. On the other hand, B s (a) is

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I Locally Convex Cones

the intersection of the sets Bvs (a) for all v ∈ V, all of which are open and closed in the respective symmetric relative v-topologies, hence in the symmetric relative topology of P by Proposition 4.19. This shows that the quasi-component of a is contained in B s (a). Hence these three components coincide. For Part (b) let v ∈ V and a ∈ P. The v-boundedness component Bvs (a) of a contains all the neighborhoods vεs (a) for ε > 0. Convexity then guarantees (see the corresponding argument in the proof of Proposition 4.17) that these neighborhoods are pathwise connected in the symmetric relative vtopology, hence P is locally connected. The components, quasi-components and the symmetric v-boundedness components of P coincide by Part (a) if we endow P with the neighborhood subsystem Vv = {αv | α > 0} . Proposition 4.22. A locally convex cone (P, V) is locally connected in its symmetric relative topology if and only if every point a ∈ P has a basis of symmetric relative neighborhoods that are contained in Bs (a). Proof. Let a ∈ P. The argument in the proof of Proposition 4.17 shows that every convex subset of B s (a) is pathwise connected, hence connected in the symmetric relative topology. On the other hand, every connected subset of P containing the element a is a subset of B s (a), the component of a by 4.21(a). Because the symmetric relative neighborhoods of a are convex, our claim follows. 4.23 Locally Convex Cones with Uniform Boundedness Components. We shall say that a locally convex cone (P, V) has uniform boundedness components if the boundedness components of P for all neighborhoods coincide, that is if Bvs (a) = B s (a) for all v ∈ V and a ∈ P. Locally convex topological vector spaces are obviously of this type as all their elements are bounded with respect to every neighborhood. Also, any locally convex cone whose neighborhood system consists of the multiples of a single neighborhood, has uniform boundedness components. Proposition 4.22 yields that a locally convex cone with uniform boundedness components is locally connected. Its global boundedness components are both open and closed in the each of the symmetric relative v-topologies (Proposition 4.19), hence also in the (global) symmetric relative topology. Similar and related notions of boundedness components in locally convex cones had previously been established in [170] and [176]. 4.24 Bounded Subsets. We shall also use notions of boundedness for subsets corresponding to those for elements of a locally convex cone (P, V). A subset A of P is called (i) bounded below if for every v ∈ V there is λ ≥ 0 such that 0 ≤ a + λv for all a ∈ A; (ii) bounded above if for every v ∈ V there is λ ≥ 0 such that a ≤ λv for all a ∈ A;

4. Boundedness and the Relative Topologies

43

(iii) bounded if it is both bounded below and above; (iv) bounded above relative to b ∈ P if for every v ∈ V there are λ, ρ ≥ 0 such that a ≤ ρb + λv for all a ∈ A; (v) relatively bounded above if it is bounded above relative to some element of P; and (vi) relatively bounded if it is both bounded below and relatively bounded above, that is if there is b ∈ P such that for every v ∈ V there are λ, ρ ≥ 0 such that 0 ≤ a + λv and a ≤ ρb + λv for all a ∈ A. All these notions do of course coincide in a locally convex topological vector space. Similar concepts may be used to deﬁne local boundedness, that is boundedness relative to a speciﬁc neighborhood v ∈ V, for subsets of P. Note that a continuous linear operator T : P → Q, where both (P, V) and (Q, W) are locally convex cones, maps bounded subsets of one of the above types in P into bounded subsets of the same type in Q. A Uniform-Boundedness-type theorem from [172] allows relative boundedness for subsets of a locally convex cone P to be characterized in terms of its dual cone P ∗ . Proposition 4.25. Let A be a subset of a locally convex cone (P, V), and let b ∈ P. If for every linear functional μ ∈ P ∗ such that μ(b) < +∞ the set μ(A) is bounded in R, then A is bounded above relative to b. Proof. Let A be a subset of P which is not bounded above relative to the element b ∈ P. Then there is v ∈ V such that the condition in 4.24(iv) does not hold for this neighborhood. We deﬁne a monotone sublinear functional p : P → R by p(a) = inf{λ + ρ | λ, ρ ≥ 0, a ≤ ρb + λv} and observe that: (i) Let c ≤ d + v for c, d ∈ P. Then d ≤ ρb + λv for λ, ρ ≥ 0 implies that c ≤ ρb + (λ + 1)v. Thus p(c) ≤ p(d) + 1, and the functional p is seen to be continuous with respect to v in the sense of Theorem 3.4 in [172]; (ii) p is unbounded on A. Assume to the contrary that there is M > 0 such that p(a) < M for all a ∈ A. Let σ ≥ 0 such that 0 ≤ b + σv. Then for every a ∈ A there are λ, ρ ≥ 0 such that a ≤ ρb + λv and λ + ρ ≤ M. Then a ≤ (ρb + λv) + (M − ρ)(b + σv) ≤ M b + M (1 + σ)v, contradiction our assumption that A ⊂ P is not bounded above relative to b. Now Theorem 3.4 from [172] yields the existence of a continuous linear functional μ ∈ v ◦ such that μ(c) ≤ p(c) for all c ∈ P, that is μ(b) ≤ 1 in particular, and such that μ is unbounded on the set A. Similar notions of boundedness will be used for nets in a locally convex cone, that is a net (ai )i∈I in P will be called bounded (below, above, relative

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I Locally Convex Cones

to an element,...) if the corresponding requirements 4.24(i) to (vi) hold for the set {ai | i ≥ i0 } for some i0 ∈ I. 4.26 Closed Convex Sets. We shall proceed making some observations regarding subsets of a locally convex cone which are closed either with respect to the lower or the upper relative topology. Lemma 4.27. Let (P, V) be a locally convex cone. Every subset of P that is closed with respect to the lower (or the upper) relative topology is decreasing (or increasing) with respect to the weak preorder. Proof. Indeed, suppose that A ⊂ P is closed with respect to the lower relative topology and let b a for some b ∈ P and a ∈ A. Then b ∈ vε (a), thus a ∈ (b)vε and (b)vε ∩ A = ∅ for all v ∈ V and ε > 0. Thus b is in the closure of A with respect to the lower relative topology which coincides with A. Similarly one argues for a subset of P which is closed with respect to the upper relative topology. For a subset A of a locally convex cone (P, V) we denote by A(l) and A its closure with respect to the lower and the upper relative topology of P, respectively. (u)

Proposition 4.28. Let A be a subset of a locally convex cone (P, V). (a) The and (b) The (c) The

set A(l) consists of all elements b ∈ P such that for every v ∈ V ε > 0 there is some a ∈ A such that b ∈ vε (a). set A(l) is convex whenever A is convex. set A(l) is bounded above whenever A is bounded above.

Proof. (a) We have b ∈ A(l) if and only if (b)vε ∩ A = ∅ for all v ∈ V and ε > 0, that is if there is a ∈ A such that b ∈ vε (a). For Part (b) suppose that A is convex and let b, b ∈ A(l) and b = αb + (1 − α)b for some 0 ≤ α ≤ 1. Given v ∈ V and ε > 0, by Part (i) there are a, a ∈ A such that b ∈ vε (a) and b ∈ vε (a ). Then b ≤ γa + εv and b ≤ γ a + εv for some 1 ≤ γ, γ ≤ 1 + ε. Set γ = (αγ + (1 − α)γ ). Then a =

αγ (1 − α)γ a + a ∈A γ γ

and b ≤ γ a + (1 + ε)v. Since 1 ≤ γ ≤ 1 + ε, this demonstrates that b ∈ A(l), and therefore this set is also convex. For Part (c) suppose that A is bounded above in the sense of 4.24(ii). Let v ∈ V and suppose that there is λ ≥ 0 such that a ≤ λv for all a ∈ A. Then for every b ∈ A(l) there is a ∈ A such that b ∈ v1 (a). This means b ≤ γa + v for some 1 ≤ γ ≤ 2, hence b ≤ (γλ + 1)v ≤ (2λ + 1)v. Our claim follows. In a similar way one proves corresponding statements for the closure of a set with respect to the upper relative topology.

4. Boundedness and the Relative Topologies

45

Proposition 4.29. Let A be a subset of a locally convex cone (P, V). (a) The and (b) The (c) The

set A(u) consists of all elements b ∈ P such that for every v ∈ V ε > 0 there is some a ∈ A such that a ∈ vε (b). set A(u) is convex whenever A is convex. set A(u) bounded below whenever A is bounded below.

Proposition 4.28(a) implies in particular that for a singleton set {a} we have b ∈ {a}(l) if and only if b ∈ vε (a) for all v ∈ V and ε > 0, that is b a. Thus {a}(l) = {b ∈ P | b a}. Likewise, Proposition 4.29(a) yields {a}(u) = {b ∈ P | a b}. Theorem 4.30. Let A be a convex subset of a locally convex cone (P, V) and let b ∈ P. Then (a) b ∈ A(l) if and only if μ(b) ≤ sup{μ(a) | a ∈ A} for all μ ∈ P ∗ . (b) b ∈ A(u) if and only if μ(b) ≥ inf{μ(a) | a ∈ A} for all μ ∈ P ∗ . Proof. Let A ⊂ P be convex and let b ∈ P. We may assume that A = ∅, because for A = ∅ our claim is trivial. (As usual, we set inf ∅ = +∞ and sup ∅ = −∞ and use the fact that for every a ∈ P there is some μ ∈ P ∗ such that μ(a) < +∞.) For Part (a), let b ∈ A(l) and let μ ∈ P ∗ , that is μ ∈ v ◦ for some v ∈ V. Given ε > 0, according to Proposition 4.28(a) there is a ∈ A and 1 ≤ γ ≤ 1+ε such that b ≤ γa+εv, hence μ(b) ≤ γμ(a)+ε ≤ γ sup{μ(a) | a ∈ A} + ε. This shows μ(a) ≤ sup{μ(a) | a ∈ A}. The proof of the converse implication will however require some advanced Hahn-Banach type arguments that had been established in the [172] and quoted earlier in Section 2: For a ﬁxed number β ∈ R consider the sublinear functional p on P deﬁned for x ∈ P as p(x) = inf{λβ | x = λa for some

a ∈ A and

λ ≥ 0},

together with the extended superlinear functional q(0) = 0 and q(x) = −∞ for x = 0. Following Theorem 2.4 (a quote of Theorem 3.1 in [172]) there is a linear functional μ ∈ P ∗ such that q ≤ μ ≤ p if and only if we can ﬁnd a neighborhood v ∈ V such that q(x) ≤ p(y) + 1 whenever x ≤ y + v for x, y ∈ P; that is in our particular case 0 ≤ λβ + 1 whenever 0 ≤ λa + v for some a ∈ A and λ ≥ 0. For this we shall have to distinguish two cases: (i) If for every v ∈ V there is a ∈ A such that 0 ≤ a + v, then we have to require that β ≥ 0. (ii) If there is v ∈ V such that 0 ≤ a + v for all a ∈ A, then for ε > 0 the condition 0 ≤ λa + εv can hold only for λ < ε .

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I Locally Convex Cones

Thus we can choose any β = −(1/ε) for the neighborhood εv ∈ V. In other words, in case (ii) for every β ∈ R we can ﬁnd a neighborhood in V such that the above condition is satisﬁed. Now we shall use Theorem 2.10 (a quote of Theorem 4.23 in [172]), which describes the range of all linear functionals μ ∈ P ∗ such that q ≤ μ ≤ p on a ﬁxed element b ∈ P. It states that if there is at least one such linear functional μ, then sup μ(b) = sup inf {p(x) − q(y) | x, y ∈ P, q(y) ∈ R, b + y ≤ x + v}.

μ∈P ∗ q≤μ≤p

v∈V

With the particular insertions for p and q from above we need to consider only the choice of y = 0 and obtain sup μ(b) = sup inf {λβ | λ ≥ 0, b ≤ λa + v

μ∈P ∗ q≤μ≤p

for some

v∈V

a ∈ A}.

Now let us assume that μ(b) ≤ sup{μ(a) | a ∈ A} holds for all μ ∈ P ∗ . As q ≤ μ ≤ p implies that sup{μ(a) | a ∈ A} ≤ β, this yields sup inf {λβ | λ ≥ 0, b ≤ λa + v v∈V

for some

a ∈ A} ≤ β

for all admissible values of β. We shall use 4.28(a) to derive b ∈ A(l) from this. Let v ∈ V and ε > 0. We choose β = 1 in the above and observe that there is a ∈ A and λ ≥ 0 such that ε b ≤ λa + v 2

and

λ ≤ 1 + ε.

If 1 ≤ λ, this satisﬁes the criterion in 4.28(a). Otherwise we proceed distinguishing the above cases: In case (i) there is a ∈ A such that 0 ≤ a +(ε/2)v. Thus ε ε b ≤ λa + v + (1 − λ) a + v ≤ a + εv 2 2 with a = λa + (1 − λ)a ∈ A, satisfying the requirement from 4.28(a). In case (ii) we may use the above inequality for β = −1 as well. There is ρ > 0 such that 0 ≤ b + ρv. Set δ = min{1/2, ε/(4ρ + 2ε)}. We ﬁnd a ∈ A and λ ≥ 0 such that ε b ≤ λ a + v 2

and

− λ ≤ −1 + δ,

that is λ ≥ 1 − δ. Next we choose 0 ≤ α ≤ 1 such that 1 − δ ≤ λ ≤ 1 holds for λ = αλ + (1 − α)λ . (Recall that we are considering the case that λ < 1, therefore such a choice of α is possible.) Then ε ε ε b ≤ α λa + v + (1 − α) λ a + v ≤ λ a + v 2 2 2

4. Boundedness and the Relative Topologies

47

with

αλ (1 − α)λ a+ a ∈ A. λ λ From 0 ≤ b + ρv we infer that 0 ≤ λ a + (ρ + ε/2)v. Our assumption that δ ≤ 1/2 guarantees 1/2 ≤ λ ≤ 1 and 1 − λ ≤ δ. Using this we infer a =

(1 − λ )(2ρ + ε) v 2λ ≤ (1 − λ )a + δ(2ρ + ε)v ε ≤ (1 − λ )a + v 2

0 ≤ (1 − λ )a +

since δ ≤ ε/(4ρ + 2ε). Now combining the above yields ε ε b ≤ λ a + v + (1 − λ )a + v ≤ a + εv, 2 2 again satisfying the requirement from 4.28(a). We conclude that b ∈ A(l), as claimed. The argument for Part (b) of the Theorem follows similar lines, but is sufﬁciently diﬀerent from the preceding one to be presented here too: If b ∈ A(u)

and if μ ∈ P ∗ , then a similar argument than before using Proposition 4.29(a) yields μ(a) ≥ inf{μ(a) | a ∈ A}. For the converse implication we will again employ Theorem 2.10. For ﬁxed numbers 0 ≤ α ∈ R and β ∈ R consider the sublinear functional p on P deﬁned for x ∈ P as p(x) = ρα if x = ρb and p(x) = +∞ else, together with the extended superlinear functional q(x) = sup{λβ | x = λa for some

a ∈ A and λ ≥ 0}.

There is μ ∈ P ∗ such that q ≤ μ ≤ p if and only if there is v ∈ V such that q(x) ≤ p(y) + 1 whenever x ≤ y + v for x, y ∈ P; that is in our particular case λβ ≤ ρα + 1 whenever λa ≤ ρb + v for some a ∈ A and λ, ρ ≥ 0. For this we shall again have to distinguish two cases: (i) If for every v ∈ V and ε > 0 there is a ∈ A and 0 ≤ δ ≤ ε such that a ≤ δb + v, then we have to require that β ≤ 0. (ii) If there are v ∈ V and ε > 0 such that a ≤ δb + v for all a ∈ A and 0 ≤ δ ≤ ε, then the above condition holds for this neighborhood v with any β ∈ R, provided that α ≥ 1/ε. Indeed, assume that λa ≤ ρb + v for some a ∈ A and λ, ρ ≥ 0, but λβ > ρα + 1. Then a≤

1 ρ b + v ≤ ρb + v. λ λ

This shows ρ/λ > ε, hence ρ > ελ > ερα + ε ≥ ρ + ε,

48

I Locally Convex Cones

a contradiction. In order to apply Theorem 2.10 we need to guarantee that μ(b) < +∞ for at least one μ ∈ P ∗ satisfying q ≤ μ ≤ p. Our insertions for p and q imply that β ≤ inf{μ(a) | a ∈ a} and μ(b) ≤ α. We shall use α = +∞ in 2.10, but the preceding discussion involving diﬀerent choices for α demonstrates that there is μ ∈ P ∗ such that q ≤ μ ≤ p and μ(b) < +∞ in case (i), for any choice of β ≤ 0 and in case (ii) for any choice of β ∈ R. Thus we may use Theorem 2.10 for inf μ(b) = inf sup {q(x) − p(y) | x, y ∈ P, p(y) ∈ R, x ≤ b + y + v},

μ∈P ∗ q≤μ≤p

v∈V

With the particular insertions for p and q from above we need to consider only the choice of y = 0 and obtain inf μ(b) = inf sup {λβ | λ ≥ 0, λa ≤ b + v

μ∈P ∗ q≤μ≤p

v∈V

for some

a ∈ A}.

Now let us assume that μ(b) ≥ inf{μ(a) | a ∈ A} holds for all μ ∈ P ∗ . This yields inf sup {λβ | λ ≥ 0, λa ≤ b + v

v∈V

for some

a ∈ A} ≥ β

for all admissible values of β. We shall use 4.29(a) to derive b ∈ A(u) from this. Let v ∈ V and ε > 0. There is ρ > 0 such that 0 ≤ b + ρv. We choose β = −1 in the above and observe that there is a ∈ A and λ ≥ 0 such that ε λa ≤ b + v 2

and

− λ ≥ −1 −

ε 2ρ

that is

λ≤1+

ε . 2ρ

If 1 ≤ λ + 1 + (ε/2ρ), we proceed as follows: ε λa ≤ b + v + (λ − 1)(b + ρv) ≤ λb + εv, 2 since

ε ε ε + (λ − 1)ρ ≤ + ρ = ε. 2 2 2ρ

Thus

ε v ≤ b + εv, λ demonstrating that a ∈ vε (b) as required in 4.29(b). Otherwise, that is if λ < 1, we continue to distinguish the above cases: In case (i) we set δ = ε/(2 − 2λ) and according to this case can ﬁnd a ∈ A such that a ≤ δ b + δv for some 0 ≤ δ ≤ δ. Thus ε + (1 − λ)δ v. a = λa + (1 − λ)a ≤ (1 + (1 − λ)δ ) b + 2 a≤b+

Since a ∈ A, and since (1 − λ)δ = ε/2 and

4. Boundedness and the Relative Topologies

49

1 ≤ 1 + (1 − λ)δ ≤ 1 + (1 − λ)δ ≤ 1 + (ε/2), this shows a ∈ vε (b) as required in 4.29(b). In case (ii) we may use the above inequality for β = +1 as well. For σ = max{1/2, 1/(1 + ε)} < 1 we ﬁnd a ∈ A and λ ≥ 0 such that ε λ a ≤ b + v 2

and

λ ≥ σ.

We can choose 0 ≤ α ≤ 1 such that σ ≤ λ ≤ 1 holds for λ = αλ+(1−α)λ . (Recall that we are considering the case that λ < 1, therefore such a choice of α is possible.) With a = we have

αλ (1 − α)λ a + a ∈A λ λ λ a ≤ b +

ε v, 2

hence

1 ε b + + v. λ 2λ Because 1 ≤ 1/λ ≤ 1/σ ≤ 1 + ε, and because ε/(2λ ) ≤ ε/2σ ≤ ε we infer that a ∈ vε (b), again satisfying the requirement from 4.29(a). We conclude that b ∈ A(u), as claimed. a ≤

Theorem 4.30 is a generalization of Theorem 3.1 in [175] as the following corollary will show. Corollary 4.31. Let (P, V) be a locally convex cone. Then a b holds for a, b ∈ P if and only if μ(a) ≤ μ(b) for all μ ∈ P ∗ . Proof. Let a, b ∈ P. We have a b if and only if a ∈ {b}(l), and if and only if b ∈ {a}(u). By Theorem 4.30, Parts (a) and (b), each of these statements holds if and only if μ(a) ≤ μ(b) for all μ ∈ P ∗ . We proceed to deﬁne neighborhoods for subsets of a locally convex cone (P, V). For a subset A ⊂ P, a neighborhood v ∈ V we deﬁne upper and lower relative neighborhoods as subsets of P by for every ε > 0 there is a ∈ A and 1 ≤ γ ≤ 1 + ε v(A) = b ∈ P such that b ≤ γa + (1 + ε)v and A v=

for every ε > 0 there is a ∈ A and 1 ≤ γ ≤ 1 + ε . b∈P such that a ≤ γb + (1 + ε)v

50

I Locally Convex Cones

Note that this notation is consistent with the one earlier introduced for elements of P, as we have a b + v if and only if a ∈ v {b} (see Lemma 3.1) and if and only if b ∈ {a} v. Lemma 4.32. Let A be a subset of a locally convex cone (P, V) and let v ∈ V. (a) The upper neighborhood v A is closed in P with respect to the lower relative topology. (b) The lower neighborhood A v is closed in P with respect to the upper relative topology. (c) If A is convex, then both v A and (A)v are convex. Proof. Let A ⊂ P and let b ∈ v A (l). Given ε > 0, set δ = min{1, ε/4}. According to 4.28(a) there is c ∈ v A such that b ∈ vδ (c), that is b ≤ γc + δv with some 1 ≤ γ ≤ 1 + δ. Moreover, we have c ≤ γ a + (1 + δ)v for some a ∈ A and 1 ≤ γ ≤ 1 + δ. Thus b ≤ (γγ )a + γ(1 + δ) + δ v ≤ γγ a + (1 + δ)2 + δ v. Since both 1 ≤ γγ ≤ (1 + δ)2 ≤ 1 + ε and (1 + δ)2 + δ ≤ 1 + ε, we infer that b ∈ v A . Similarly one veriﬁes Part (b) of the Lemma. For Part (c) suppose that A is convex and let b, b ∈ v A and b = αb + (1 − α)b for some 0 ≤ α ≤ 1. Given v ∈ V and ε > 0 there are a, a ∈ A such that b ≤ γa + (1 + ε)v and b ≤ γ a + (1 + ε)v for some 1 ≤ γ, γ ≤ 1 + ε. Set γ = (αγ + (1 − α)γ ). Then a =

αγ (1 − α)γ a + a ∈A γ γ

and b ≤ γ a + (1 + ε)v. Since 1 ≤ γ ≤ 1 + ε, this demonstrates that b ∈ v A , and therefore this set is also convex. Similarly one argues for the lower neighborhood A v. Theorem 4.33. Let A be a convex subset of a locally convex cone (P, V), let v ∈ V and b ∈ P. Then ◦ (a) b ∈ v A if and only if μ(b) ≤ sup{μ(a) | a ∈ A} + 1 for all μ ∈ v◦ . (b) b ∈ A v if and only if μ(b) ≥ inf{μ(a) | a ∈ A} − 1 for all μ ∈ v . Proof. We may again assume that A = ∅. For Part (a), suppose that b ∈ v A and let μ ∈ v ◦ . Given ε ≥ 0 there is a ∈ A such that b ≤ γa+(1+ε)v, hence μ(b) ≤ γ μ(a) + (1 + ε) ≤ γ sup{μ(a) | a ∈ A} + (1 + ε) for some 1 ≤ γ ≤ 1+ε. This shows μ(b) ≤ sup{μ(a) | a ∈ A}+1 since ε > 0 was arbitrarily chosen. In order to prove the converse implication we consider V) containing both P and the neighborhood a full locally convex cone (P,

4. Boundedness and the Relative Topologies

51

The lower neighborhood vˆ A formed in P is system V. Then A ⊂ P. larger than v A formed in P, but we v A = vˆ A ∩ P. Thus if have b ∈ v A for b ∈ P we also have b ∈ vˆ A . Since vˆ A is a convex subset and closed with respect to the lower relative topology, according to of P ∗ such that μ(b) > sup{μ(c) | c ∈ v A }. Theorem 4.30 there is μ ∈ P This implies in particular that sup{μ(c) | c ∈ v A } is ﬁnite, and since a + v ∈ v A whenever a ∈ A we have μ(a + v) = μ(a) + μ(v) < +∞, hence μ(v) < +∞. If μ(v) = 0, then λμ ∈ v ◦ for all λ ≥ 0, and we may choose λ such that (λμ)(b) > sup{(λμ)(c) | c ∈ v A } + 1 ≥ sup{(λμ)(a) | a ∈ A} + 1. If μ(v) > 0, we set λ = 1/μ(v) and have again λμ ∈ v ◦ . Then for every a ∈ A we have a + v ∈ v A , hence (λμ)(b) > (λμ)(a + v) = (λμ)(a) + 1 and therefore (λμ)(b) > sup{(λμ)(a) | a ∈ A} + 1. ∗ to P is an element of v ◦ ⊂ Since the restriction of the functional λμ ∈ P ∗ P , this proves our claim for Part (a). The argument for Part (b) uses the easily veriﬁed fact that b ∈ A v implies that b + v ∈ A(u). Indeed, if b + v ∈ A(u), then by 4.29(b) for every ε > 0 there is a ∈ A such that a ≤ γ(b + v) + (ε/2)v with some 1 ≤ γ ≤ 1 + (ε/2). This shows a ≤ γb + γ + (ε/2) v ≤ a ≤ γb + (1 + ε)v, hence b ∈ A v. The remainder of the argument is similar to that in Part (a). Theorem 4.33 is a generalization of Theorem 3.2 in [175] as the following corollary will show. Corollary 4.34. Let (P, V) be a locally convex cone. Then a b + v holds for a, b ∈ P and v ∈ V if and only if μ(a) ≤ μ(b) + 1 for all μ ∈ v ◦ . Proof. Let a, b ∈ P and v ∈ V We have a b + v if and only if a ∈ v {b} and if and only if b ∈ {a} v. By Theorem 4.33, Parts (a) and (b), each of these statements holds if and only if μ(a) ≤ μ(b) + 1 for all μ ∈ P ∗ . Corollary 4.35. Let (P, V) be a locally convex cone. Let a, b ∈ P and v ∈ V such that the element a is v-bounded. Then a b + v holds if and only if μ(a) ≤ μ(b) + 1 for all extreme points μ of v ◦ .

52

I Locally Convex Cones

Proof. All left to show is the following: Let a, b ∈ P and v ∈ V such that a is v-bounded. Then μ(a) < +∞ for all μ ∈ v ◦ . Thus the function μ → μ(b) − μ(a) : v ◦ → R is aﬃne and continuous with respect to the topology w(P ∗ , P) (see Section 2). According to Lemmas II.4.4 and II.4.5 in [100] this function attains its minimum value at some extreme point of v ◦ . If a b + v, then according to Corollary 4.34 this minimum value is less than −1. Our claim follows. Remark 4.36. The following counterexample will demonstrate that the statement of Theorem 4.30 does in general not hold true for convex subsets A ⊂ P which are closed for the given lower or upper topologies rather than for the (coarser) upper or lower relative topologies. For this, let (P, V) be the locally convex cone of all continuous real-valued and bounded below functions on the interval [0, +∞), endowed functions v > 0 with the positive constant as its neighborhood system V. see Example 1.4(e) . Let the subset A ∈ P consist of all functions in g ∈ P with the following properties: (i) g(x) ≤ x for all x ∈ [0, +∞), and (ii) there is M ≥ 0 and α < 1 such that g(x) ≤ αx for all x ∈ [M, +∞). We claim that A is closed in the given lower topology. Indeed, let f ∈ P be in the closure of A. Then, given v > 0 there is g ∈ (f )v ∩ A, that is f ≤ g + v, hence f (x) ≤ g(x) + v ≤ x + v for all x ∈ [0, +∞). Thus f (x) ≤ x for all x ∈ [0, +∞), since v > 0 was arbitrarily chosen, hence (i) holds for f. For (ii) let g ∈ A such that f (x) ≤ g(x)+1 for all x ∈ [0, +∞), and let M ≥ 0 and α < 1 such that g(x) ≤ αx for all x ∈ [M, +∞). Choose N = max{M, 2/(1 − α)}. Then for all x ∈ [N, +∞) we have 2/(1 − α) ≤ x, hence 1 ≤ (1 − α)x/2 and f (x) ≤ g(x) + 1 ≤ αx +

(1 + α) (1 − α) x≤ x. 2 2

Since (1 + α)/2 < 1, this shows f ∈ A, conﬁrming that A is closed with respect to the lower topology. The set A ⊂ P is however not closed with respect to the coarser lower relative topology as the function f (x) = x is contained in A(l). Indeed, given v > 0 and ε > 0, set α = 1/(1 + ε) < 1. Then f (x) = x = (1 + ε)αx ≤ (1 + ε)αx + εv

for all

x ∈ [0, +∞).

This shows g ∈ (f )vε ∩ A = ∅, where g(x) = αx. We therefore have μ(f ) ≤ sup{μ(g) | g ∈ A} for all μ ∈ P ∗ , but f ∈ A. Examples 4.37. (a) Let P = R, endowed with the neighborhood system V = {ε ∈ R | ε > 0} see Example 1.4(a) . For the neighborhood v = 1 and ε > 0 the relative neighborhoods of an element a ∈ R are vε (a) = − ∞, (1 + ε)a + ε or vε (a) = − ∞, a + ε

4. Boundedness and the Relative Topologies

if a ≥ 0 or if a < 0, respectively. Thus (a)vε = a−ε or 1+ε , +∞

53

(a)vε = a − ε, +∞

if a ≥ ε or if a < ε, respectively. This yields vεs (a) = a−ε vεs (a) = a − ε, (1 + ε)a + ε , 1+ε , (1 + ε)a + ε , or

vεs (a) = a − ε, a + ε

if a ≥ ε, if 0 ≤ a < ε , or if a < 0, respectively. The upper, lower and symmetric relative topologies of R therefore coincide with the corresponding given topologies. see 1.4(a) . The symmetric relative topology, in particular, is the usual topology on R with +∞ as an isolated point. (b) Let P = R+ = {a ∈ R | a ≥ 0}, endowed with the neighborhood system V = {0} see Example 1.4(b) . For the only neighborhood v = 0 ∈ V and ε > 0 the relative neighborhoods of an element a ∈ R+ are a a vε (a) = 0, (1 + ε)a , (a)vε = 1+ε , +∞ and vεs (a) = 1+ε , (1 + ε)a . The symmetric relative topology therefore coincides with the Euclidean topology on (0, +∞), but renders 0 ∈ P and +∞ ∈ P as isolated points. Recall from Example 1.4(b) that the symmetric given topology on R+ , in contrast, is the discrete topology. For the boundedness components of R+ we have B(a) = [0, +∞),

(a)B = (0, +∞]

and

Bs (a) = (0, +∞)

(0)B = [0, +∞]

and

B s (0) = {0},

for a ∈ (0, +∞), B(0) = {0}, and B(+∞) = [0, +∞],

(+∞)B = {∞}

and

B s (+∞) = {∞}.

As claimed, the symmetric boundedness components furnish a partition of P = R+ into disjoint subsets that are both open and closed in the symmetric relative topology. (c) Let us consider Example 1.4(e) with the special insertions for P = R generated by a family Y of subsets of the and the neighborhood system V is spanned by the R-valued domain X as elaborated in 1.4(e). Recall that V functions vˆY ∈ V, corresponding to some Y ∈ Y, and such that vˆY (x) = 1 is the locally for x ∈ Y and vˆY (x) = +∞, else. Thus FVb (X, R), V convex cone of all bounded below (on the sets in Y) R-valued functions on X, carrying the topology of uniform convergence on the sets in Y. For a the vˆY -boundedness function f ∈ FVb (X, R) and a neighborhood vˆY ∈ V,

54

I Locally Convex Cones

component BvsˆY (f ) consists of all g ∈ FVb (X, R) such that αf (x) − β ≤ g(x) ≤ γf (x) + δ holds with some constants α, β, γ, δ > 0 for all x ∈ Y. Thus, obviously, (ˆ vY )sε (g) ⊂ BvsˆY (f ) for all ε > 0 whenever g ∈ BvsˆY (f ). This observation conﬁrms that the component BvsˆY (f ) is both open and closed in the symconvergence metric relative vˆY -topology, which is the topology of uniform

on Y. Yet the (global) boundedness component B s (f ) = Y ∈Y BvsˆY (f ) is in general only closed in the symmetric relative topology, which is the topology of uniform convergence on all sets Y ∈ Y. However, if the set X itself is contained in Y, then the multiples of the neighborhood vˆX form already a basis ˆ and the vˆX -boundedness components coincide with the global ones. for V, Following Proposition 4.22, FVb (X, R) is locally connected in this case. Its boundedness components therefore coincide with the components and quasicomponents in the symmetric relative topology (Proposition 4.21) and are both open and closed. If, for another special case, Y consists of all ﬁnite subsets of X, then for Y ∈ Y the above condition yields that two functions f, g ∈ FVb (X, R) are contained in the same vˆY -boundedness component if and only if they take the value +∞ at exactly the same points of Y. The symmetric relative vˆY -topology is the topology of pointwise convergence on the set Y in this case. Correspondingly, the global boundedness components consist of functions that take the value +∞ at exactly the same points of X, and the symmetric relative topology is the topology of pointwise convergence on X. If X itself is an inﬁnite set, then the global boundedness components are seen to be closed but not open in this topology. (d) Let (P, V) be a locally convex cone and let Q be the family of all non-empty convex subsets of P which are closed with respect to the lower relative topology. (See 4.26 to 4.35 before.) If we use the standard multiplication for sets by non-negative scalars and a slightly modiﬁed addition, that is for A, B ∈ Q, A ⊕ B = (A + B)(l) then Q becomes a cone. Indeed, since the set A + B is obviously again convex, so is its closure with respect to the lower relative topology by Proposition 4.28(b). The neutral element of Q is given by {0}(l) = {b ∈ P | b 0}. We use the set inclusion as the order on Q and deﬁne neighborhoods corresponding to those in P : We set A≤B⊕v if A ⊂ v B for A, B ∈ Q and v ∈ V, that is if for every a ∈ A and ε > 0 there is b ∈ B such that a ≤ γb + (1 + ε)v for some 1 ≤ γ ≤ 1 + ε. First we observe that for every A ∈ Q and a ﬁxed element a ∈ A v ∈ V there is λ ≥ 0 such that 0 ≤ a + λv. Since {0}(l) = {b ∈ P | b 0}, this yields {0}(l) ≤ A ⊕ (λ + 1)v. Indeed, for every b 0, we have b ≤ v, hence

4. Boundedness and the Relative Topologies

55

b ≤ a+(λ+1)v. Thus every element A ∈ Q is seen to be bounded below and (Q, V) satisﬁes the requirements for a locally convex cone. Next we observe that the weak preorder on (Q, V) coincides with the given order. Indeed, suppose that A B, and let a ∈ A. Given v ∈ V and ε > 0 we set δ = min{ε/3, 1} and have A ≤ γB ⊕ δv for some 1 ≤ γ ≤ 1 + δ. According to Lemma 3.1 there is 1 ≤ γ ≤ 1 + δ such that a ≤ (γ γ)b + (1 + δ)δv for some b ∈ B. Since (1 + δ)δ ≤ ε, this yields a ≤ (γ γ)b + εv, and since 1 ≤ γγ ≤ (1 + δ)2 ≤ 1 + ε, we have a ∈ vε (b) and infer from (i) that a ∈ B(l) = B, hence A ≤ B. Therefore A B holds if and only if A ≤ B. A similar argument shows that A B ⊕ v holds for A, B ∈ Q and v ∈ V if and only if A ≤ B ⊕ v. An element A ∈ Q is bounded above in Q if for every v ∈ V there is λ ≥ 0 such that A ≤ λv, that is a ≤ (λ + 1)v holds for all a ∈ A, that is if the set A ⊂ P is bounded above in P in the sense of 4.25(ii). (e) Similarly, but less intuitively we may consider the family Q of all convex subsets of a locally convex cone P which are closed with respect to the upper relative topology and bounded below in the sense of 4.25(i). (See 4.26 to 4.35 before.) We use the standard multiplication for sets by non-negative scalars and the addition A ⊕ B = (A + B)(u)

for A, B ∈ Q.

Since the sum of two bounded below convex subsets of P is obviously again bounded below and convex, Proposition 4.29(b) and (c) guarantees that the set A ⊕ B is indeed also bounded below and convex. Thus Q is a cone with the neutral element {0}(u) = {b ∈ P | 0 b}. In this example we use the inverse set inclusion as the order on Q, that is A≤B

if B ⊂ A

and deﬁne neighborhoods corresponding to those in P by A≤B⊕v if B ⊂ A v for A, B ∈ Q and v ∈ V, that is if for every b ∈ B, and ε > 0 there is a ∈ A such that a ≤ γb + (1 + ε)v for some 1 ≤ γ ≤ 1 + ε. Because for every A ∈ Q and v ∈ V there is λ > 0 such that 0 ≤ a + λv for all a ∈ A, we have {0}(u) ≤ A ⊕ λv, and every element A ∈ Q is bounded below. Hence (Q, V) is a locally convex cone. A similar argument than in (d) yields that (Q, V) carries its weak preorder. Note that other than in (d) the empty set is a member of Q, indeed its maximal element. We set A ⊕ ∅ = ∅, α · ∅ = ∅ and 0 · ∅ = {0}(u) for all A ∈ Q and α > 0. An element A ∈ Q is bounded above in Q if for every v ∈ V there is λ ≥ 0 such that A ≤ λv, that is there is a ∈ A such that a ≤ λv. Note that in both Examples (d) and (e) the given locally convex cone P may be considered as a subcone of Q via the embedding a → {a}. This is

56

I Locally Convex Cones

an embedding of (P, V) into (Q, V) in the sense of 2.2, provided that P is endowed with the weak preorder, that is {a} ≤ {b} + v holds if and only if a b + v for a, b ∈ P and v ∈ V see also 2.2(iii) . Remarks 4.38. (a) As a consequence of the last observation in 4.37(a) and of Proposition 4.5 we infer that a continuous linear functional μ on a locally convex cone (P, V) is still continuous if we endow P with its relative topologies. More precisely: Let μ ∈ v ◦ , that is the polar of some neighborhood v ∈ V. Then μ is a continuous linear operator from (P, V0 ) to R, where V0 consists of the multiples of the single neighborhood v. As shown in 4.37(a), the relative topologies on R coincide with the given ones as described in Example 1.4(a). Thus according to 4.5, the functional μ is also continuous if we endow P with either the upper, lower or symmetric relative v-topology and, correspondingly, R with its given upper, lower or symmetric topology. (b) We noted earlier that for a locally convex cone (P, V) the mapping (α, a) → αa : [0, +∞) × P → P, is generally not continuous with respect to any of the given topologies of R and P. However, if we endow P with either of the relative topologies, this mapping is continuous at all points (α, a) ∈ [0, +∞) × P such that either α > 0 or a ∈ P is bounded. This was established in Proposition 4.2(iii). Now using 4.37(b) we realize that this mapping is continuous at all points of [0, +∞) × P if we consider the symmetric relative topology of R+ see 4.37(b) and any of the relative topologies on P instead. Indeed, the symmetric relative topology of R+ coincides with the usual topology of R on (0, +∞), hence continuity at all points (α, a) ∈ [0, +∞) × P such that α > 0 follows from 4.2(iii). Continuity at the points (0, a) for all a ∈ P, on the other hand is obvious, since 0 is an isolated element in the symmetric relative topology of R+ .

5. Locally Convex Lattice Cones Our upcoming integration theory for cone-valued functions in Chapter II deals with locally convex cones that contain suprema and inﬁma for suﬃciently many of their subsets. Let us recall the classical concepts: A topological vector lattice is a vector lattice and a locally convex topological vector space E over R that possesses a neighborhood base of solid sets. (See for example Chapter V.7 in [185], also [132] or [184]. Recall that a subset A of E is called solid if b ∈ A whenever |b| ≤ |a| for b ∈ E and a ∈ A.) Some of the following deﬁnitions and results are adaptations of the corresponding classical ones. The presence of unbounded elements and the general unavailability of negatives in locally convex cones, however, requires a more delicate approach.

5. Locally Convex Lattice Cones

57

5.1 Locally Convex Lattice Cones. We shall say that a locally convex cone (P, V) is a locally convex ∨-semilattice cone if its order is antisymmetric and if for any two elements a, b ∈ P their supremum a ∨ b exists in P and if (∨1) (a + c) ∨ (b + c) = a ∨ b + c holds for all a, b, c ∈ P. (∨2) a ≤ c + v and b ≤ c + w for a, b, c ∈ P and v, w ∈ V implies that a ∨ b ≤ c + (v + w). Likewise, (P, V) is a locally convex ∧-semilattice cone if its order is antisymmetric and if for any two elements a, b ∈ P their inﬁmum a ∧ b exists in P and if (∧1) (a + c) ∧ (b + c) = a ∧ b + c holds for all a, b, c ∈ P. (∧2) c ≤ a + v and c ≤ b + w for a, b, c ∈ P and v, w ∈ V implies that c ≤ a ∧ b + (v + w). If both sets of the above conditions hold, then (P, V) is called a locally convex lattice cone. In case that (P, V) is indeed a locally convex topological vector space, the existence of suprema implies the existence of inﬁma and vice versa, as a ∧ b = − (−a) ∨ (−b) . Conditions (∨1) and (∨2) then are equivalent to (∧1) and (∧2) and consistent with the above mentioned deﬁnition of a topological vector lattice. Indeed, a ≤ c + v and b ≤ c + w means that a ≤ c + s b ≤ c + t in this case, for some elements s and t of the neighborhoods v and w, respectively. Because these neighborhoods are supposed to be solid, we have s ∨ 0 ≤ v and t ∨ 0 ≤ w as well. Now a ≤ c + s ∨ 0 + t ∨ 0 and b ≤ c + s ∨ 0 + t ∨ 0 implies a ∨ b ≤ c + s ∨ 0 + t ∨ 0 ≤ c + (v + w) as required in (∨1). Proposition 5.2. Let (P, V) be a locally convex ∨- (or ∧-) semilattice cone. The lattice operation (a, b) → a ∨ b (or (a, b) → a ∧ b) is a continuous mapping from P × P to P if P is endowed with the symmetric relative topology. Proof. Suppose that (P, V) is a locally convex ∨-semilattice cone, and let a ∈ vε (b) and c ∈ vε (d) for a, b, c, d ∈ P, v ∈ V and ε > 0. There is λ ≥ 0 such that both 0 ≤ b + λv and 0 ≤ d + λv. Then a ≤ (1 + ε)b + ε(1 + λ)v and c ≤ (1 + ε)d + ε(1 + λ)v by Lemma 4.1(b). Thus a ≤ (1 + ε)(b ∨ d) + ε(1 + λ)v

and

c ≤ (1 + ε)(b ∨ d) + ε(1 + λ)v,

hence a ∨ c ≤ (1 + ε)(b ∨ d) + 2ε(1 + λ)v by (∨2). This shows a ∨ c ∈ v(2ε(1+λ)) (b ∨ d). Similarly, using 4.1(c) one veriﬁes that a ∈ (b)vε and c ∈ (d)vε implies a ∨ c ∈ (b ∨ d)v(2ε(1+λ+ε)) (b ∨ d).

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Combining these observations for both the upper and lower relative neighborhoods then demonstrates that a ∈ vεs (b) and c ∈ vεs (d) implies a ∨ c ∈ s (b ∨ d), hence our claim. A similar argument yields our claim for v(2ε(1+λ+ε)) locally convex ∧-semilattice cones. Proposition 5.3. Let (P, V) be a locally convex lattice cone. Then a + b = a ∨ b + a ∧ b for all a, b ∈ P. Proof. We observe that a + b ≤ inf a + a ∨ b , b + a ∨ b = a ∧ b + a ∨ b by (∧1), and by (∨1) a ∨ b + a ∧ b = sup a + a ∧ b , b + a ∧ b ≤ a + b. As the order of P is supposed to by antisymmetric, this yields our claim. Proposition 5.3 implies in particular that a = a ∨ 0 + a ∧ 0 for all elements a of a locally convex lattice cone. Examples of locally convex lattice cones include classical topological vector lattices and the cones R and R+ from Examples 1.4(a) and (b). If (P, V) is a is a neighborlocally convex ∨- or ∧-semilattice cone, if X is a set, and if V hood system consisting of V ∪{∞})-valued functions on X, then the locally convex cone FVb (X, P), Vˆ of P-valued functions from Example 1.4(e) is also a semilattice cone of the same type. Suprema and inﬁma are formed pointwise in this case. The cones (lp , Vp ) from 1.4(f) are locally convex lattice cones. The locally convex cone of all non-empty convex subsets of some locally convex topological vector space E see Example 1.4(c) is antisymmetrically ordered by set inclusion (we assume that equality is the order in E) and indeed a ∨-semilattice cone. The supremum of two convex subsets of E is the convex hull of their union while inﬁma, that is intersections, do not always exist. Requirements (∨1) and (∨2) are readily checked. 5.4 Locally Convex Complete Lattice Cones. Later in this text, in particular when developing our integration theory, we shall consider substantially stronger properties concerning the lattice operations of a locally convex cone. We shall require the existence of suprema and inﬁma for bounded and bounded below subsets, respectively. This assumption corresponds to the notion of order completeness for ordered vector spaces. Moreover, the upper or lower neighborhoods are supposed to be closed for suprema or inﬁma of their subsets, respectively. This requirement corresponds to the properties of M-topologies in locally convex vector lattices. We shall say that a locally convex cone (P, V) is a locally convex semilattice cone if P carries the weak preorder (that is the given order coincides with the weak preorder for the elements and the neighborhoods in P), this order is antisymmetric and if

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59

1 Every non-empty subset A ⊂ P has a supremum sup A ∈ P and sup(A + b) = sup A + b holds for all b ∈ P. 2 Let ∅ = A ⊂ P, b ∈ P and v ∈ V. If a ≤ b + v for all a ∈ A, then sup A ≤ b + v. In particular, every -semilattice cone P contains a largest element, that is +∞ = sup P, which can be adjoined as a maximal element to any locally convex cone with the convention that a + ∞ = +∞, α · (+∞) = +∞, 0 · (+∞) = 0 and that a ≤ +∞ forall a ∈ P and α > 0 . Likewise, (P, V) is said to be a locally convex -semilattice cone if P carries the weak preorder, this order is antisymmetric and if 1 Every subset A ⊂ P that is bounded below has an inﬁmum inf A ∈ P and inf(A + b) = inf A + b holds for all b ∈ P. 2 Let A ⊂ P be bounded below, b ∈ P and v ∈ V. If b ≤ a + v for all a ∈ A, then b ≤ inf A + v. These requirements are obviously stronger then the corresponding ones in 4.23, so every locally convex - (or -) semilattice cone is also a ∨- (or ∧-) semilattice cone. The assumptions 2 and 2 signify that the upper or lower neighborhoods in P are closed for suprema or inﬁma of their subsets, respectively. If (P, V) is a full cone, then 2 is evident, and 2 follows 1 . Recall from our convention in 4.24(i) that a subset A of P is said from to be bounded below if for every v ∈ V there is λ ≥ 0 such that 0 ≤ a + λv for all a ∈ A. This condition does in general not imply the existence of a lower bound in P. However, if A has a lower bound b ∈ P, that is b ≤ a for all a ∈ A, then A is bounded below in the above sense. Indeed, for every v ∈ V there is λ ≥ 0 such that 0 ≤ b + λv, hence 0 ≤ a + λv holds for all a ∈ A. Note that the empty set ∅ ⊂ P is bounded below, and we have inf ∅ = +∞ (see the remark above). Combining both of the above notions, we shall say that a locally convex cone (P, V) is a locally convex complete lattice cone if P is both a -semilattice cone and a -semilattice cone. Corresponding to a family {Ai }i∈I of non-empty subsets of a locally con vex -semilattice cone P we denote the subset Ai = ai (ai )i∈I ∈ Ai , ⊂ P. i∈I

i∈I

i∈I

Lemma 5.5. Let (P, V) be a locally convex -semilattice cone. Let A, B and {Ai }i∈I be non-empty subsets of P. Then (a) sup(A + B)= sup A+ sup B. (b) sup Ai = sup Ai = sup sup Ai | i ∈ I . i∈I

i∈I

i∈I

Proof. For Part (b) we observe that for every a ∈ i∈I Ai there is (ai )i∈I ∈ i∈I Ai such that a is one of the projections of (ai )i∈I onto the factor

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spaces Ai . This yields a ≤

ai , hence Ai ≤ sup Ai . sup i∈I

i∈I

i∈I

For every (ai ) i∈I ∈ i∈I Ai on the other hand, we have ai ≤ sup Ai for all i ∈ I, hence i∈I ai ≤ supi∈I sup Ai | i ∈ I and sup Ai ≤ sup sup Ai | i ∈ I . i∈I

i∈I

Finally, since sup Ai ≤ sup

i∈I Ai

holds for all i ∈ I, we infer that

sup sup Ai | i ∈ I ≤ sup Ai . i∈I

i∈I

Our claim in Part (b) now follows from the requirement that the order in P is antisymmetric. For Part (a) we argue as follows: If A and B are non-empty subsets of P, then we use Part (b) and 1 for sup(A + B) = sup (A + b) b∈B

= sup sup(A + b) | b ∈ B b∈B = sup sup A + b | b ∈ B b∈B

= sup A + sup B. Similarly, for a family {Ai }i∈I of subsets of a locally convex cone such that i∈I Ai is bounded below in P we denote Ai = ai (ai )i∈I ∈ Ai , ⊂ P i∈I

i∈I

-semilattice

i∈I

and obtain in analogy to Lemma 5.5: Lemma 5.6. Let (P, V) be a locally convex -semilattice cone.Let A, B and {Ai }i∈I be bounded below subsets of P and suppose that i∈I Ai is also bounded below. Then (a) inf(A + B)= inf A+ inf B. (b) inf Ai = inf Ai = inf inf Ai | i ∈ I . i∈I

i∈I

i∈I

Remarks and Examples 5.7. (a) Every locally convex -semilattice cone P contains also suprema for all of its non-empty subsets. Indeed, the set of all

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upper bounds for a non-empty subset A of P is bounded below, and its inﬁmum is the supremum of A in P. Requirement 1 does however not necessarily follow (see Example (e) below). Likewise, every locally convex -semilattice cone has inﬁma for subsets with lower bounds in P. (Recall the before mentioned subtle diﬀerence between “bounded below” and “having a lower bound”.) But again, requirement 1 does not follow (see (d) below). (b) The locally convex cones R and R+ Examples 1.4(a) and (b) are of course complete lattices. (c) If (P, V) is a locally convex -semilattice (or -semilattice) lat tice cone, if X is a set, and system consisting of if V is a neighborhood V∪{∞} -valued functions see Example 1.4(e) , then the locally convex cone ˆ of P-valued functions from 1.4(e) is also a locally convex (X, P), V F V b -semilattice (or -semilattice) lattice cone, provided that for every x ∈ X such that vˆ(x) ≤ v. (Using Lemma 3.2, this conand v ∈ V there is vˆ ∈ V dition guarantees that FVb (X, P), Vˆ carries its weak preorder.) Suprema and inﬁma are formed pointwise. For P = R in particular, FVb (X, R), Vˆ is a locally convex complete lattice cone, provided that for each x ∈ X there such that vˆ(x) < +∞. is vˆ ∈ V (d) Example 4.37(d) yields a locally convex -semilattice cone. The cone (Q, V) of all non-empty closed (with respect to the lower relative topology) convex subsets of a locally convex cone (P, V) is ordered by set inclusion and carries the weak preorder which is antisymmetric see 4.37(d) . For a non-empty family A ⊂ Q its supremum is given by sup A = conv

(l) A ,

A∈A

where conv(C) denotes the convex hull of a set C ⊂ P, and the closure is meant with respect to the lower relative topology of P. Condition ( 1) can be readily checked: Let B ∈ P. Clearly A ⊕ B ⊂ sup A ⊕ B for all A ∈ A, hence sup{A ⊕ B | A ∈ A} ≤ sup A ⊕ B. For the converse inequality let c ∈ sup A ⊕ B = conv A∈A A + B (l) . Then for everylower relative neighborhood (c)vε there is d ∈ (c)vε ∩ conv A∈A A + B . This means n d = i=1 αi ai + b for some n ai ∈ Ai ∈ A, b ∈ B and 0 ≤ αi such that n α = 1. Thus d = i i=1 i=1 αi (ai + b) ∈ sup{A ⊕ B | A ∈ A}. This implies c ∈ sup{A ⊕ B | A ∈ A} as well, since this set is closed in the lower topology. Our claim follows. (e) A similar argument shows that Example 4.37(e) yields a locally convex -semilattice cone. In this case Q consists of all bounded below closed (with respect to the upper relative topology) convex subsets of P and is ordered by the inverse set inclusion. For a bounded below family A ⊂ Q its inﬁmum is given by (u) , A inf A = conv A∈A

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where the closure is meant with respect to the upper relative topology of P. It is easily checked that for such a bounded below family A ⊂ Q the convex hull of its union is again a bounded below subset of P, hence by Proposition 4.28 also the closure of the latter with respect to the upper relative topology. (f) Let X be a topological space, and let P be the cone of all Rvalued lower semicontinuous functions on X, where R is endowed with the usual, that is the one-point compactiﬁcation topology. P is endowed with the pointwise operations and order and neighborhoods v ∈ V for P are given by the strictly positive constant functions. Because the pointwise inﬁmum of any two functions as well as the pointwise supremum of any non-empty family of functions in P is again an R-valued and lower semicontinuous function, (P, V) forms a locally convex lattice as well as a -semilattice cone, however in general not a locally convex complete lattice cone. Similarly, the cone of all R-valued bounded below upper semicontinuous functions on X forms a locally convex lattice and -semilattice cone. 5.8 Zero Components. Throughout the following we shall assume that (P, V) is a locally convex -semilattice cone. We deﬁne the zero component of an element a of a locally convex -semilattice cone P by O(a) = inf b ≥ 0 | a ∈ B(b) . This expression is well deﬁned, and O(a) ≥ 0 for all a ∈ P. Recall from Proposition 4.10 that a ∈ B(b) if and only if for every v ∈ V there are α, β ≥ 0 such that a ≤ αb+βv. If (a)B does not contain a positive element, then O(a) = inf ∅ = +∞ ∈ P. The introduction of zero components is especially useful for the investiga tion of variations of the cancellation law in -semilattice cones. Proposition 5.9. Let (P, V) be a locally convex -semilattice cone, and let a, b, c ∈ P and v ∈ V. If a + c v b + c, then a v b + O(c). Proof. Let a, b, c ∈ P and v ∈ V and suppose that a + c v b + c. As we observed before, the weak local preorder v is compatible with the algebraic operations in P. Following Lemma I.4.1 in [100] if applied to this order, the above implies a + ρc b + ρc for all ρ > 0. There is λ > 0 such that both 0 ≤ b + λv and 0 ≤ c + λv. Thus 0 ≤ (b + ρc) + 2λv for all 0 < ρ ≤ 1. Next we recall that a + ρc v b + ρc means that a + ρc ∈ vε (b + ρc) for all ε > 0. Using Lemma 4.1(b) we infer that a + ρc ≤ (1 + ε)(b + ρc) + ε(1 + 2λ)v holds for all ε > 0 and 0 < ρ ≤ 1. Thus a ≤ a + ρ(c + λv) ≤ (1 + ε)(b + ρc) + (ε + 2ελ + ρλ)v. Let d ≥ 0 such that c ∈ B(d). Then c ≤ αd + βv holds for some α, β ≥ 0. Consequently, for all ρ > 0 such that ρ ≤ max {(ε/λ), (1/α), (2ε/β)} we

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63

have ρc ≤ (ρα)d + (ρβ)v ≤ d + 2εv since d ≥ 0, and

(ε + 2ελ + ρλ)v ≤ 2ε(1 + λ)v,

hence a ≤ (1 + ε)(b + ρc) + 2ε(1 + λ)v ≤ (1 + ε)(b + d) + 2ε(2 + λ)v. Now we may use rules 1 and 2 and take the inﬁmum over the righthand side of this inequality with respect to all d ≥ 0 such that c ∈ B(d). This yields a ≤ (1 + ε) b + O(c) + 2ε(2 + λ)v. This last inequality holds true for all ε > 0 and therefore demonstrates a v b + O(c). Proposition 5.10. Let (P, V) be a locally convex let a, b, c ∈ P.

-semilattice cone, and

(a) If a + c ≤ b + c, then a ≤ b + O(c). (b) If a ∈ B(b), then O(a) ≤ O(b) (c) If a is bounded, then O(a) = 0. Proof. Let a, b, c ∈ P. Recall that a b, that is a ≤ b in the case of a completely ordered locally convex cone which is supposed to carry its weak global preorder, means that a v b holds for all v ∈ V. This yields Part (a) as an immediate consequence of 5.9. For Part (b) suppose that a ∈ B(b). Then for every c ≥ 0 such that b ∈ B(c) we have B(b) ⊂ B(c) by 4.10(ii), hence a ∈ B(c) as well. This yields O(a) ≤ O(b). Part (c) follows from Part (b) with b = 0. Proposition 5.11. Let (P, V) be a locally convex -semilattice cone, and let a, b, ∈ P. Then (a) O(a + b) = O(a) + O(b). (b) O(αa) = αO(a) = O(a) for all α > 0. (c) If αa = a for all α > 0, then O(a) = a. Proof. Let a, b, ∈ P. Part (b) is obvious since for every α > 0 and every c ∈ P we have αa ∈ B(c) if and only if a ∈ B(c) by 4.11(a). For Part (a) let a ∈ B(c) and b ∈ B(d) for c, d ≥ 0. Then a + b ∈ B(c + d) by 4.11(c). This shows O(a + b) ≤ O(a) + O(b). For the converse, given v ∈ V there is λ ≥ 0 such that 0 ≤ b+λv. Hence a ≤ (a+b)+λv, and we infer that a ∈ B(a+b). Thus O(a) ≤ O(a + b) by 5.10(b), and likewise O(b) ≤ O(a + b). This yields O(a) + O(b) ≤ 2O(a + b) = O(a + b).

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For Part (c) let a ∈ P such that αa = a for all α ≥ 0. For every v ∈ V there is λ > 0 such that 0 ≤ a + λv, hence 0 ≤ (1/λ)a + v = a + v. This shows 0 ≤ a, since P carries the weak preorder. Thus O(a) ≤ a. If on the other hand a ∈ B(c) for some c ≥ 0, then there are α, β ≥ 0 such that a ≤ αc + βv. Since εαc ≤ c for all 0 < ε ≤ 1/(α + 1), this implies a = εa ≤ εαc + εβv ≤ c + εβv for all such ε. This yields a ≤ c since P carries the weak preorder, and we also have a ≤ O(a). Proposition 5.11(b) implies in particular that a linear functional μ ∈ P ∗ can attain only the values 0 or +∞ at a zero component. Some additional properties can be derived if P contains also suprema of its elements, that is if (P, V) is also a locally convex lattice or indeed a locally convex complete lattice cone see Example 5.7(f) . Lemma 5.12. Suppose (P, V) is a locally convex lattice and -semilattice cone. Then the zero component of an element a ∈ P can be alternatively expressed as O(a) = inf ε (a ∨ 0) . ε>0

Proof. Let a ∈ P. Then 0 ≤ a ∨ 0 and a ≤ a ∨ b. Thus a ∈ B(a ∨ b). This implies a ∈ B ε (a ∨ 0) for all ε > 0 by 4.11(c). Hence inf b≥0| a ∈ B(b) ≤ inf ε>0 ε (a ∨ 0) . For the converse inequality let b ≥ 0 such that a ∈ B(b). Given v ∈ V and ε > 0 there are α, β ≥ 0 such that a ≤ αb + βv see 4.10(iii) . Condition (∨2) then yields a ∨ 0 ≤ αb + 2βv. ε 1 , α+1 } we have Thus for 0 < δ ≤ min{ 2β+1 δ(a ∨ 0) ≤ δαb + 2δβv ≤ b + εv, since b ≥ 0 and δα ≤ 1 implies (δα)b ≤ b. This shows inf ε>0 ε (a ∨ 0) ≤ b + εv, hence inf ε (a ∨ 0) ≤ inf b ≥ 0 | a ∈ B(b) + εv ε>0

by 2 . Because this holds for all v ∈ V and for all ε > 0, and because P carries the weak preorder, we conclude that inf ε (a ∨ 0) ≤ inf b ≥ 0 | a ∈ B(b) . ε>0

Proposition 5.13. Suppose (P, V) is a locally convex lattice and semilattice cone. Let a, b, c ∈ P and v ∈ V. (a) If a ∈ Bv (b), then O(a) v O(b) and b + O(a) v b. (b) If a is v-bounded, then O(a) v 0.

-

5. Locally Convex Lattice Cones

65

Proof. Let a, b, c ∈ P and v ∈ V. For Part (a) , suppose that a ∈ Bv (b). There are α, β > 0 such that a≤ αb + βv and λ ≥ 0 such that 0 ≤ a + λv, hence also 0 ≤ a ∧ 0 + λv by 2 . Then b + O(a) ≤ b + ε (a ∨ 0) ≤ b + ε (a ∨ 0) + ε (a ∧ 0) + λv = b + εa + ελv ≤ (1 + εα)b + ε(β + λ)v. for all ε > 0 by Lemma 5.12. This shows b + O(a) v b. Furthermore, using the cancellation rule from Proposition 5.9 for the element b in O(a) + b v 0 + b yields O(a) v O(b) as claimed. Part (b) follows from Part (a) with b = 0. Proposition 5.14. Suppose (P, V) is a locally convex lattice and semilattice cone. Then b + O(a) = b holds for all a, b ∈ P whenever a ∈ B(b). Proof. Let a, b ∈ P such that a ∈ B(b). Then a ∈ Bv (b), hence b+O(a) v b by Proposition 5.13, for all v ∈ V. Thus b+O(a) b. Since P carries the weak preorder which is supposed to be antisymmetric, and since b b+O(a) is evident, our claim follows. Proposition 5.15. Let (P, V) be a locally convex complete lattice cone, and let A, B be non-empty subsets of P. Then (a) inf(A ∨ B) = inf A ∨ inf B if both A and B are bounded below. (b) sup(A ∧ B) ≤ sup A ∧ sup B ≤ sup(A ∧ B) + O (sup(A ∨ B)) . Proof. We ﬁrst observe that inf A ∨ inf B ≤ a ∨ b

and

a ∧ b ≤ sup A ∧ sup B

holds for all a ∈ A and b ∈ B. Thus inf A ∨ inf B ≤ inf(A ∨ B)

and

sup(A ∧ B) ≤ sup B ∧ sup A.

For Part (a) we assume that both sets A and B are bounded below and use Proposition 5.3 for inf(A ∨ B) + inf(A ∧ B) ≤ inf{a ∨ b + a ∧ b | a ∈ a, b ∈ B} = inf(A + B) = inf A + inf B = inf A ∨ inf B + inf A ∧ inf B. As inf(A ∧ B) = inf A ∧ inf B, the cancellation law in Proposition 5.10(a) yields inf(A ∨ B) ≤ inf A ∨ inf B + O inf(A ∧ B) .

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Similarly, one obtains sup A ∧ sup B ≤ sup(A ∧ B) + O (sup(A ∨ B)) , that is Part (b). Finally, as inf(A ∧ B) = inf A ∧ inf B ≤ inf A ∨ inf B, Proposition 5.14 shows inf A ∨ inf B + O inf A ∧ inf B = inf A ∨ inf B. This completes our proof of Part (a).

We proceed to reﬁne the cancellationrules in Proposition 5.10 further. Let (P, V) be a locally convex lattice and -semilattice cone. We deﬁne the zero component of an element a ∈ P relative to b ∈ P by O(a b) = inf c ≥ 0 | c + O(b) ≥ O(a) . Obviously, O(a 0) = O(a). Also, O(αa βb) = αO(a b) = O(a b) holds for all α, β > 0. Proposition 5.16. Let (P, V) be a locally convex lattice and -semilattice cone, and let a, b, c ∈ P. (a) 0 ≤ O(a b) ≤ O(a) ≤ O(a b) + O(b). (b) If a + c ≤ b + c, then a ≤ b + O(c b). (c) If a ∈ B(b), then O(a b) = 0 and b + O(c) = b + O(c a). Proof. Part (a) follows directly from the deﬁnition of O(a b) together with 2 . For (b) we recall that a + c ≤ b + c implies a ≤ b + O(c) by 5.10(a). As O(c) ≤ O(c b) + O(b) by Part (a) and b + O(b) = b by 5.14, our claim follows. For (c), let a ∈ B(b). Then O(a) ≤ O(b) by 5.10(b), and we may use c = 0 in the deﬁnition of O(a b). Thus indeed O(a b) = 0. Furthermore, we have b + O(c) ≤ b + O(c a) + O(a) = b + O(a) + O(c a) = b + O(c a) by Part (a) and 5.14.

Examples 5.17. (a) If P = R or P = R+ see 1.4(a) and 1.4(b) , then O(a) = 0 for all a< +∞, and O(+∞) = +∞. (b) Consider FVb (X, P), Vˆ , where (P, V) is a locally convex is a neighborhood system consisting semilattice cone, X a set, and V of V ∪ {∞} -valued functions see Example 1.4(e) such that for every such that vˆ(x) ≤ v see 5.7(c) . For x ∈ X and v ∈ V there is vˆ ∈ V f ∈ FVb (X, P) then O(f ) is the mapping x → O f (x) . For P = R, in particular, the zero component of an R-valued function f ∈ FVb (X, R) is the mapping O f (x) = 0 if f (x) < +∞, and O f (x) = +∞ else. The same

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67

observation applies to the second part of Example 5.7(f), that is the cone of R-valued bounded below upper semicontinuous functions on a topological space with the positive constants as neighborhoods. This was seen to be an example of a locally convex lattice and -semilattice cone. (c) Let us consider Example 4.37(e) see also 5.7(e) , that is the locally convex -semilattice cone (Q, V) of all convex subsets locally convex cone (P, V), which are bounded below and closed with respect to the upper relative topology. Recall that the order in Q is the inverse set inclusion and the neighborhoods are given by A ≤ B ⊕ v for A, B ∈ Q and v ∈ V, if for every b ∈ B, and ε > 0 there is a ∈ A such that a ≤ γb + (1 + ε)v for some 1 ≤ γ ≤ 1 + ε. The closed convex subsets (including the empty set) of {0}(u) = {b ∈ P | 0 b} are the positive elements in Q. We claim that for an element A ∈ Q we have

O(A) = {b 0 | Bv (b) ∩ A = ∅

for all

v ∈ V}.

We shall argue for this using the following steps: Let B denote the set on the right-hand side of the above equation. (i) The set B ⊂ P is convex. Indeed, let b1 , b2 ∈ B, 0 ≤ λ1 , λ2 ≤ 1 such that λ1 +λ2 = 1 and b = λ1 b1 +λ2 b2 . Given v ∈ V there are a1 ∈ Bv (b1 )∩A and a2 ∈ Bv (b2 ) ∩ A. Set a = λ1 a1 + λ2 a2 ∈ A and choose α1 , α2 , β, ρ ≥ 0 such that a1 ≤ α1 b1 + βv,

a2 ≤ α2 b2 + βv,

0 ≤ b1 + ρv

and

0 ≤ b2 + ρv.

Setting α = max{α1 , α2 } we have a1 ≤ α1 b1 + βv + (α − α1 )(b1 + ρv) + α1 ρv = αb1 + (β + αρ)v and, likewise a2 ≤ α2 b1 + βv + (α − α2 )(b2 + ρv) + α2 ρv = αb2 + (β + αρ)v. Thus a ≤ λ1 αb1 + (β + αρ)v + λ2 αb1 + (β + αρ)v = αb + (β + αρ)v. We infer that a ∈ Bv (b) ∩ A, hence Bv (b) ∩ A = ∅. Since this holds for all v ∈ V and since b 0 is evident from b1 , b2 0, we conclude that b ∈ B. (ii) The set B ⊂ P is closed with respect to the upper topology. Indeed, let c ∈ B(u) and let v ∈ V. There is b ∈ v1 (c)∩B, that is b ≤ γc+v for some 1 ≤ γ ≤ 2. There is a ∈ Bv (b) ∩ A, that is a ≤ αb + βv for some α, β ≥ 0. Combining these yields a ≤ αγc + (α + β)v. This shows Bv (c) ∩ A = ∅ for all v ∈ V. Furthermore, since B ⊂ {0}(u) = {b ∈ P | 0 b} which is closed with respect to the upper relative topology, we have c ∈ {0}(u) as well, hence c 0. Together with the above this yields c ∈ B. Since B ⊂ P is obviously

68

I Locally Convex Cones

bounded below (we have 0 ≤ b + v for all b ∈ P), we conclude from (i) and (ii) that B ∈ Q. (iii) We have A ∈ B {b}(u) for all b ∈ B. Indeed, let v ∈ V. Given b ∈ B there is some a ∈ Bv (b) ∩ A, that is there are α, β, λ ≥ 0 such that a ≤ αb + βv and 0 ≤ b + λv. Then for every c ∈ {b}(u), that is b c, we have b ∈ v1 (c), hence b ≤ 2c + (2 + λ)v (see Lemma 4.1(c) with ε = 1). This yields a ≤ 2αc + (2α + λα + β)v and A ≤ 2α {b}(u) ⊕ (2α + λα + β)v, hence A ∈ B {b}(u) . Consequently, O(A) ≤ inf {b}(u) | b ∈ B = conv b∈B {b}(u) (u) = B. (iv) On the other hand, let C ∈ Q such that C ≥ 0, that is C ⊂ {0}(u), and A ∈ B(C). Let c ∈ C. Given v ∈ V there are α, β ≥ 0 such that A ≤ αC ⊕ βv. According to our deﬁnition of the neighborhoods in Q see 4.37(e) , for ε = 1 we ﬁnd a ∈ A such that a ≤ γ(αc) + 2(βv) with some 1 ≤ γ ≤ 2. This yields Bv (b) ∩ A = ∅ for all v ∈ V, hence c ∈ B since c 0. Thus C = conv c∈C {c}(u) (u) ⊂ B. This shows O(A) ⊂ B, that is O(A) ≥ B, and our claim follows. In particular, we have O(A) = {0}(u) if and only if Bv (0) ∩ A = ∅ for all v ∈ V, that is if and only if for every v ∈ V there are a ∈ A and λ ≥ 0 such that a ≤ λv, that is if and only if the element A ∈ Q is bounded above see 4.37(e) . For a concrete example let P be the cone of all real-valued bounded below continuous functions on the open interval (0, 1), endowed with the positive constants as neighborhoods see 1.4(e) and let Q be as before. Consider the subset 1 C = f ∈ P f (x) ≥ − 2 for all x ∈ (0, 1) . x This set is convex, bounded below and closed with respect to the upper relative topology, hence C ∈ Q. For a function g ≥ 0 in P, we have B(g) ∩ C = ∅ if and only if there are α, β ≥ 0 such that 1/x ≤ αg(x) + β for all x ∈ (0, 1), that is if and only if the inferior limit of xg(x) at 0 is greater than 0. Thus O(C) = {g ∈ P g ≥ 0 and lim xg(x) > 0}. x→0

Now according to the cancellation rule in Proposition 5.10(a), if A, B ∈ Q such that A + C ≤ B + C, that is B + C ⊂ A + C, then A ≤ B + O(c), that is B + O(C) ⊂ A.

5. Locally Convex Lattice Cones

69

5.18 Order Convergence. We proceed to deﬁne order convergence for nets in a locally convex complete lattice cone (P, V). A net (ai )i∈I in P is called bounded below if there is i0 ∈ I such that the set {ai | i ≥ i0 } is bounded below in the sense of 4.24(i). We deﬁne the superior and inferior limits of a bounded below net (ai )i∈I in P by lim ai = sup inf ak and lim ak = inf sup ak . i∈I

k≥i

i∈I

i∈I

i∈I

k≥i

Because the order of P is supposed to be antisymmetric, both limits are uniquely deﬁned. Obviously, limi∈I ai ≤ limi∈I ai . If limi∈I ai and limi∈I ai coincide, we shall denote their common value by limi∈I ai and say that the net (ai )i∈I is order convergent. Obviously, every increasing or decreasing bounded below net is order convergent in this sense, converging towards the supremum or the inﬁmum of the set of its elements, respectively. Lemma 5.19. Let (P, V) be a locally convex complete lattice cone, and let (ai )i∈I and (bi )i∈I be bounded below nets in P. Then lim ai + lim bi ≤ lim(ai + bi ) ≤ lim ai + lim bi ≤ lim(ai + bi ) ≤ lim ai + lim bi . i∈I

i∈I

i∈I

i∈I

i∈I

i∈I

i∈I

i∈I

Proof. For any bounded below net (ci )i∈I in P, for i ∈ I, let (c)

si = inf ck (c)

(c)

and

k≥i

Si

= sup ck . k≥i

(c)

The nets (si )i∈I and (Si )i∈I are increasing and decreasing, respectively, and (c) (c) and lim ci = inf Si . lim ci = sup si i∈I

i∈I

i∈I

i∈I

Now, using the nets (ai )i∈I , (bi )i∈I and (ai + bi )i∈I in place of (ci )i∈I we observe that (a+b)

(a)

(b)

≥ si + si

(a+b)

(a)

Si ≤ Si for all i ∈ I. For every k ∈ I we have by 1 si

(a)

and

(b)

(a)

(b)

(a)

(b)

+ Si

(b)

sk + sup si = sup(sk + si ) ≤ sup(sl + sl ), i∈I

(a)

(b)

(a)

i∈I

(b)

as sk + si ≤ sl + sl

l∈I

whenever i, k ≤ l. This shows (a)

(b)

(a)

(b)

lim ai + lim bi = sup sk + sup sk ≤ sup(sl + sl ) = lim(ai + bi ), i∈I

i∈I

k∈I

i∈I

l∈I

i∈I

the ﬁrst part of our claim. A similar argument using the decreasing nets (c) (Si )i∈I yields

70

I Locally Convex Cones (a)

(b)

(a)

lim ai + lim bi = inf Sk + inf Sk ≥ inf (Sl i∈I

i∈I

i∈I

k∈I

l∈I

(b)

+ Sl ) = lim(ai + bi ). i∈I

Finally, for all i, l ∈ I and j ≥ i, l we have (a+b)

si

(a)

= inf (ak + bk ) ≤ inf (Sl k≥i

k≥j

(a)

+ bk ) = S l

(a)

+ inf bk ≤ Sl k≥j

+ lim bi , i∈I

hence (a+b)

lim(ai + bi ) = sup si i∈I

(a)

≤ inf Sl l∈I

i∈I

+ lim bi = lim ai + lim bi . i∈I

i∈I

i∈I

A similar argument shows that lim ai + lim bi ≤ lim(ai + bi ). i∈I

i∈I

i∈I

Note that Lemma 5.19 implies in particular that lim(a + bi ) = a + lim bi i∈I

i∈I

and

lim(a + bi ) = a + lim bi i∈I

i∈I

holds for and a bounded below net (bi )i∈I . We shall use Condi a ∈ P tions 2 and 2 for a comparison of the inferior and superior limits of nets: Lemma 5.20. Let (P, V) be a locally convex complete lattice cone, let (ai )i∈I and (bj )j∈J be nets in P, and let v ∈ V. (a) If for every i0 ∈ I there is j0 ∈ J such that for every j ≥ j0 there is i ≥ i0 such that ai ≤ bj + v, then limi∈I ai ≤ limj∈J bj + v. (b) If for every j0 ∈ J there is i0 ∈ I such that for every i ≥ i0 there is j ≥ j0 such that ai ≤ bj + v, then limi∈I ai ≤ limj∈J bj + v. (c) If I = J and if there is i0 ∈ I such that ai ≤ bi + v for all i ≥ i0 , then limi∈I ai ≤ limi∈I bi + v and limi∈I ai ≤ limi∈I bi + v. (d) If (ail )l∈L is a subnet of (ai )i∈I , then limi∈I ai ≤ liml∈L ail and liml∈L ail ≤ limi∈I ai . Proof. (a) Given i0 ∈ I choose j0 ∈ J as in the assumption of Part (a). Then inf i≥i0 ai ≤ bj + v for all j ≥ j0 , hence inf ai ≤ inf bj + v ≤ lim bj + v

i≥i0

j≥j0

j∈J

by 2 . Thus by 2 we have limi∈I ai ≤ limj∈J bj + v as well. The argument for Part (b) is similar. The assumptions for Part (c) yield those for Parts (a) and (b) with j0 = i0 and j = i. Part (d) follows from (a) and (b) if we set J = L and bl = aij .

5. Locally Convex Lattice Cones

71

Lemma 5.20(c) yields in particular that limi∈I ai ≤ limi∈I bi for order convergent nets (ai )i∈I and (bi )i∈I whenever ai ≤ bi for all i ∈ I. Part (d) implies that every subnet of an order convergent net is again order convergent with the same limit. Lemma 5.21. Let (P, V) be a locally convex complete lattice cone. Let (ai )i∈I be a bounded below net in P, and let (αi )i∈I be a bounded net of non-negative reals such that lim αi > 0. Then i∈I

lim αi i∈I

lim ai ≤ lim(αi ai ) ≤ lim(αi ai ) ≤ i∈I

i∈I

i∈I

lim αi i∈I

lim ai . i∈I

Proof. Obviously the net (αi ai )i∈I is also bounded below in P. We set α = lim αi > 0. Given v ∈ V there is λ ≥ 0 such that 0 ≤ ai + λv i∈I

for all i ∈ I. For ε > 0 we set γ = 1 + ε and ﬁnd i0 ∈ I such that (1/γ)α ≤ αi ≤ γα for all i ≥ i0 . Thus αi ai +

α λv ≤ αi (ai + λv) ≤ γα(ai + λv), γ

hence, using the cancellation law for positive elements (see Lemma I.4.2 in [100]) 1 αi ai ≤ γαai + αλ γ − v + εv ≤ γαai + ε(2αλ + 1)v. γ Using Lemma 5.20(c) we infer that lim αi ai ≤ γ α lim ai + ε(2αλ + 1)v. i∈I

i∈I

Since the latter holds for all ε > 0 and since P carries the weak preorder, we conclude that lim αi ai ≤ α lim ai . i∈I

i∈I

The ﬁrst part of the inequality in our claim follows in a similar fashion.

Proposition 5.22. Let (P, V) be a locally convex complete lattice cone. Let (ai )i∈I and (bi )i∈I be order convergent nets in P, and let (αi )i∈I be a bounded net of non-negative reals such that lim αi > 0. Then i∈I

lim(ai + bi ) = lim ai + lim bi i∈I

i∈I

i∈I

and

lim(αi ai ) = i∈I

lim αi i∈I

lim ai . i∈I

The latter is an obvious consequence of our previous results 5.19 and 5.21. Note that the requirement that limi∈I αi > 0 can not be omitted if the elements of the net (ai )i∈I are not bounded in P : In the locally convex complete lattice cone an = +∞ and αn = (1/n). Then R choose lim (αn an ) = +∞, but lim αn lim an = 0.

n→∞

n→∞

n→∞

72

I Locally Convex Cones

The following will provide a useful criterion for the convergence of a given net. Proposition 5.23. Let (P, V) be a locally convex complete lattice cone, and let (ai )i∈I be a bounded below net in P. If for every v ∈ V there is a convergent net (bi )i∈I in P such that (ai + bi )i∈I is convergent and the limit of (bi )i∈I is v-bounded, then the net (ai )i∈I is also convergent. Proof. Let (ai )i∈I be a net in P and for v ∈ V let (bi )i∈I be as stated. We use Lemma 5.19 for lim ai + lim bi ≤ lim(ai + bi ) ≤ lim ai + lim bi . i∈I

i∈I

i∈I

i∈I

i∈I

As b = lim bi is v-bounded, following Proposition 5.13(b) we have O(b) ≤ εv i∈I

for all ε > 0, hence lim ai ≤ lim ai + εv. i∈I

i∈I

by Proposition 5.10(a). Because this holds for all v ∈ V and ε > 0 and because P is a complete lattice cone, we infer that lim ai ≤ lim ai i∈I

holds as claimed.

i∈I

Proposition 5.24. Let (P, V) be a locally convex complete lattice cone, and let (ai )i∈I be a bounded below net in P. Then and lim O(ai ) ≤ O lim ai lim O(ai ) ≤ O lim ai . i∈I

i∈I

i∈I

i∈I

Proof. Let (ai )i∈I be a bounded below net, let v ∈ V and λ ≥ 0 such that 0 ≤ ai + λv for all i ≥ i0 ∈ I . Then O(ai ) ≤ ε(ai + λv) for all i ≥ i0 and ε > 0, hence lim O(ai ) ≤ ε lim ai + ελv ≤ ε sup lim ai , 0 + ελv i∈I

i∈I

i∈I

by 5.20(a). Taking the inﬁmum over all ε > 0 on the right-hand side we obtain lim O(ai ) ≤ O lim ai + O(λv) ≤ O lim ai + v. i∈I

i∈I

i∈I

Because this last inequality holds for all v ∈ V and because P carries the weak preorder, we conclude that lim O(ai ) ≤ O lim ai i∈I

i∈I

5. Locally Convex Lattice Cones

73

holds as claimed. A similar argument demonstrates the same inequality for the superior limits. A simple example can show that equality does in general not hold in the expressions of Proposition 5.24: The locally convex cone P = R is a complete lattice. Order convergence in R means convergence with respect to its usual one-point compactiﬁcation topology, which at the point +∞ diﬀers from the symmetric topology of R as a locally convex cone. For each n ∈ N let an = n ∈ R. Then lim an = +∞ with respect to order convergence (but n→∞

not with respect to the topology). We therefore have O(an ) = 0 symmetric for all n ∈ N, but O lim an = +∞. n→∞ We proceed to investigate continuity of the lattice operations with respect to order convergence (c.f. Proposition 5.2). Proposition 5.25. Let (P, V) be a locally convex complete lattice cone and let (ai )i∈I and (bi )i∈I be convergent nets in P. Then (a) lim(ai ∨ bi ) = lim ai ∨ lim bi . i∈I i∈I i∈I (b) lim(ai ∧ bi ) ≤ lim ai ∧ lim bi ≤ lim(ai ∧ bi ) + O lim(ai ∨ bi ) . i∈I

i∈I

i∈I

i∈I

i∈I

Proof. (a) Let (ai )i∈I and (bi )i∈I be convergent nets. Then sup al ∨ sup bj . lim(ai ∨ bi ) = inf sup(al ∨ bl ) ≤ inf i∈I

i∈I

i∈I

l≥i

j≥i

l≥i

Because for any choice of i, k ∈ I and any p ∈ I such that both i ≤ p and k ≤ p we have sup al ∨ sup bj ≤ sup al ∨ sup bj , j≥p

l≥p

l≥i

j≥k

we realize that sup al ∨ sup bj ≤ inf sup al ∨ sup bj . inf i∈I

i,k∈I

j≥i

l≥i

l≥i

j≥k

Now we use Proposition 5.15(a) for inf sup al ∨ sup bj = inf sup al ∨ inf sup bj = lim ai ∨ lim bi .

i,k∈I

l≥i

i∈I

j≥k

l≥i

k∈I

i∈I

j≥k

i∈I

Both nets (ai )i∈I and (bi )i∈I are supposed to be convergent. So we have lim ai ∨ lim bi = lim ai ∨ lim bi ≤ lim(ai ∨ bi ). i∈I

i∈I

i∈I

i∈I

i∈I

Summarizing, the above yields lim(ai ∨ bi ) ≤ lim ai ∨ lim bi ≤ lim(ai ∨ bi ) i∈I

i∈I

i∈I

i∈I

74

I Locally Convex Cones

as claimed in Part (a). Similarly, one veriﬁes Part (b): The inequality lim(ai ∧ bi ) ≤ lim ai ∧ lim bi = lim ai ∧ lim bi i∈I

i∈I

i∈I

i∈I

i∈I

is obvious. Next we use Part (a), Proposition 5.3 and the limit rules from Lemma 5.17 for lim ai ∧ lim bi +lim(ai ∨ bi ) = lim ai ∧ lim bi + lim ai ∨ lim bi i∈I

i∈I

i∈I

i∈I

i∈I

i∈I

i∈I

= lim ai + lim bi = lim(ai + bi ) i∈I

i∈I

i∈I

= lim(ai ∧ bi + ai ∨ bi ) i∈I

≤ lim(ai ∧ bi ) + lim(ai ∨ bi ) i∈I

i∈I

= lim(ai ∧ bi ) + lim(ai ∨ bi ). i∈I

i∈I

Now the cancellation rule from Lemma 5.9(a) yields the remaining part of (b). ∞ 5.26 Series. A series i=1 ai with terms ai in a locally convex complete lattice cone (P, V) is said to be convergent with limit s ∈ P if the se n = a of its partial sums is order convergent to s. We write quence s n i i=1 ∞ a = s in this case. Convergence of a series requires in particular that i i=1 the sequence of its partial sums is bounded below (see 5.18). Proposition 5.27. Let (P, V) be a locally ∞ convex complete lattice cone and let ai , bi ∈ P for i ∈ N. If the series i=1 ai is convergent and if ai ≤ bi ∞ for all i ∈ N, then the series i=1 bi is also convergent. Proof. Let ai , bi ∈ P such that ai ≤ bi for all i ∈ N. Let sn = ni=1 ai n ∞ ∞ and rn = i=1 bi be the partial sums of the series i=1 ai and i=1 bi , ∞ and let s = i=1 ai . Then sn ≤ rn for all n ∈ N, hence s ≤ lim rn . For n→∞

m ≥ n we have rn + sm = rn + sn +

m !

ai ≤ rn + sn +

i=n+1

m !

bi = rm + sn .

i=n+1

For a ﬁxed n ∈ N and m → ∞ this leads to rn + s = rn + lim sm = lim (rn + sm ) ≤ lim (sn + rm ) = lim rm + sn . m→∞

m→∞

m→∞

Now we let n → ∞ and obtain

lim rn + s = lim (rn + s) ≤ lim

n→∞

m→∞

n→∞

n→∞

lim rm + sn m→∞

= lim rm + lim sn = lim rm + s. m→∞

n→∞

m→∞

5. Locally Convex Lattice Cones

75

The cancellation law from Proposition 5.10(a) now yields lim rn ≤ lim rn + O(s).

n→∞

n→∞

But s ≤ lim rn , as we observed before, and therefore lim rn + O(s) = n→∞

n→∞

lim rn by Proposition 5.19. This yields n→∞

lim rn ≤ lim rn ,

n→∞

n→∞

hence convergence of the sequence (rn )n∈N , that is the partial sums of the ∞

series i=1 bi . ∞ We shall say that a series i of a loi=1 Ai of non-empty subsets A ∞ cally convex complete lattice cone P is convergent if the series i=1 inf Ai ∞ converges in P. In this case, all series a , for any choice of elements i=1 i Proposition 5.27, and we shall denote the set of ai ∈ Ai , are convergent by ∞ all limits of these series by i=1 Ai . Proposition 5.28. Let (P, V) be a locally convex complete lattice cone and ∞ let A be a convergent series of non-empty subsets of P. Then i i=1 ∞ { ∞ (a) i=1 sup Ai = sup i=1 Ai } . ∞ ∞ ∞ ∞ inf A ≤ inf { (b) i i=1 i=1 Ai } ≤ i=1 inf Ai + O (inf { i=1 Ai }) . ∞ Proof. Let i=1 Ai be a convergent series of non-empty subsets of P. We = sup Ai and si = shall consider Parts (a) and (b) simultaneously. Let Si ∞ on the series of sets, and inf Ai for all i ∈ N. By our assumption ∞ i=1 Ai ∞ ∞ s , S and following Proposition 5.27, all the series i i i=1 i=1 i=1 ai for in P. Moreover, for any choice of any choice of ai ∈ Ai are convergent n n n elements ai ∈ Ai , for i ∈ N, as i=1 si ≤ i=1 ai ≤ i=1 Si holds for all n ∈ N, we have ∞ ∞ ∞ ! ! ! si ≤ ai ≤ Si , i=1

and therefore ∞ !

si ≤ inf

i=1

∞ !

i=1

Ai

≤ sup

∞ !

∞ !

Si ≤ sup

i=1

Ai

≤

Ai

≤

i=1

i=1

inf

∞ !

i=1

Thus all left to show is that

and

i=1

∞ !

∞ !

Si .

i=1

Ai

i=1 ∞ ! i=1

" si + O inf

∞ ! i=1

# Ai

.

76

I Locally Convex Cones

For this, let us ﬁx n ∈ N and choose arbitrary elements ai , bi ∈ Ai . We set ci = bi for i = 1, . . . , n and ci = ai , else. Then, obviously, for every m ≥ n we have n m m n ! ! ! ! bi + ai = ci + ai . i=1

i=1

i=1

i=1

We let m tend to inﬁnity and obtain n !

bi +

∞ !

i=1

ai =

i=1

∞ !

n !

ci +

i=1

ai .

i=1

As ci ∈ Ai for all i ∈ N, this yields inf

∞ !

Ai

i=1

+

n !

ai ≤

n !

i=1

bi +

i=1

∞ !

ai ≤ sup

∞ !

i=1

Ai

+

i=1

n !

ai .

i=1

As sup { ni=1 Ai } = ni=1 Si and inf { ni=1 Ai } = ni=1 si by Lemma 5.6(a), variation of the elements b1 , . . . , bn yields inf

∞ !

Ai

+

i=1 n !

and

Si +

i=1

n ! i=1 ∞ !

ai ≤

n ! i=1

ai ≤ sup

i=1

si +

∞ !

ai

i=1

∞ !

Ai

+

i=1

n !

ai .

i=1

Now we let n tend to inﬁnity and infer that

∞ ∞ ∞ ∞ ! ! ! ! Ai + ai ≤ si + ai inf i=1 ∞ !

and

i=1

Si +

i=1

∞ !

i=1

ai ≤ sup

i=1

i=1

∞ !

Ai

+

i=1

∞ !

ai .

i=1

Finally, we take the inﬁmum over all choices for the elements ai ∈ Ai in this last pair of inequalities and obtain

∞

∞

∞ ∞ ! ! ! ! Ai + inf Ai ≤ si + inf Ai inf i=1

and

∞ ! i=1

i=1

Si + inf

∞ ! i=1

Ai

i=1

≤ sup

∞ ! i=1

i=1

Ai

+ inf

∞ ! i=1

Ai

.

5. Locally Convex Lattice Cones

77

Now the cancellation rule in Proposition 5.10(a) yields

∞ " ∞ # ∞ ! ! ! inf Ai ≤ si + O inf Ai i=1 ∞ !

and

i=1

Si ≤ sup

∞ !

i=1

i=1

"

+ O inf

Ai

i=1

∞ !

# Ai

.

i=1

∞ As inf { ∞ i=1 Ai } ≤ sup { i=1 Ai } , Proposition 5.14 yields

∞ " ∞ #

∞ ! ! ! sup Ai + O inf Ai Ai . = sup i=1

i=1

This demonstrates our claim.

i=1

5.29 Order Continuous Linear Operators and Functionals. Let (P, V) and (Q, W) be locally convex complete lattice cones. We shall say that a continuous linear operator T : P → Q is order continuous if it is continuous with respect to order convergence, that is if T lim ai = lim T (ai ) i∈I

i∈I

holds for every order convergent net (ai )i∈I in P. The limits refer to order convergence in P and Q, respectively. Sums and non-negative multiples of order continuous linear operators are again order continuous. We are particularly interested in order continuous linear functionals in P ∗ , that is order continuous linear operators from P into the locally convex complete lattice cone R. They form a subcone of P ∗ . For every bounded below net (ai )i∈I in P and every order continuous linear operator T : P → Q we have T lim ai = T lim inf ak = lim T inf ak ≤ lim inf T (ak ) = lim T (ai ) i∈I

i∈I k≥i

i∈I

k≥i

i∈I k≥i

i∈I

and, likewise T lim ai = T lim sup ak = lim T sup ak ≥ lim sup T (ak ) = lim T (ai ), i∈I

that is

i∈I k≥i

i∈I

k≥i

i∈I k≥i

i∈I

T lim ai ≤ lim T (ai ) ≤ lim T (ai ) ≤ T lim ai . i∈I

i∈I

i∈I

i∈I

5.30 Lattice Homomorphisms. Let both (P, V) and (Q, W) be locally convex ∨- (or ∧-)semilattice cones. A continuous linear operator T : P → Q is called a ∨- (or ∧-)semilattice homomorphism if it is compatible with the lattice operations in P and Q, that is if T (a ∨ b) = T (a) ∨ T (b) or T (a ∧ b) = T (a) ∧ T (b)

78

I Locally Convex Cones

holds for all a, b ∈ P. If both (P, V) and (Q, W) are locally convex lattice cones and T : P → Q is both a ∨- and a ∧-semilattice homomorphism, then T is called a lattice homomorphism. Non-negative multiples of lattice homomorphism are again lattice homomorphisms, but sums are generally not. Linear operators that are both order continuous and lattice homomorphisms are of particular interest. Suppose that both (P, V) and (Q, W) are locally convex complete lattice cones. A continuous linear operator T : P → Q is an order continuous lattice homomorphism if and only if T (sup A) = sup T (A)

and

T (inf B) = inf T (B)

holds for all non-empty subsets A and bounded below subsets B of P, that is if and only if T preserves that lattice operations of P and Q. Indeed, sup A or inf B is the limit with respect to order convergence of the net of suprema or inﬁma of ﬁnite subsets of A or B, respectively. Since an order continuous lattice homomorphism T : P → Q preserves ﬁnite suprema and inﬁma as well as order convergence, we conclude that T preserves inﬁnite suprema and inﬁma as well. Conversely, if T preserves the suprema and inﬁma of subsets of P, then we have and T lim ai = lim T (ai ) T lim ai = lim T (ai ) i∈I

i∈I

i∈I

i∈I

for every bounded below net (ai )i∈I in P. Thus T maps order convergent nets in P into order convergent nets in Q and is therefore an order continuous lattice homomorphism. Examples 5.31. (a) Theorem II.6.7 in [100] states that for every neighborhood v ∈ V in an M-type locally convex ∨- (or ∧-)semilattice cone (P, V) all the extreme points of its polar v ◦ ⊂ P ∗ are ∨- (or ∧-)semilattice homomorphisms from P into R. (b) Let (P, V) be a locally convex cone with dual P ∗ and let (Q, V) be the cone of all non-empty convex subsets of P which are closed with respect to the lower topology see Example 4.37(d) . In 5.7(d) we showed that (Q, V) is a locally convex -semilattice cone ordered by the set inclusion. There is a natural embedding μ → μ ˜ : P ∗ → Q∗ , where μ ˜(A) = sup{μ(a) | a ∈ A} for μ ∈ P ∗ and A ∈ Q. Indeed, if μ ∈ v ◦ for some v ∈ V, then A ≤ B ⊕ v for A, B ∈ Q means that for every a ∈ A and ε ≥ 0 there is b ∈ B such that a ≤ γb + (1 + ε)v see 4.37(e) with some 1 ≤ γ ≤ 1 + ε. This yields μ(a) ≤ γμ(b) + (1 + ε) ≤ γ μ ˜(b) + (1 + ε) for all ε > 0, hence μ(a) ≤ μ ˜(B) + 1. We infer μ ˆ(A) ≤ μ ˜(B) + 1, and conclude that μ ˜ ∈ v ◦ ⊂ Q∗ . Moreover, μ ˜ is a ∨-semilattice homomorphism even with respect to arbitrary suprema in Q : Let A be a subset of Q and

5. Locally Convex Lattice Cones

79

let c be an element of conv A , the convex hull of the union of all A∈A n α a for some ai ∈ Ai ∈ A and αi ≥ 0 such elements of A. Then c = i=1 i i n α = 1. Thus that i=1 i μ(c) =

n !

αi μ(ai ) ≤

i=1

n !

αi μ ˜(Ai ) ≤ sup μ ˜(A). A∈A

i=1

Since the functional μ : P → R is also continuous with respect to the lower relative topology on P, we conclude that

(l) A μ ˜(sup A) = sup μ(a) a ∈ conv A∈A

A = sup μ(a) a ∈ conv A∈A

≤ sup μ ˜(A). A∈A

The converse inequality is obvious. (c) Similarly one argues for the locally convex cone (Q, V) of all bounded below convex subsets of P which are closed with respect to the upper topology see Examples 4.37(e) and 5.7(e) . In this case (Q, V) is a locally convex -semilattice cone, ordered by the inverse set inclusion. There is a natural embedding μ → μ ˜ : P ∗ → Q∗ , where μ ˜(A) = inf{μ(a) | a ∈ A} for μ ∈ P ∗ and A ∈ Q. As similar argument as in (b) shows that μ ˜(inf A) = inf μ ˜(A) A∈A

holds for every bounded below family of sets A ⊂ Q. (d) Let (P, V) be a locally convex ∨- (or ∧-)semilattice cone, X a set, of P-valued functions and consider the locally convex cone FVb (X, P), V consisting of V ∪ {∞} on X, endowed with system V the neighborhood -valued functions. Example 1.4(e) . This was seen to be again a locally convex ∨- (or ∧-)semilattice cone, provided that for every x ∈ X and such that vˆ(x) ≤ v see 5.7(c) . For μ ∈ v ◦ ⊂ P ∗ , v ∈ V there is vˆ ∈ V and x ∈ X such that vˆ(x) ≤ v, the mapping a neighborhood vˆ ∈ V μx : FVb (X, P) → R such that μx (f ) = μ f (x) for all f ∈ FVb (X, P) is a continuous linear functional on FVb (X, P) see 2.1(f) , more precisely: an element of vˆ◦ . Moreover, if μ is a ∨- (or ∧-)semilattice homomorphism for P, then μx is a semilattice homomorphism of the same type for FVb (X, P). 5.32 Functionals Supporting the Separation Property. Corollary 4.34 (see also the Separation Theorem 3.2 in [175]) guarantees that in a locally convex cone the neighborhoods with respect to the weak preorder are completely

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I Locally Convex Cones

determined by their polars, that is a b + v holds for a, b ∈ P and v ∈ V if and only if μ(a) ≤ μ(b) + 1 for all μ ∈ v ◦ . In this vein, for a locally convex cone (P, V) we shall say that a subset Υ of P ∗ supports the separation property for P if for a, b ∈ P and v ∈ V such that a b + v there is α ≥ 0 and μ ∈ Υ ∩ (αv ◦ ) such that μ(a) > μ(b) + α. This property implies in particular that the functionals in Υ determine the weak preorder of P, that is a b holds for a, b ∈ P if and only if μ(a) ≤ μ(b) for all μ ∈ Υ. Indeed, the latter implies that a b + v for all v ∈ V, which by Lemma 3.2(a) yields a b. Examples 5.33. (a) In Examples 1(a) and (b), that is for P = R or P = R+ the dual cone contains all positive reals, and the set Υ = {1} supports the separation property. (b) If V consists of the multiples of a single neighborhood v, then we may choose Υ = {μ ∈ P ∗ | ψv (μ) = 0 or ψv (μ) = 1}, where ψv (μ) = inf{α ≥ 0 | μ ∈ αv ◦ }. (In case that v ∈ P, we have ψv (μ) = μ(v).) Indeed, if a ≤ b + (ρv) for a, b ∈ P and ρv ∈ P, then by Corollary 4.34 there is μ ∈ (ρv)◦ = (1/ρ)v ◦ such that μ(a) > μ(b) + 1. This implies ψv (μ) ≤ 1/ρ. If ψv (μ) = 0, then μ ∈ Υ ∩ (ρv)◦ as required. Otherwise, we set α = 1/ψv (μ) > 0 and ν = αμ ∈ Υ and observe that both ν ∈ α(ρv)◦ and ν(a) > ν(b) + α, again satisfying the requirement. If in addition all elements of P are bounded, that is for example, if P is normed vector space, then according to Corollary 4.35 we may further reduce the size of Υ and choose Υ = Ex(v ◦ ), that is the set of all extreme points of the w(P ∗ , P)-compact convex set v ◦ . Obviously, ψv (μ) = 1 holds for every μ ∈ Ex(v ◦ ). (c) More generally, a locally convex cone (P, V) is said to be tightly covered by its bounded elements see II.2.13 in [100] if for all a, b ∈ P and v ∈ V such that a b + v there is some bounded element a ∈ P such that a a and a b + v. In this case, if V0 is a subcollection of V such that every v ∈ V is a multiple of some v0 ∈ V0 , then according to Corollary II.4.7 in [100] the set Υ = Ex(v0◦ ) v0 ∈V0

supports the separation property for P. (d) Let (P, V) be a locally convex cone with dual P ∗ and let (Q, V) be the cone of all non-empty convex subsets of P which are closed with respect to the lower topology see Example 4.37(d) . For every μ ∈ P ∗ the formula μ ˜(A) = sup{μ(a) | a ∈ A}

for

A∈Q

5. Locally Convex Lattice Cones

81

deﬁnes an element μ ˜ ∈ Q∗ , more precisely, μ ˜ ∈ v ◦ whenever μ ∈ v ◦ (see Example 5.31(b) before). Now Theorem 4.33 guarantees that the set Υ = {˜ μ | μ ∈ P ∗ } ⊂ Q∗ supports the separation property for Q. Indeed, if A B ⊕ v for A, B ∈ Q and v ∈ V, then there is a ∈ A such that a ∈ v B see 4.37(d) . Following ˜ ∈ v ◦ such that Theorem 4.33(a) then there is μ ∈ v ◦ , hence μ μ(a) > sup{μ(b) | b ∈ B} + 1 = μ ˜(B) + 1. Thus μ ˜(A) = sup{μ(a) | a ∈ A} > μ ˜(B) + 1. (e) Similarly, if (Q, V) is the locally convex cone of all bounded below convex subsets of P which are closed with respect to the upper topology see Example 4.37(e) , then for every μ ∈ P ∗ the formula μ ˜(A) = inf{μ(a) | a ∈ A}

for

A∈Q

μ ˜ ∈ Q∗ , more precisely, μ ˜ ∈ v ◦ whenever μ ∈ v ◦ deﬁnes an element see 5.31(c) . Then Υ = {˜ μ | μ ∈ P ∗ } ⊂ Q∗ supports the separation property for Q. Indeed, if A B ⊕ v for A, B ∈ Q and v ∈ V, then there is b ∈ B such that b ∈ A v see 4.37(e) . Following ˜ ∈ v ◦ such that μ(b) < Theorem 4.33(b) then there is μ ∈ v ◦ , hence μ inf{μ(b) | b ∈ B} − 1 = μ ˜(A) − 1. Thus μ ˜(B) = inf{μ(b) | b ∈ B} < μ ˜(A) − 1, that is μ ˜(A) > μ ˜(B) + 1. (f) Let (P, V) be a locally convex cone, X a set, and consider the lo of P-valued functions on X, where the cally convex cone FVb (X, P), V is generated by a family of V ∪ {∞} -valued funcneighborhood system V tions on X as elaborated in Example 1.4(e). For every μ ∈ v◦ ⊂ P ∗ for the formula v ∈ V, and x ∈ X such that vˆ(x) ≤ v for vˆ ∈ V, μx (f ) = μ f (x) for f ∈ FVb (X, P) deﬁnes a continuous linear functional on FVb (X, P) see 2.1(f) , more precisely: We have μx ∈ vˆ◦ . Let us denote by X0 the subset of all x ∈ X such If Υ ⊂ P ∗ supports the separation that vˆ(x) = ∞ for at least one vˆ ∈ V. property for P, then Υ = {μx | μ ∈ Υ, x ∈ X0 } ⊂ FVb (X, P)∗ supports the separation property for FVb (X, P). Indeed, if f g + vˆ for then there is x ∈ X such that f (x) ≤ g(x) + f, g ∈ FVb (X, P) and vˆ ∈ V, vˆ(x). This implies vˆ(x) = ∞, hence v = vˆ(x) ∈ V. Following our assump tion there is α ≥ 0 and μ ∈ Υ ∩ (αv ◦ ) such that μ f (x) > μ g(x) + α.

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We therefore have μx ∈ Υ ∩ (αˆ v ◦ ) and μx (f ) > μx (g) + α, as required. In case that P = R or P = R+ we may choose Υ = {1} see 5.33(a) . Then Υ consists of all point evaluations at the points x ∈ X such that vˆ(x) < +∞ for at least one of the R+ -valued neighborhood functions vˆ ∈ V. The presence of suitable subsets of P ∗ supporting the separation property permits a strengthening of certain statements for the general case. The following Propositions 5.34 and 5.35 will improve on Proposition 5.15(b) and Propositions 5.25(b) and 5.28(b) under these circumstances. Recall that a subset A of an ordered cone P is said to be directed upward (or downward) if for a, b ∈ A there is c ∈ A such that both a ≤ c and b ≤ c (or c ≤ a and c ≤ b.) Proposition 5.34. Let (P, V) be a locally convex complete lattice cone, and suppose that the order continuous lattice homomorphisms support the separation property for P. Then (a) sup(A ∧ B) = sup A ∧supB for non-empty subsets A, B of P. (b) lim(ai ∧ bi ) = lim ai ∧ lim bi i∈I

i∈I

i∈I

for convergent nets (ai )i∈I and (bi )i∈I in P. Proof. Let Υ be the subset of all order continuous lattice homomorphisms in P ∗ and suppose that Υ supports the separation property for P. For Part (a), let A, B be non-empty subsets of P. In Proposition 5.15(b) we already demonstrated sup(A ∧ B) ≤ sup A ∧ sup B. For the converse inequality, it suﬃces to verify that μ(sup A ∧ sup B) ≤ μ sup(A ∧ B) holds for all μ ∈ Υ (see 5.32). For this, let μ ∈ Υ. Then μ(sup A ∧ sup B) = μ(sup A) ∧ μ(sup B) = sup μ(A) ∧ sup μ(B) , since μ is an order continuous lattice homomorphism. We may assume that μ(sup A) ≤ μ(sup B). Then for every a ∈ A and ε > 0 there is b ∈ B such that μ(a) ≤ μ(b) + ε. Thus also μ(a) ≤ μ(a ∧ b) + ε. This shows sup μ(A) ∧ sup μ(B) = sup μ(A) ≤ μ sup(A ∧ B) + ε and veriﬁes our claim. For Part (b), let (ai )i∈I and (bi )i∈I be convergent nets in P. In the light of 5.25(b) and our assumption on Υ it suﬃces to verify that lim ai ∧ lim bi μ ≤ μ lim(ai ∧ bi ) , i∈I

i∈I

i∈I

5. Locally Convex Lattice Cones

that is

83

lim μ(ai ) ∧ lim μ(bi ) ≤ lim μ(ai ) ∧ μ(bi ) i∈I

i∈I

i∈I

holds for all μ ∈ Υ. For this, given a functional μ ∈ Υ, we may assume that limi∈I μ(ai ) ≤ limi∈I μ(bi ). Then for every ε > 0 there is i0 ∈ I such that μ(ai ) ≤ μ(bi ) + ε for all i ≥ i0 . This implies μ(ai ) ≤ μ(ai ) ∧ μ(bi ) + ε. Thus limi∈I μ(ai ) ≤ limi∈I μ(ai ) ∧ μ(bi ) + ε. This yields our claim. Proposition 5.35. Let (P, V) be a locally convex complete lattice cone. If the order continuous lattice homomorphisms (or the order continuous linear functionals) support the separation property for P, then

∞ ∞ ! ! inf Ai = inf Ai i=1

for every convergent series downward) subsets of P.

i=1

∞ i=1

Ai of non-empty (or non-empty directed

Proof. Let Υ be the subset of all order continuous lattice homomorphisms (or order continuous linear functionals) in P ∗ and suppose that Υ supports the separation property for P. In the second case we assume in addition that the sets Ai ⊂ P are directed downward. Thus, in each of the cases for Υ we have μ inf Ai = inf {μ(Ai )} for all i ∈ N and μ ∈ Υ. The order continuity of the functionals μ then yields "∞ # ∞ ∞ ! ! ! μ inf Ai = μ inf Ai = inf μ(Ai ) . i=1

∞

i=1

i=1

Likewise, since the sets i=1 Ai are seen to be directed downward in the second case for Υ, we have # " ∞ ! Ai μ inf i=1

"

= inf

μ

∞ ! i=1

#

∞ ! ai ai ∈ Ai = inf μ(ai ) i=1

ai ∈ Ai .

Given μ ∈ Υ we choose ai ∈ Ai such that μ(ai ) ≤ inf {μ(Ai )} + 2−i . Then

∞ ∞ ∞ ! ! ! μ(ai ) ai ∈ Ai ≤ μ(ai ) ≤ inf {μ(Ai )} + 1, inf i=1

i=1

i=1

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I Locally Convex Cones

hence

" μ inf

∞ !

#

" ≤μ

Ai

i=1

∞ !

# inf Ai

+ 1.

i=1

Because Υ supports the separation property for P, this shows

∞ ∞ ! ! inf Ai ≤ inf Ai + v i=1

for all v ∈ V, hence inf

∞ ! i=1

i=1

Ai

≤

∞ !

inf Ai ,

i=1

since P carries its weak preorder. The reverse inequality was established in Proposition 5.28(b). We shall say that a subcone N of P is a locally convex lattice subcone of (P, V) if a ∨ b ∈ N and a ∧ b ∈ N whenever a, b ∈ N . Likewise, N is a locally convex complete lattice subcone of (P, V) if sup A ∈ N and inf B ∈ N whenever A, B ⊂ N , A is not empty and B is bounded below. The suprema and inﬁma are taken in P. A family A of subsets of P will be called sup-bounded below if the set {sup A | A ∈ A} is bounded below in P. This implies in particular that ∅ ∈ A and that inf{sup A | A ∈ A} exists in P. Proposition 5.36. Let (P, V) be a locally convex complete lattice cone, and suppose that the order continuous lattice homomorphisms support the separation property for P. Let N be a subcone of P. The smallest locally convex complete lattice subcone of P that contains N consists of all elements a ∈ P which can be expressed in the following way: a = inf{sup A | A ∈ A}, where A is a sup-bounded below family of subsets of N . Proof. Let (P, V) be a locally convex complete lattice cone. Corresponding to a sup-bounded below family A of non-empty subsets of P let us deﬁne the element aA ∈ P by aA = inf{sup A | A ∈ A}. For families A and B of this type and α ≥ 0 we denote αA = {αA | A ∈ A} and A + B = {A + B | A ∈ A, B ∈ B}. It is evident from Lemmas 5.5 and 5.6 that these are again sup-bounded below families of subsets of P. We also use 5.5 and 5.6 for the following observations: (i)

αaA = α inf{sup A | A ∈ A} = inf{sup αA | A ∈ A} = aαA .

5. Locally Convex Lattice Cones

aA + aB

(ii)

85

= inf sup A | A ∈ A + inf sup B | B ∈B = inf sup A + sup B | A ∈ A, B ∈ B = inf sup(A + B) | A ∈ A, B ∈ B = inf sup C | C ∈ (A + B) = a(A+B) .

Let {Ai }i∈I be a collection of sup-bounded families Ai of subsets of P. In a ﬁrst instance, suppose that this collection is not empty, and let A = {∪i∈I Ai | (Ai )i∈I ∈ i∈I Ai }, that is the elements A of A are all unions of the type A = ∪i∈I Ai , where Ai ∈ Ai . (The Axiom of Choice is required for this construction.) This family A is also sup-bounded below. Indeed, given v ∈ V and a ﬁxed k ∈ I there is λ ≥ 0 such that 0 ≤ sup Ak + λv for all Ak ∈ Ak . Thus for every A ∈ A we have Ak ⊂ A for some Ak ∈ Ak , hence 0 ≤ sup Ak + λv ≤ sup A + λv. We claim that (iii)

sup aAi = sup inf{sup Ai | Ai ∈ Ai } = inf{sup A | A ∈ A} = aA . i∈I

i∈I

Indeed, for every ﬁxed i ∈ I and every A ∈ A there is some Ai ∈ Ai such that Ai ⊂ A. This shows inf{sup Ai | Ai ∈ Ai } ≤ sup A for all A ∈ A, hence inf{sup Ai | Ai ∈ Ai } ≤ inf{sup A | A ∈ A} holds for all i ∈ I. This yields sup inf{sup Ai | Ai ∈ Ai } ≤ inf{sup A | A ∈ A}. i∈I

For the converse inequality we will have to use the fact that the lattice operations are formed in a locally convex complete lattice cone for which the order continuous lattice homomorphisms support the separation property, that is it suﬃces to verify that μ inf{sup A | A ∈ A} ≤ μ sup inf{sup Ai | Ai ∈ Ai } i∈I

holds for every order continuous lattice homomorphism μ ∈ P ∗ . For this assume to the contrary that there is ρ < μ inf{sup A | A ∈ A} = inf{sup μ(A) | A ∈ A}, and that μ sup inf{sup Ai | Ai ∈ Ai } = sup inf{sup μ(Ai ) | Ai ∈ Ai } < ρ i∈I

i∈I

holds for some order continuous lattice homomorphism μ ∈ P ∗ . This means inf{sup μ(Ai ) | Ai ∈ Ai } < ρ for all i ∈ I, hence sup μ(Ai ) < ρ for some Ai ∈ Ai . We use these sets Ai for A = ∪i∈I Ai ∈ A. We have sup μ(A) = sup{μ(Ai ) | i ∈ I} ≤ ρ, contradicting the assumption that ρ < sup μ(A) holds for all A ∈ A. This yields our claim. In a second instance, suppose that the set {aAi }i∈I is bounded below in P. Then the family A = ∪i∈I Ai is also sup-bounded below. Indeed, given v ∈ V

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I Locally Convex Cones

there is λ ≥ 0 such that 0 ≤ aAi + λv for all i ∈ I. Thus for every A ∈ A we have A ∈ Ai for some i ∈ I and therefore 0 ≤ aAi + λv ≤ sup A + λv. Now we infer that inf aAi = inf inf{sup Ai | Ai ∈ Ai } (iv) i∈I i∈I = inf sup A | A ∈ ∪i∈I Ai = aA the subset of P consistNow let N be a subcone of P and denote by N ing of all elements aA , where A is an sup-bounded below family of subsets is a locally convex of N . The preceding arguments (i) to (iv) yield that N lattice subcone of P. Since aA = a for every a ∈ N with A = {a} , we . On the other hand, every locally convex lattice subcone of P have N ⊂ N that contains N , necessarily contains also all elements aA ∈ P of this type. is indeed the smallest locally convex complete lattice subcone of P Thus N that contains N . For the following recall the notations from Example 1.4(e). We observed is a locally convex complete lattice cone for any before that FVb (X, R), V choice of the set X and the neighborhood system that for every V, provided such that vˆ(x) < +∞ see 5.7(c) . The point evaluax ∈ X there is vˆ ∈ V tions at the points x ∈ X are order continuous lattice homomorphisms and according to Example 5.33(f) support the separation property for FVb (X, R). the Thus for every locally convex complete lattice subcone of FVb (X, R), V order continuous lattice homomorphisms support the separation property. For an inverse implication recall the deﬁnition of an embedding in 2.2: Proposition 5.37. Let (P, V) be a locally convex complete lattice cone. If the set Υ of all order continuous lattice homomorphisms in P ∗ supports the separation property, then (P, V) can be embedded into the locally convex , endowed with a suitable system V of R+ complete lattice of FVb (Υ, R), V -valued neighborhood functions. This embedding is one-to-one and preserves the lattice operations. Proof. Let Υ be the set of order continuous lattice homomorphisms in P ∗ . Recall that αμ ∈ Υ whenever μ ∈ Υ and α ≥ 0. With every element a ∈ P we associate the function ϕa : Υ → R such that ϕa (μ) = μ(a)

for all μ ∈ Υ.

The mapping Φ : P → F(Υ, R) such that Φ(a) = ϕa for all a ∈ P is obviously linear, monotone, and since P carries the weak preorder which is supposed to be antisymmetric, Φ is also one-to one. Since the elements of Υ are all order continuous lattice homomorphisms in P ∗ , for every subset A of P and μ ∈ Υ we have

5. Locally Convex Lattice Cones

87

μ(sup A) = sup{μ(a) | a ∈ A} = sup{ϕa (μ) | a ∈ A} = sup ϕa (μ). Thus Φ(sup A) = sup Φ(A). Likewise, we have Φ(inf B) = inf Φ(B) for every bounded below subset B of P. Recall that the lattice operations are carried out pointwise in F(Υ, R.) Corresponding to the neighborhoods v ∈ V we consider the R-valued functions ψv on Υ such that ψv (μ) = inf{α > 0 | μ ∈ αv ◦ } for all μ ∈ P ∗ . As usual, we set inf ∅ = +∞, but observe that for every μ ∈ Υ there is v ∈ V such that ψv (μ) < +∞. Note that ψv = ϕv in case that v ∈ P. We also note that the family of all functions ψv for v ∈ V is not necessarily closed for the pointwise addition of its functions. For this reason we refer to the last remark in Example 1.4(e) relating to the construction of a locally convex cone of cone-valued functions. For F(Υ, R) we use the abstract neighborhood system V = {ˆ v | v ∈ V} with the addition ⊕ and multiplication by scalars carried over by the corresponding operations in V, that is u ˆ ⊕ vˆ = u + v and αˆ v = α $v for u, v ∈ V and α > 0. corresponds to the family {ψv | v ∈ V} of The neighborhood system V R+ -valued neighborhood functions which deﬁne the neighborhoods vˆ ∈ V for F(Υ, R) by f ≤ g + vˆ if f (μ) ≤ g(μ) + ψv (μ) for all μ ∈ Υ see 1.4(e) for functions f, g ∈ F(Υ, R). As required, we have ψ(αv) = αψv and ψ(u+v) ≥ ψu + ψv for all u, v ∈ V and α > 0. The ﬁrst of these claims is obvious. For the second one, let μ ∈ Υ and let σ < ψu (μ) and ρ < ψv (μ). Since both μ ∈ σu◦ and μ ∈ ρv ◦ , there are a, b, c, d ∈ P such that a ≤ b+u and μ(a) > μ(b) + σ as well as c ≤ d + v and μ(c) > μ(d) + σ. Then from (a + c) ≤ (b + d) + (u + w) and μ(a + c) > μ(b + d) + (σ + ρ) we conclude that μ ∈ (σ + ρ)(u + v)◦ . This shows ψ(u+v) (μ) ≥ (σ + ρ), yielding our claim. is a locally convex lattice cone in the sense of 1.4(e). Thus FVb (Υ, R), V we Moreover, since ψv (μ) < +∞ holds for all μ ∈ P ∗ with some vˆ ∈ V, ∗ established in Example 5.7 that FVb (P , R), V is indeed a locally convex complete lattice cone. We claim that Φ(P) is contained in FVb (Υ, R), V and that a≤b+v if and only if Φ(a) ≤ Φ(b) + vˆ holds for a, b ∈ P and v ∈ V. Indeed, suppose that a ≤ b + v. Then for every μ ∈ Υ and α > 0 such that μ ∈ αv ◦ we have μ(a) ≤ μ(b) + α, that is ϕa (μ) ≤ ϕb (μ) + ψv (μ), hence Φ(a) ≤ Φ(b) + ψv . Conversely, if a ≤ b + v, then following our assumption that Υ supports the separation property, there is μ ∈ v ◦ ∩ Υ such that μ(a) > μ(b) + 1. The former implies ψv (μ) ≤ 1, hence ϕa (μ) > ϕb (μ) + ψv (μ) and therefore Φ(a) Φ(b) + ψv . We infer in particular that the functions Φ(a) ∈ F(Υ, R) are bounded below

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Indeed, given a ∈ P and v ∈ V there relative to the neighborhoods in V. is λ ≥ 0 such that 0 ≤ a + λv, hence 0 ≤ Φ(a) + λˆ v . Therefore the element Φ(a) is contained in FVb (Υ, R) as claimed. Finally we establish that the linear operator Φ : P → FVb (Υ, R) is an embedding in 2.2 of the locally convex complete lattice cone the sense of . Indeed, we set Φ(v) = vˆ for v ∈ V towards the (P, V) into FVb (Υ, R), V extension . Φ : (P ∪ V) :→ FVb (Υ, R) ∪ V and by the above a ≤ b+v holds for a, b ∈ P and v ∈ V if Then Φ(V) = V, and only if Φ(a) ≤ Φ(b) + Φ(v), as required in 2.2. Moreover, since the (weak pre-)order of the locally convex complete lattice cone P is antisymmetric, its symmetric topology is Hausdorﬀ by Proposition 4.8. Lemma 2.3 therefore yields that the operator Φ : P → FVb (Υ, R) is one-to-one, as claimed. We shall demonstrate in 5.57 below that every locally convex cone (P, V) can be canonically embedded into a locally convex complete lattice cone for which the set of order continuous lattice homomorphisms in P ∗ supports the separation property. 5.38 Almost Order Convergent Nets. The concept of order convergence can in some cases be meaningfully extended to nets that are not necessarily bounded below. We shall say that a net (ai )i∈I in a locally convex complete lattice cone (P, V) is almost order convergent towards a ∈ P if for every k ∈ I the net (ai ∨ ak )i∈I is order convergent and if lim lim (ai ∨ ak ) = a. k∈I

i∈I

The net (ai ∨ ak )i∈I is of course bounded below for any choice of k ∈ I. Indeed, given v ∈ V there is λ ≥ 0 such that 0 ≤ ak + λv ≤ (ai ∨ ak ) + λv for all i ∈ I. Lemma 5.39. Let (P, V) be a locally convex complete lattice cone. A bounded below net (ai )i∈I in P is order convergent if and only if it is almost order convergent with the same limit. Proof. Let (ai )i∈I be a bounded below net in P. If (ai )i∈I is order convergent and limi∈I ai = a, then lim(ai ∨ ak ) = lim ai ∨ ak = a ∨ ak i∈I

i∈I

for all k ∈ I by Proposition 5.25(a). Therefore lim lim (ai ∨ ak ) = lim(a ∨ ak ) = a ∨ lim ak = a k∈I

i∈I

k∈I

k∈I

again by 5.25(a), and we infer that (ai )i∈I is almost order convergent towards a. On the other hand, for every b ∈ P we have

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89

lim(ai ∨ b) = sup inf (aj ∨ b) = sup inf aj ∨ b) j≥i j≥i i∈I i∈I i∈I = sup inf aj ∨ b = lim ai ∨ b j≥i

i∈I

i∈I

by Proposition 5.15 and Lemma 5.5. Similarly one realizes that lim(ai ∨ ak ) = inf sup(aj ∨ b) = inf sup aj ∨ b) i∈I

i∈I

j≥i

=

inf sup aj i∈I

j≥i

i∈I

∨b=

j≥i

lim ai ∨ b i∈I

If the net (ai )i∈I is almost order convergent towards a ∈ P, this yields lim(ai ∨ ak ) = lim ai ∨ ak = lim ai ∨ ak i∈I

i∈I

i∈I

for all k ∈ I. Thus, again using the above a = lim lim (ai ∨ ak ) i∈I k∈I lim ai ∨ ak = lim i∈I k∈I = lim ai ∨ lim ak = lim ai , i∈I

as well as

k∈I

i∈I

a = lim lim (ai ∨ ak ) i∈I k∈I lim ai ∨ ak = lim i∈I k∈I = lim ai ∨ lim ak = lim ai . i∈I

This yields limi∈I ai = a.

k∈I

i∈I

Examples 5.40. Let P be the cone of all bounded below R-valued functions on [0, +∞), endowed with the pointwise operations and order, and the positive constant functions v > 0 as its neighborhood system V see Example 1.4(e) . (P, V) is a locally convex complete lattice cone, and order convergence in P implies pointwise convergence on [0, +∞) for the functions involved. Pointwise convergence, on the other hand does not require that a net in P is bounded below and therefore does not always imply order convergence. Let us illustrate this in a simple example: For n ∈ N let fn ∈ P such that fn (x) = −n for 0 < x ≤ 1/n, and fn (x) = 0 else. The sequence (fn )n∈N converges pointwise to 0 ∈ P, but it is not bounded below in P and therefore not order convergent. However, for every m ∈ N the sequence

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(fn ∨ fm )n∈N is bounded below and converges pointwise, hence in order towards 0 ∈ P. We infer that (fn )n∈N is almost order convergent towards 0 ∈ P. In fact, it can be easily veriﬁed that pointwise convergence coincides with almost order convergence in this example (see Proposition 5.51 below). We proceed probing diﬀerent patterns of convergence in a locally convex complete lattice cone (P, V). For a net (ai )i∈I in P, convergence with respect to the symmetric relative topology of P towards a ∈ P means that for every v ∈ V and ε > 0 there is i0 ∈ I such that ai ∈ vεs (a) for all i ≥ i0 . (ai )i∈I is a Cauchy net if for every v ∈ V and ε > 0 there is i0 ∈ I such that ai ∈ vε (ak ) for all i, k ≥ i0 . Obviously, convergence implies that (ai )i∈I is a Cauchy net. The converse, that is topological completeness holds also true: Proposition 5.41. Every locally convex complete lattice cone is complete with respect to the symmetric relative topology. Proof. Suppose that (ai )i∈I is a Cauchy net in P. We shall ﬁrst demonstrate that (ai )i∈I is order convergent. Let v ∈ V and 0 < ε ≤ 1. There is i0 ∈ I such that ai ∈ vε (ak ) for all i, k ≥ i0 . Choose λ ≥ 0 such that 0 ≤ ai0 +λv. Following Lemma 4.1(b) and (c) this implies ai ≤ (1 + ε)ai0 + ε(1 + λ)v

and

ai0 ≤ (1 + ε)ai + ε(2 + λ)v

for all i ≥ i0 . This shows in particular that (ai )i∈I is bounded below and also that ai ≤ (1 + ε)2 ak + 3ε(2 + λ)v for all i, k ≥ i0 . This shows lim ai ≤ (1 + ε)2 lim ak + 3ε(2 + λ)v. i∈I

k∈I

As this holds for all v ∈ V and 0 < ε ≤ 1, and as P carries the weak preorder which is supposed to be antisymmetric, we infer that limi∈I ai = limk∈I ak , hence order convergence towards an element a ∈ P. Moreover, the above shows that ai ≤ (1 + ε2 )a + 3ε(2 + λ)v

and

a ≤ (1 + ε2 )ai + 3ε(2 + λ)v

holds for all i ≥ i0 . Thus the net (ai )i∈I converges to a in the symmetric relative topology as well.

In fact, we just veriﬁed that every Cauchy net, hence every convergent net in the symmetric relative topology of (P, V) is indeed order convergent with the same limit. We shall formulate this as a separate proposition: Proposition 5.42. Let (P, V) be a locally convex complete lattice cone. Convergence of a net (ai )i∈I in P towards a ∈ P in the symmetric relative topology implies order convergence towards a.

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91

While convergence in the symmetric relative topology implies order convergence, the converse is not necessarily true, as a simple example can show: In the locally convex complete lattice cone R order convergence means convergence in the usual (one-point compactiﬁcation) topology of R which for the element +∞ does not coincide with the symmetric relative topology of R. The sequence (n)n∈N , for example, is order convergent towards +∞ ∈ R, but does not converge in the symmetric relative topology, as +∞ is an isolated point in this topology. 5.43 Order Topology. While order convergence in a locally convex complete lattice cone (P, V) does not necessarily correspond to a topology on P in the sense that order and topological convergence for nets coincide (see 1.1.9 in [132]), there is a ﬁnest topology O(P) on P with the following properties: (OT1) Every very element of P admits a basis of both convex and order convex neighborhoods. The neighborhoods in the basis for 0 ∈ P are also balanced. (OT2) The mappings (a, b) → a + b, (a, b) → a ∨ b and (a, b) → a ∧ b from P 2 into P are continuous. (OT3) The mapping (α, a) → αa : [0, +∞) × P → P is continuous at all points (α, a) ∈ [0, +∞) × P such that either α > 0 or a ∈ P is bounded. (OT4) All almost order convergent nets in P are topologically convergent with the same limit. Indeed, let T be the family of all topologies on P with these properties. These topologies need not be Hausdorﬀ. Therefore T is not empty as it contains the discrete topology. Let O(P) be the supremum of this family in the lattice of topologies on P. A neighborhood basis in O(P) for a point a ∈ P is generated by the intersections of ﬁnitely many neighborhoods for a taken from topologies in T. This shows that O(P) again satisﬁes (OT1) to (OT4), hence is the ﬁnest topology with these properties. We shall call O(P) the (strong) order topology on P. Note that O(P) is not necessarily a locally convex cone topology. For P = R, for example, the order topology is the usual topology of R where +∞ is not an isolated point. In Proposition 4.2 we veriﬁed that the symmetric relative topology of P satisﬁes (OT1), (OT2) and (OT3), however it does not meet (OT4) in general. Proposition 5.44. Let (P, V) be a locally convex complete lattice cone. The order topology O(P) on P is coarser than the symmetric relative topology. Proof. We observed in Proposition 5.42 that convergence for a net in the symmetric relative topology implies order convergence, hence convergence in O(P). Since the closure in any topology of a given subset A of P can be described as the set of all limit points of convergent nets in this subset, Proposition 5.42 implies that the closure of A with respect to the symmetric

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relative topology is contained in the closure of A with respect to O(P). We infer that O(P) is generally coarser than the symmetric relative topology. Lemma 5.45. Let (P, V) be a locally convex complete lattice cone and let P0 be the subcone of all invertible elements of P. The mapping (α, a) → αa : R × P0 → P0 is continuous with respect to the order topology O(P). Proof. We shall make this argument in several short steps: First suppose that ai → 0 for ai ∈ P0 in any topology satisfying (OT1) to (OT4). Given a neighborhood U in the basis for 0 there is i0 such that ai ∈ U for all i ≥ i0 . This implies −ai ∈ U as well since U is supposed to be balanced by that ai → a for ai , a ∈ P0 . Then (OT1). Thus (−ai ) → 0. Next suppose ai + (−a) → 0 by (OT2), hence (−ai ) + a → 0 by the preceding step, and (−ai ) → (−a) by (OT2). In a third step, suppose that αi → α ∈ R for 0 ≤ αi ∈ R and ai → a for ai , a ∈ P0 . Then αi ai → αa by (OT3) since every invertible element is bounded. Now in the fourth and ﬁnal step of our argument, let αi → α in R and ai → a for ai , a ∈ P0 . Let βi = αi ∨ 0 and γi = −(αi ∧ 0). Then βi , γi ≥ 0 and αi = βi − γi . We have βi ai → βa and γi (−ai ) → γ(−a), where β = α ∨ 0 and γ = −(α ∧ 0), by the second and third steps of our argument. Thus αi ai = βi ai + γi (−ai ) → βa + γ(−a) = αa, again by (OT2), as claimed.

Proposition 5.46. Let (P, V) and (Q, W) be a locally convex complete lattice cones. An order continuous lattice homomorphism T : P → Q is also continuous with respect to the respective order topologies O(P) and O(Q). Proof. Let T : P → Q be an order continuous lattice homomorphism, consider the order topology O(Q) on Q and let τ be the initial topology induced on P by T, that is τ is the coarsest topology on P for which the mapping T : P → Q is continuous. The sets in τ then are just the inverse images under T of the sets in O(Q). It is straightforward to verify that τ satisﬁes the requirements (OT1) to (OT4): For a ∈ P the element T (a) ∈ Q admits a basis of neighborhoods in O(Q) satisfying (OT1). Their inverse images under T have the same properties and form a neighborhood basis for a in τ. Next suppose that ai → a and bi → b in τ. Then T (ai ) → T (a) and T (bi ) → T (b) in O(Q). Thus T (ai ) + T (bi ) → T (a) + T (b) = T (a + b) since O(Q) satisﬁes (OT2). Because every neighborhood of a + b in τ is the inverse image under T of a neighborhood of T (a + b), this shows that (ai + bi ) → (a + b) in τ. Similarly one veriﬁes the continuity of the mappings (a, b) → a ∨ b, (a, b) → a ∧ b and (α, a) → αa with respect to τ. For (OT4) let (ai )i∈I be an almost order convergent net in P with limit a ∈ P. Then T (a) = T lim lim (ai ∨ ak ) = lim lim T (ai ) ∨ T (ak ) k∈I

i∈I

k∈I

i∈I

5. Locally Convex Lattice Cones

93

since T is an order continuous lattice homomorphism. The net T (ai ) i∈I is therefore almost order convergent with limit T (a) in Q. As O(Q) satisﬁes (OT4), this implies T (ai ) → T (a) in O(Q), and therefore ai → a in τ, since the neighborhoods of a in τ are inverse images under T of neighborhoods of T (a) in O(Q). Summarizing, we have veriﬁed that the topology τ on P satisﬁes conditions (OT1) to (OT4) and is therefore coarser then the order topology O(P). Hence the operator T : P → Q is also continuous if we endow P with O(P). Proposition 5.47. Let (P, V) be a locally convex complete lattice cone and let N be a subcone of P. Then the closure N of N with respect to O(P) is again a subcone of P. If N is a lattice subcone of P, then N is a complete lattice subcone of P. Proof. The ﬁrst part of the claim follows directly from (OT2) and (OT3). For the second part suppose that N is a lattice subcone of P and let a, b ∈ N . There are nets (ai )i∈I and (bj )j∈J in N converging in the order topology towards a and b, respectively. Then the net (ai ∨ bj )(i,j)∈I×J in N converges to a ∨ b by (OT3). Thus a ∨ b ∈ N . Similarly one shows that a ∧ b ∈ N , hence N is also a lattice subcone of P. Now let A be a non-empty subset of N . For every ﬁnite subset i = {a1 , . . . , an } of A set ai = a1 ∨ . . . ∨ an ∈ N . Then sup A = limi∈I ai , where I is the collection of all ﬁnite subsets of A , ordered by set inclusion. This shows sup A ∈ N by (OT4). Similarly one shows that inf B ∈ N whenever B is a bounded below subset of N . Thus N is indeed a complete lattice subcone of P. Proposition 5.48. Let (P, V) be a locally convex complete lattice cone and let N be a complete lattice subcone of P. The restriction of O(P) to N is coarser than the order topology O(N ) of N . Proof. This follows from the easily veriﬁable fact that the restriction of O(P) to the complete lattice subcone N satisﬁes the requirements (OT1) to (OT4). 5.49 Weak Order Convergence. Weak order convergence for a net (ai )i∈I in a locally convex complete lattice cone (P, V) means that μ(ai ) i∈I converges towards μ(a) in R (with respect to order convergence) for every order continuous lattice homomorphism μ ∈ P ∗ . This notion of convergence results from the weak order topology o(P, P ∗ ) on P which is generated by the (both convex and order convex) neighborhoods

|μ (b) − μ (a)|≤ 1 , if μ (a) < +∞ i i , VΥo (a) = b ∈ P i μi (b)≥ 1 , if μi (a) = +∞ for an element a ∈ P, corresponding to a ﬁnite set Υ = {μ1 , . . . , μn } of order continuous lattice homomorphisms in P ∗ . Like the order topology O(P), this is in general not a locally convex cone topology.

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Proposition 5.50. Let (P, V) be a locally convex complete lattice cone. The weak order topology o(P, P ∗ ) on P is coarser than the order topology O(P) and also coarser than the weak topology σ(P, P ∗ ). Proof. Requirements (OT1) to (OT4) from 5.40 are readily checked for the weak order topology: (OT1) and the ﬁrst part of (OT2) are self evident. The second part of (OT2) follows from the easily veriﬁed fact that μ(a ∨ b ) ≤ μ(a ∨ b) + 1 holds whenever μ(a ) ≤ μ(a) + 1 and μ(b ) ≤ μ(b) + 1 for elements a, a , b, b ∈ P and an order continuous lattice homomorphism μ ∈ P ∗ . Similarly one argues for the third part of (OT2). For (OT4) let (ai )i∈I be an almost order convergent net in P with limit a ∈ P. Then μ(a) = μ lim lim (ai ∨ ak ) = lim lim μ(ai ) ∨ μ(ak ) k∈I

i∈I

k∈I

i∈I

for every order continuous lattice homomorphism μ ∈ P ∗ . The limit on the right-hand side is taken with respect to the usual (that is the order) topology of R. The net (ai )i∈I is therefore also convergent with respect to the weak order topology. We infer that o(P, P ∗ ) is generally coarser than the order topology O(P). The second statement of Proposition 5.50 follows immediately from a comparison of the respective neighborhoods in 4.6 and in 5.49: For a ∈ P and a ﬁnite set Υ = {μ1 , . . . , μn } of order continuous lattice homomorphisms in P ∗ we have VΥs (a) ⊂ VΥo (a). Thus σ(P, P ∗ ) is indeed ﬁner than o(P, P ∗ ). Proposition 5.51. Let (P, V) be a locally convex complete lattice cone, and suppose that the order continuous lattice homomorphisms support the separation property for P. Then the order and the weak order topologies coincide on P and are Hausdorﬀ. A net in P is convergent in the (weak) order topology if and only if it is almost order convergent. Proof. Let (P, V) be a locally convex complete lattice cone such that the order continuous lattice homomorphisms support the separation property for P. Let us ﬁst argue that the weak order topology is Hausdorﬀ. Indeed, for distinct elements a, b ∈ P we have either a b or b a, since the order of P is supposed to be antisymmetric. Thus a b + v or b a + v for some v ∈ V by Lemma 3.2. Then there exists an order continuous linear functional μ ∈ v ◦ such that μ(a) > μ(b) + 1 or μ(b) > μ(a) + 1, respectively. For o o (a) and V{δμ} (b) are seen to suitable ε, δ > 0 then the neighborhoods V{εμ} be disjoint. Next we shall verify the last statement of our claim: Let (ai )i∈I be a net in P. If (ai )i∈I is almost order convergent, then it is convergent with the same limit in O(P) by (OT4), hence weakly order convergent since the weak order topology is coarser than O(P). For the converse suppose that (ai )i∈I is weakly order convergent toward a ∈ P. Then for every b ∈ P and every order continuous lattice homomorphism μ ∈ P ∗ we have

5. Locally Convex Lattice Cones

95

μ lim(ai ∨ b) = lim μ(ai ) ∨ μ(b) i∈I i∈I = lim μ(ai ) ∨ μ(b) i∈I

= μ(a) ∨ μ(b) = μ(a ∨ b). This shows limi∈I (ai ∨ b) = (a ∨ b) since the weak order topology was seen to be Hausdorﬀ. Similarly one veriﬁes that limi∈I (ai ∨ b) = (a ∨ b), hence lim(ai ∨ b) = (a ∨ b). i∈I

For b = ak in particular, this renders limi∈I (ai ∨ ak ) = (a ∨ ak ) for every k ∈ I. Repeating this argument with b = a and ak in place of ai then yields lim(a ∨ ak ) = (a ∨ a) = a. k∈I

We thus veriﬁed that the net (ai )i∈I is almost order convergent towards a ∈ P. This completes our argument for convergent nets and also implies the ﬁrst part of our claim. Indeed, since the closed sets in any given topology can be described in terms of limits of convergence nets alone, having the same notion of convergence for nets means that the topologies involved coincide. Proposition 5.52. Let (P, V) be a locally convex complete lattice cone such that the order continuous lattice homomorphisms support the separation property for P, and let N be a complete lattice subcone of P. Then N is closed in O(P). The order topology O(N ) of N coincides with the restriction of O(P) to N . Proof. Let (P, V) be a locally convex complete lattice cone and let N be a complete lattice subcone of P. Because the restriction to N of an order continuous lattice homomorphism on P is an order continuous lattice homomorphism on N , under the assumptions of the Proposition these functionals support the separation property for both P and N . The conclusions of Proposition 5.51 therefore apply to both of these cones. Let (ai )i∈I be a net in N . We observe the following: If (ai )i∈I is almost order convergent as a net in N with limit a ∈ N , then it is also almost order convergent as a net in P with the same limit. Conversely, if (ai )i∈I is almost order convergent in as a net in P with limit a ∈ P, then a ∈ N , and (ai )i∈I is also almost order convergent as a net in N with the same limit. This is an immediate consequence of the fact that the subcone N contains the inﬁma and suprema of its sets as elements, hence the limits of its order convergent nets. Now both of our claims follow, since the convergent nets in the order topologies of P and of N coincide with the almost order convergent nets in P and N , respectively.

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In Proposition 5.37 we established that every locally convex lattice cone can be represented as a as a cone of R-valued functions on some set X. The preceding considerations now allow us to identify the weak and strong order topologies as the topology of pointwise convergence in this representation. be a complete lattice subcone of F (X, R), V Proposition 5.53. Let (P, V) Vb consisting of R-valued funcfor some set X and a neighborhood system V Then the tions such that vˆ(x) < +∞ for every x ∈ X with some vˆ ∈ V. order topology, the weak order topology and the topology of pointwise convergence on X (with respect to the usual topology of R ) all coincide on P and are Hausdorﬀ. Proof. Under the assumptions of the Proposition, the order continuous lat- tice homomorphisms support the separation property for FVb (X, R), V The cosee 5.33(f) , hence also for the complete lattice subcone (P, V). incidence of the order and the weak order topology was established in Proposition 5.51. Since for every x ∈ X the point evaluation f → f (x) is an order continuous lattice homomorphism on P see 5.31(d) , week order convergence for a net in P implies pointwise convergence on X. A pointwise convergent net, on the other hand is seen to be almost order convergent and therefore convergent in the order topology. The three notions of convergence, hence the respective topologies therefore coincide. 5.54 Extension of Linear Operators. A short inspection of the HahnBanach type extension results for linear functionals in [172] (see also Section 2) shows that they are still valid if the range R for the functionals is replaced by some locally convex cone (Q, W), provided that (i) (Q, W) is full and a complete lattice cone, (ii) all elements of Q, with the exception of the element +∞ = sup Q, are invertible, (iii) the neighborhood system W consists of all (strictly) positive multiples of a single neighborhood w ∈ W. Requirement (ii) means of course that Q is a Dedekind complete Riesz space with an adjoint maximal element +∞. Results about the extension of monotone linear operators between vector spaces and Dedekind complete Riesz spaces are due to Kantoroviˇc [96] and [98] and can for example be found in Section 1.5 of [132]. Without furnishing the details of this, we reformulate Corollary 4.1 in [172] (see also Corollary 2.7 before). Theorem 5.55. Let (N , V) be a subcone of the locally convex cone (P, V). Suppose that (Q, W) is a full locally convex complete lattice cone, that all elements of Q other than +∞ are invertible, and that W = {αw | α > 0} for some w ∈ W. Then every continuous linear operator T : N → Q can be extended to a continuous linear operator T : P → Q.

5. Locally Convex Lattice Cones

97

Unfortunately, a similar result is not generally available if the locally convex complete lattice cone (Q, W) does not meet the stringent additional requirements of Theorem 5.55. However, we have the following: Theorem 5.56. Let N be a subcone of the locally convex cone (P, V) and let (Q, W) be a locally convex complete lattice cone. Every continuous linear operator T : N → Q can be uniquely extended to N , the closure of N in P with respect to the symmetric relative topology. Proof. Let T : N → Q be a continuous linear operator and let a ∈ N . There is a net (ai )i∈I in N converging to a in the symmetric relative topology. Given w ∈ W and ε > 0 there is v ∈ V such that T (b) ≤ T (c) + w whenever b ≤ c + v for b, c ∈ N . Because (ai )i∈I is a Cauchy net in N , there is i0 ∈ I such that ai ∈ vε (ak ) for all i, k ≥ i0 . This implies T (ai ) ∈ wε T (ak ) for all i, k ≥ i0 , hence T (ai ) i∈I is a Cauchy net in Q as well. Proposition 5.41 shows that this net converges in Q. Moreover, if (bj )j∈J is a second net in N converging toward the same element a, given w ∈ W and ε > 0 we choose v ∈ V as above and ﬁnd i0 ∈ I and j0 ∈ J such that both ai ∈ vε (bj ) and bj ∈ vε (ai ), hence T (ai ) ∈ wε T (bj ) and T (bj ) ∈ vε T (ai ) , for all i ≥ i0 and j ≥ j0 . This shows that both nets T (ai ) i∈I and T (bj ) j∈J have the same limit in Q which we denote T (a). It is now straightforward to verify that this procedure results in a bounded linear extension T : N → Q of the operator T. Uniqueness of this extension is obvious. 5.57 The Standard Lattice Completion of a Locally Convex Cone. We proceed to establish that every locally convex cone (P, V) can be canonically embedded into a locally convex complete lattice cone. For this, we use a representation for (P, V) as a cone of R-valued functions on its dual cone P ∗ , in analogy to the construction that we employed in the proof of Proposition 5.37: With the element a ∈ P we associate the R-valued function ϕa on P ∗ such that ϕa (μ) = μ(a)

for all

μ ∈ P ∗.

The mapping Φ : P → F(P ∗ , R) such that Φ(a) = ϕa for all a ∈ P is linear and monotone. Corresponding to the neighborhoods v ∈ V we consider the R-valued functions Φ(v) = ψv on P ∗ such that ψv (μ) = inf{α > 0 | μ ∈ αv ◦ } = {ˆ are v | v ∈ V}. The neighborhoods vˆ ∈ V for all μ ∈ P ∗ and denote V ∗ deﬁned for F(P , R) by f ≤ g + vˆ

if

f (μ) ≤ g(μ) + ψv (μ)

for all μ ∈ P ∗

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We have ψ(αv) = αψv see 1.4(e) for functions f, g ∈ F(P ∗ , R) and vˆ ∈ V. and ψ(u+v) ≥ ψu + ψv for all u, v ∈ V and α > 0 (see the proof of 5.37 for is a locally convex cone in the sense of 1.4(e), details). Thus FVb (P ∗ , R), V and a complete lattice since ψv (μ) < +∞ holds for all μ ∈ P ∗ with some We claim that Φ(P) ⊂ F (P ∗ , R) and that vˆ ∈ V. Vb ab+v

if and only if

ϕa ≤ ϕb + ψv .

holds for a, b ∈ P and v ∈ V. Indeed, suppose that a b + v. Then for every μ ∈ P ∗ and α > 0 such that μ ∈ αv ◦ we have μ(a) ≤ μ(b) + α, that is ϕa (μ) ≤ ϕb (μ) + ψv (μ), hence ϕa ≤ ϕb + ψv . Conversely, if a b + v, then following Corollary 4.34 there is μ ∈ v ◦ ⊂ P ∗ such that μ(a) > μ(b)+1. The former implies ψv (μ) ≤ 1, hence ϕa (μ) > ϕb (μ) + ψv (μ) and therefore ϕa ϕb + ψv . We infer in particular that the functions Φ(a) = ϕa are contained in FVb (P ∗ , R) for all a ∈ P. Indeed, given a ∈ P and v ∈ V there is λ ≥ 0 such that 0 a + λv, hence 0 ≤ Φ(a) + λˆ v . Therefore the element Φ(a) is contained in FVb (P ∗ , R) as claimed. Finally we establish that the linear operator Φ : P → FVb (P ∗ , R) is an embedding in the sense , provided that of 2.2 of the locally convex cone (P, V) into FVb (P ∗ , R), V we consider (P, V) in its weak preorder. Indeed, we set Φ(v) = vˆ for v ∈ V towards the extension . Φ : (P ∪ V) :→ FVb (Υ, R) ∪ V and by the above a b + v holds for a, b ∈ P and v ∈ V Then Φ(V) = V, if and only if Φ(a) ≤ Φ(b) + Φ(v), as required in 2.2. the smallest locally convex complete lattice subFinally, we denote by P ∗ cone of FVb (P , R) that contains the embedding Φ(P) of P. proposition 5.36 consists of all functions in F (P ∗ , R) that can be expressed speciﬁes that P Vb in the following way: ϕA = inf sup Φ(A) | A ∈ A . where A is family of subsets of P such that Φ(A) = {Φ(A) | A ∈ A} is sup V) the standard lattice bounded below in FVb (P ∗ , R) (see 5.36). We call (P, completion of the locally convex cone (P, V). According to Proposition 5.52, of F (P ∗ , R) is closed in the order topology O F (P ∗ , R) , the subcone P Vb Vb According coincides with the restriction of O F (P ∗ , R) to P. and O(P) Vb to Proposition 5.53 the order topology, the weak order topology and the and are Hausdorﬀ. topology of pointwise convergence on X all coincide on P is Moreover, the restriction of this topology to the subcone Φ(P) of P ∗ generally coarser than the image of the weak topology σ(P, P ) see 4.6 is the dual cone on Φ(P). Indeed, while the domain of the functions ϕ ∈ P P ∗ of P, pointwise convergence for a net (ϕai )i∈I of R-valued functions in

5. Locally Convex Lattice Cones

99

Φ(P) is treated diﬀerently from weak convergence for the corresponding net (ai )i∈I in P if the function value +∞ ∈ R is involved. The order topology of R , which is used for pointwise convergence of the functions is coarser than the given (locally convex cone) topology of R at this point (see 4.6 and 5.40). However, if all elements of P are bounded, that is for example in the case of a vector space, then continuous linear functionals take only ﬁnite values on P, hence the elements of Φ(P) take only ﬁnite values as functions on P ∗ . In this case the order topology, the weak order topology, the weak topology and the topology of pointwise convergence all coincide on Φ(P). The embedding of a locally convex cone (P, V) into some locally convex complete lattice cone is of course not unique. However, the standard lattice V) of (P, V) is distinguished by the fact that every contincompletion (P, uous linear operator from P into some locally convex complete lattice cone (Q, W) can be extended to an order continuous lattice homomorphism from V) into (Q, W). More precisely: (P, Proposition 5.58. Let (P, V) be a locally convex cone, and let Φ be the Let (Q, W) canonical embedding of P into its standard lattice completion P. be a locally convex complete lattice cone such that the order continuous lattice homomorphisms support the separation property for Q. For every continuous linear operator T : P → Q there exists an order continuous lattice homo → Q such that T = T ◦ Φ. Moreover, if v ∈ V and morphism T : P w ∈ W are such that a ≤ b + v implies T (a) ≤ T (b) + w for a, b ∈ P, then ϕ ≤ ψ + Φ(v) implies T(ϕ) ≤ T(ψ) + w for ϕ, ψ ∈ P. Proof. Let (P, V), (Q, W) and T : P → Q be as stated. The adjoint op erator T ∗ : Q∗ → P ∗ is deﬁned as follows see II.2.15 in [100]: For any ν ∈ Q∗ deﬁne the linear functional T ∗ (ν) on P by T ∗ (ν) a = ν T (a) for all a ∈ P. More precisely: If ν ∈ w◦ for some w ∈ W and if v ∈ V is such that T (a) ≤ T (b) + w whenever a ≤ b + v for a, b ∈ P, then T ∗ (ν) ∈ v ◦ . Indeed, a ≤ b + v for a, b ∈ P implies that T ∗ ν (a) = ν T (a) ≤ ν T (b) + 1 = T ∗ ν (b) + 1. Now let A be a family of subsets of P such that Φ(A) = {Φ(A) | A ∈ A} is sup-bounded below in FVb (P ∗ , R). We claim that the family T (A) = {T (A) | A ∈ A} is sup-bounded below in Q : Indeed, given w ∈ W there is v ∈ V is λ ≥ such that T ∗ (w◦ ) ⊂ v ◦ . There 0 such that 0 ≤ sup Φ(A) + λv for all A ∈ A. This means μ sup Φ(A) ≥ −λ for all μ ∈ v ◦ . For an order continuous lattice homomorphism ν ∈ w◦ set μ = T ∗ (ν) ∈ v ◦ . Then for A ∈ A we have ν sup T (A) = sup ν (T (A) = sup μ(A) = μ sup Φ(A) ≥ −λ. This shows 0 ≤ sup T (A) + λv, since the order continuous lattice homomorphisms are supposed to support the separation property for Q. Our claim has

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I Locally Convex Cones

and ϕ therefore been veriﬁed. Now consider the elements of ϕA ∈ P %A ∈ Q deﬁned as ϕA = inf sup Φ(A) | A ∈ A and ϕ %A = inf sup T (A) | A ∈ A , For every μ ∈ P ∗ , where Φ denotes the canonical embedding of P into P. that is the domain of the functions in P, and for every order continuous lattice homomorphism ν ∈ Q∗ we calculate ϕA (μ) = inf sup Φ(A) | A ∈ A (μ) = inf sup μ(A) | A ∈ A and ν(ϕ %A ) = sup inf ν T (A) | A ∈ A = sup inf T ∗ (ν) A | A ∈ A = ϕA (T ∗ (ν) . Thus, if w ∈ W, if v ∈ V is such that T (a) ≤ T (b) + w whenever a ≤ b + v for a, b ∈ P, and if ϕA ≤ ϕB + Φ(v) for such families A and B of subsets of P, then ν(ϕ %A ) = ϕA (T ∗ (ν) ≤ ϕB (T ∗ (ν) + ψv T ∗ (ν) ≤ ν(ϕ %B ) + 1 ◦ ∗ holds for all order continuous ∗ lattice homomorphisms ν ∈ w , since T (ν) ∈ ◦ v implies that ψv T (ν) ≤ 1. This shows

ϕ %A ≤ ϕ %B + w, since these functionals are supposed to support the separation property for Q. In particular, we infer that ϕ %A = ϕ %B whenever ϕA = ϕB . This fol and Q carry their respective weak lows from the fact that both cones P preorders, which are supposed to be antisymmetric. We are now prepared → Q. For a family A of subsets of P such to deﬁne the operator T : P that Φ(A) = {Φ(A) | A ∈ A} is sup-bounded below in FVb (P ∗ , R) and the we set corresponding element ϕA ∈ P %A T ϕA = ϕ and observe the following: (i) The operator T is linear. Indeed, we note that Φ(αA) = αΦ(A) and T (αA) = αT (A) as well as Φ(A+B) = Φ(A)+Φ(B) and T (A+B) = T (A)+ T (B) holds for any such families A and B of subsets of P and α ≥ 0. Then the arguments in Parts (i) and (ii) of the proof for Proposition 5.36 yield that ϕA + ϕB = ϕ(A+B)

and

ϕ %A + ϕ %B = ϕ %(A+B) .

5. Locally Convex Lattice Cones

101

Thus %(A+B) = ϕ %A + ϕ %B = T ϕA + T ϕB . T ϕA + ϕB = T ϕ(A+B) = ϕ Likewise, αϕA = ϕαA and αϕ %A = ϕ %αA yields T αϕA = αT ϕA for all α ≥ 0. (ii) We observed before that, given w ∈ W and v ∈ V such that a ≤ b+v for a, b ∈ P implies T (a) ≤ T (b)+w, then ϕA ≤ ϕB +Φ(v) for ϕA , ϕB ∈ P implies T(ϕA ) = ϕ %A ≤ ϕ %B + w = T(ϕB ) + w. This entails continuity for the operator T. and ϕ %A = (iii) Let a ∈ P and set A = {a} . Then ϕA = Φ(a) ∈ P T (a). Thus T ◦ Φ (a) = T(ϕA ) = ϕ %A = T (a). This shows T = T ◦ Φ. (iv) Let {Ai }i∈I be a collection of such families Ai of subsets of P. In a ﬁrst instance, suppose that this collection is not empty, and let A = {∪i∈I Ai | the type (Ai )i∈I ∈ i∈I Ai }, that is the elements A of A are all unions of = {∪ Φ(A ) | (A ) ∈ A = ∪i∈I Ai , where Ai ∈ Ai . Then Φ(A) i∈I i i i∈I i∈I Ai } and T (A) = {∪i∈I T (Ai ) | (Ai )i∈I ∈ i∈I Ai }. Therefore Part (iii) in the proof for Proposition 5.36 yields sup ϕAi = ϕA

sup ϕ %Ai = ϕ %A .

and

i∈I

This shows

i∈I

T sup ϕAi = sup T ϕAi . i∈I

i∈I

In a second instance, suppose that the set {aAi }i∈I is bounded below in P, and let A = ∪i∈I Ai . Then Φ(A) = {∪i∈I Φ(Ai ) | i ∈ I} and T (A) = {∪i∈I T (Ai ) | i ∈ I}. Part (iv) in the proof for Proposition 5.36 yields inf ϕAi = ϕA i∈I

Thus

inf ϕ %Ai = ϕ %A .

and

i∈I

T inf ϕAi = inf T ϕAi i∈I

i∈I

→ Q is therefore an order continuous holds as well. The operator T : P lattice homomorphism. For Q = R in particular, Proposition 5.38 states that for v ∈ V and ◦ every linear functional μ ∈◦ v on P there is an order continuous lattice such that μ = μ ˆ ◦ Φ. We proceed to homomorphism μ ˆ ∈ Φ(v) on P V) of a locally convex demonstrate that the standard lattice completion (P,

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I Locally Convex Cones

cone (P, V) is indeed the only (up to embedding) locally convex complete lattice cone which contains an embedding of P and satisﬁes this property. % V) % be a loProposition 5.59. Let (P, V) be a locally convex cone, let (P, cally convex complete lattice cone such that the order continuous lattice ho% Suppose that there is momorphisms support the separation property for P. % an embedding Ψ : P → P with respect to the weak preorder of P and that ◦ for every v ∈ V and every linear functional μ◦ ∈ v on P there is an order % such that μ = μ ˜ ◦ Ψ. continuous lattice homomorphism μ ˜ ∈ Ψ(v) on P : P → P, % where (P, V) denotes the Then there exists an embedding Ψ standard lattice completion of (P, V). This embedding preserves the lattice and P. % operations for P Proof. We shall use the notations from the proof of the preceding proposition, the canonical embedding of (P, V) in particular we denote by Φ : P → P V). Now suppose that the linear into its standard lattice completion (P, % is also an embedding in the sense of 2.2, that is there operator Ψ : P → P is an extension % ∪ V) % Ψ : (P ∪ V) → (P with the required properties. According to Proposition 5.58 there exists an :P →P % such that Ψ = Ψ ◦Φ order continuous lattice homomorphism Ψ and ϕA ≤ ϕB + Φ(v)

implies that

A ) ≤ Ψ(ϕ B ) + Ψ(v) Ψ(ϕ

and v ∈ V. As before we abbreviate vˆ = Φ(v) ∈ V for for ϕA , ϕB ∈ P v ∈ V and use this notation for the extension : (P ∪ V) → (P % ∪ V) % Ψ v ) = Ψ(v) for all v ∈ V. Clearly Ψ( V) = Ψ(V) = V, % and rewriting setting Ψ(ˆ the above yields that ϕA ≤ ϕB + vˆ

implies that

A ) ≤ Ψ(ϕ B ) + Ψ(ˆ v) Ψ(ϕ

and vˆ ∈ V. All left to verify for Ψ : P → P % to be an for ϕA , ϕB ∈ P embedding is the reverse implication in the above. For this, suppose that A ) ≤ Ψ(ϕ B ) + Ψ(ˆ v ) and let μ ∈ v ◦ . Following our assumption there Ψ(ϕ ◦ % such that is an order continuous lattice homomorphism μ ˜ ∈ Ψ(v) on P μ=μ ˜ ◦ Ψ. Then B ) + 1. A) ≤ μ ˜ Ψ(ϕ μ ˜ Ψ(ϕ On the other hand, we have

5. Locally Convex Lattice Cones

103

ϕA (μ) = inf sup μ(A) | A ∈ A = inf sup μ ˜ Ψ(A) | A ∈ A =μ ˜ inf sup Ψ(A) | A ∈ A and Φ(A) | A ∈ A inf sup Ψ(A) | A ∈ A = inf sup Ψ inf sup Φ(A) | A ∈ A =Ψ A) = Ψ(ϕ ◦ Φ by 5.58 and since Ψ : P → P % is an order continuous since Ψ = Ψ A ) and, ˜ Ψ(ϕ lattice homomorphism. Combining the above yields ϕA (μ) = μ B ) . Therefore likewise ϕB (μ) = μ ˜ Ψ(ϕ ϕA (μ) ≤ ϕB (μ) + 1 holds for all μ ∈ v ◦ . Form this we infer that ϕA (μ) ≤ ϕB (μ) + ψv (μ) holds for all μ ∈ P ∗ , and conclude that ϕA ≤ ϕB + vˆ, as claimed. Remarks 5.60. (a) We shall make extensive use of the standard lattice com V) of a locally convex cone (P, V) in the integration theory for pletion (P, cone-valued functions in Chapters II and III. However, many of the results It is will refer only to the order closure of the embedding of P into P. therefore useful to observe that the elements of this order closure can be in be an terpreted as elements of some second dual of P. Indeed, let ϕ ∈ P coinelement of this closure. Since convergence in the order topology of P ∗ cides with pointwise convergence on P , there is a net (ai )i∈I in P such from above converge pointwise to ϕ. Thus for that the functions ϕai ∈ P ∗ μ, ν ∈ P we have ϕ(μ + ν) = lim ϕai (μ + ν) = lim ϕai (μ) + lim ϕai (ν) = ϕ(μ) + ϕ(ν) i∈I

i∈I

i∈I

and μ ∈ P ∗ and by (OT2). Since ϕ(αμ) = αϕ(μ) holds for all ϕ ∈ P α ≥ 0, the function ϕ is an R-valued linear functional on P ∗ , that is an element of P ∗∗ , the dual cone of P ∗ under its ﬁnest locally convex topology which renders all linear functionals on P ∗ continuous see 7.3(i) below . the functional ϕ is bounded below on all Moreover, as an element of P, polars of neighborhoods in V. (b) If (P, V) is a locally convex vector space over K = R or K = C in its symmetric (modular) topology see Example 1.4(d) , that is a locally convex topological vector space, then the dual cone P ∗ of P consist of the real parts μ of all continuous K-linear functionals μK in the vector space dual PK∗ of P see 2.1(d) . Similarly, in 2.1(d) we established a correspondence between real-valued linear (with respect to multiplication by non-negative

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scalars) functionals ϕ on P ∗ and K-valued K-linear functionals ϕK on PK∗ . This correspondence is given by ϕ(μ) = e ϕK (μK ) for all μ ∈ P ∗ , and ϕK (μK ) = ϕ(μ)

or

ϕK (μK ) = ϕ(μ) − i ϕ(iμ)

for all μB ∈ PK∗ in the real or complex case, respectively. If the real-linear that is for example if ϕ is contained functional ϕ on P ∗ is contained in P, then in the order closure of the embedding of P see Part (a) before into P, ◦ ∗ ϕ is bounded below on the polars v ⊂ P of all neighborhoods v ∈ V. Therefore the corresponding K-linear functional ϕK in the second vector space dual PK∗∗ of P is also bounded on all polars of neighborhoods in V. In the case of a normed space (P, ), for example, the latter implies that ϕK is an element of the (strong) second vector space dual of P. (c) For a concrete example to (b) let P = K endowed with the Euclidean topology, that is the neighborhood system V = {εB | ε > 0}, where B is the unit ball in K. The vector space dual PK of K then is of course K itself, which corresponds to the dual cone P ∗ of K as a locally convex cone as elaborated in 2.1(d), that is every z ∈ K deﬁnes a real-linear functional in P ∗ via a → e(za) : K → R. On the other hand, every real-valued linear functional ϕ on P ∗ = K corresponds to an element z ∈ K, that is the second vector space dual of K, by z = ϕ(1)

or

z = ϕ(1) − iϕ(i)

in the real or complex case, respectively. (d) If under the assumptions of (b), (Q, W) is a second locally convex is vector space over K, then we shall say that a linear operator T : Q → P K-linear if (i) T f (μ + ν) =T f (μ) + T f (ν) and (ii) T αf (μ) = T f (αμ) holds for all f ∈ Q, μ, ν ∈ P ∗ and α ∈ K. In this case T corresponds to a K-linear operator T% : Q → PK∗∗ . 5.61 Simpliﬁed Standard Lattice Completion. It is often preferable to realize a lattice completion of a locally convex cone (P, V) as a cone of R-valued functions on a suitable subset of P ∗ rather than on the whole of P ∗ . For this we use a subset Υ of P ∗ which supports the separation property for P in the sense of 5.32. (Following Corollary 4.34 this holds true Υ and V Υ the restrictions to Υ of the for Υ = P ∗ ). Let us denote by P and of the associated neighborhood functions in V. Then functions in P

5. Locally Convex Lattice Cones

105

Υ , V Υ ) is again a full locally convex complete lattice cone. Consider the (P : P → P Υ and its composition Ψ = Ψ ◦ Φ with the restriction map Ψ Υ is an canonical embedding Φ of P into P. We claim that Ψ : P → P embedding of P into PΥ if we consider P in its weak preorder. Indeed, if a ) ≤ Ψ(ϕ b )+ψv holds as well in a b+ψv for a, b ∈ P and v ∈ V, then Ψ(ϕ PΥ , that is Ψ(a) ≤ Ψ(b) + Ψ(v). (We use the earlier notations.) Conversely, if a b + v, then by our assumption there is α ≥ 0 and μ ∈ Υ ∩ αv ◦ such that μ(a) > μ(b) + α. Then ψv (μ) ≤ α, hence ϕa (μ) > ϕb (μ) + ψv (μ). This a ) ≤ Ψ(ϕ b ) + ψv , that is Ψ(a) ≤ Ψ(b) + Ψ(v). Thus Φ : P → P Υ shows Ψ(ϕ is an embedding in the sense of 2.2 as claimed. Since the lattice operations are performed pointwise, we have Ψ(sup A) = sup Ψ(A) for every non-empty subset A of P and Ψ(inf A) = inf Ψ(A) for every non-empty bounded The operator below subset A of P. :P →P Υ Ψ is therefore a surjective order continuous lattice homomorphism, but not is also continuous with necessarily an embedding. Because the operator Ψ and P Υ (Proposition 5.46), the image respect to the order topologies on P of the order closure of Φ(P) in P is contained in the order clounder Ψ is an sure of Ψ(P) in PΥ . According to the preceding Proposition 5.59, Ψ embedding and indeed an isomorphism if and only if for every v ∈ V and every linear functional μ ∈ v ◦ on P there is an order continuous lattice ◦ Υ such that μ = μ on P ˜ ◦ Ψ. This condihomomorphism μ ˜ ∈ Ψ(v) ∗ tion is satisﬁed if for every μ ∈ P there is ν ∈ Υ and α ≥ 0 such that Υ as it μ = αν. In this case the conclusion of Proposition 5.58 applies to P does to P. Υ ) Υ , V We shall at times us such a simpliﬁed standard lattice completion (P → P Υ in order to and the order continuous lattice homomorphism Ψ : P represent and visualize results that were obtained in the standard lattice V) of a locally convex cone (P, V). completion (P, Examples 5.62. In the preceding Examples 5.33 we investigated a range of locally convex cones (P, V) and identiﬁed subsets Υ of the dual cone which support the separation property. All of these choices are suitable for the Υ ). Let us Υ , V construction of a simpliﬁed standard lattice completion (P elaborate on the most important of these situations. (a) For P = R with the usual order and neighborhood system V = {ε ∈ R | ε > 0} the dual cone is P ∗ = {α ∈ R | α ≥ 0}. Therefore the standard of R is the cone of all linear R-valued functions on lattice completion R ∗ P . This can be visualized more easily if we choose the subset Υ = {1} of Υ then coincides with R. P ∗ for the above simpliﬁed construction, since P Following the remark in 5.61, we realize that the standard lattice completion of R is isomorphic to R.

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(b) If (P, ) is a normed vector space see 5.33(b) , then we may choose the dual unit sphere for Υ ⊂ P ∗ . According to our preceding remark, the Υ ) then is isomorphic to the standard lattice comΥ , V lattice completion (P pletion (P, V). Alternatively, we may choose Υ = Ex(B), that is the set of all extreme points of the dual unit ball B in P ∗ . However, the conclusion Υ for the latter choice of Proposition 5.58 does not generally apply to P of Υ. In both cases the lattice completion PΥ of P consists of a cone of R-valued bounded below functions on Υ, endowed with the topology of uniform convergence. (c) For a special case of 5.33(f) consider the locally convex cone FbY (X, R), VY of R-valued functions on a set X endowed with the topology of uniform convergence on the sets in a family Y of subsets of X see 1.4(e) . Then

Y ⊂ FbY (X, R)∗ , Υ = εx x ∈ Y ∈Y

where εx denotes the point evaluation at x ∈ X, supports the separation property for FbY (X, P) see 5.33(f) . The corresponding lattice completion (X, R) of F (X, R) then consists of R-valued bounded below functions F bY

Υ

bY

on Υ, endowed with the topology of uniform convergence. (d) For a special case of (c) let X be a compact set and let P = C(X) be the space of all continuous real-valued functions on X, endowed with the pointwise operations and order. The neighborhood system V consisting of all positive constants generates the topology of uniform convergence. The set Υ of all point evaluations εx for x ∈ X supports the separation property, V) of (P, V) can be realized as a cone of R and the lattice completion (P, valued functions on X. (e) In Section 7 below we shall provide another example, that is cones H(N , M) of linear operators from a cone N into a second cone M, endowed with suitable locally convex cone topologies, where the canonical choice for Υ , M) is a proper subset rather than the whole for a lattice completion H(N dual cone of H(N , M). (f) Let P = K, endowed with the Euclidean topology, that is the neighborhood system V = {εB | ε > 0}, where B is the unit ball in K (see the preceding Remark 5.60(c)). The vector space dual PK of K then is K itself which corresponds to the dual cone P ∗ of K as a locally convex cone as elaborated in 2.1(d) and in 5.60(c). Υ of K For the construction of a simpliﬁed standard lattice completion K we choose Υ = Γ, the unit circle in K. It is straightforward to verify that Υ consists of all bounded below R-valued functions on Γ, endowed with K Υ can the (strictly) positive constants as neighborhoods. A function ϕ ∈ K ∗ be canonically extended to a real-linear functional on all of P = K if and only if it takes only ﬁnite values in R and if

6. Quasi-Full Locally Convex Cones n !

107

αi ϕ(γi ) = 0

i=1

n holds whenever i=1 αi γi = 0 for αi ∈ R and γi ∈ Γ. In the real case, that is for K = R, the latter requires just that ϕ(−1) = −ϕ(1). If the above condition holds, then the corresponding K-linear functional ϕK in the second vector space dual of K, that is K itself, is represented by the number or ϕK = ϕ(1) − iϕ(i) ∈ C ϕK = ϕ(1) ∈ R in the real or in the complex case, respectively.

6. Quasi-Full Locally Convex Cones In Section 1, a locally convex cone (P, V) was deﬁned to be a subcone of a full locally convex cone, inheriting both the order and the algebraic structure from the latter. Using only the convex quasiuniform structure of P (see I.3), a procedure described in Chapter I.5 of [100] allows to recover such a full locally convex cone containing P. However this construction is rather unwieldy and far from unique. In situations like in our upcoming measure and integration theory we shall require more immediate access to a canonically constructed full locally convex cone, containing the given cone of interest. This will be possible for a restricted class of locally convex cones which we shall deﬁne and describe in the following. 6.1 Quasi-Full Locally Convex Cones. In a locally convex cone (P, V) the scalar multiples and sums for neighborhoods in V are not necessarily reﬂected in the corresponding operations for their upper, lower or symmetric neighborhoods as subsets of P. In general we only have α v(a) = αv (αa) and u(a) + v(b) ⊂ u + v (a + b) for u, v ∈ V, a, b ∈ P and α > 0, as well as similar relations for the lower and symmetric neighborhoods. Stronger links for the addition are however desirable in some cases. In this vein, we shall say that a locally convex cone (P, V) is quasi-full if for a, b ∈ P and u, v ∈ V (QF1) a ≤ b + v for a, b ∈ P and v ∈ V if and only if a ≤ b + s for some s ∈ P such that s ≤ v, and (QF2) a ≤ u + v for a ∈ P and u, v ∈ V if and only if a ≤ s + t for some s, t ∈ P such that s ≤ u and t ≤ v.

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These conditions can be reformulated as v(a) = v(0) + a and u + v (0) = ↓ u(0) + v(0) , where ↓A = {b ∈ P | b ≤ a for some a ∈ A } denotes the decreasing hull of a subset A of P. Indeed, the ﬁrst statement is clearly equivalent to (QF1), whereas ↓ u(0) + v(0) ⊂ u + v (0) always holds as the latter set is decreasing. The reverse inclusion is equivalent to (QF2). Obviously, every full cone, that is every locally convex cone that contains its neighborhoods as elements, is quasi-full. Most importantly, every ordered locally convex topological vector space (P, V), where V denotes a basis of balanced convex neighborhoods of the origin, is seen to be a quasi-full locally convex cone in this sense. Recall from Example 1.4(c) that the cone topologies on P are deﬁned for elements a, b ∈ P and V ∈ V by a≤b+V

if

a − b ≤ s for some

s ∈ V.

(QF1) is evident, since s ∈ V implies s ≤ V. For (QF2), let a ≤ (U + V ) for a ∈ P and U, V ∈ V. Then a ≤ s + t for some s ∈ U and t ∈ V, since the addition in V is the usual addition for subsets of P. As s ≤ U and t ≤ V, this yields (QF2). Recall that equality is a possible choice for the order on P. In fact, quasi-full locally convex cones are close to locally convex topological vector spaces in the sense that the neighborhoods of every element a ∈ P are already determined by the neighborhoods of the element 0 ∈ P. The sum of two neighborhoods in V coincides with the usual sum of the corresponding subsets of P, that is u(a) + v(b) = u + v (a + b) and (a)u + (b)v = (a + b) u + v for u, v ∈ V and a, b ∈ P. Another advantage of quasi-full locally convex cones is that for complete lattice structures in the sense of Section 5.4, Condition 2 transfers from zero-neighborhoods to general ones and from individual neighborhoods to their sums, thus needs to be checked only for a subsystem of zeroneighborhoods that span the entire neighborhood system. Indeed, suppose that (P, V) is a quasi-full locally convex cone that contains suprema of nonempty sets, and that Condition 2 holds with b = 0 and the neighborhoods u and v in V, that is for a non-empty subset A ⊂ P, a ≤ v for all a ∈ A implies sup A ≤ v, and a ≤ u for all a ∈ A implies sup A ≤ u. Now if there is b ∈ P such that a ≤ b + (u + v) for all a ∈ A, then a ≤ b + sa + ta for some sa ≤ u and ta ≤ v. Then s = supa∈A sa ≤ u and t = supa∈A ta ≤ v by our assumption on u and v. This shows that a ≤ b+(s+t) for all a ∈ A, hence sup A ≤ b+(s+t) ≤ b+(u+v), as claimed. If (P, V) contains both suprema of non-empty and inﬁma of bounded be low subsets andsatisﬁes 1 , then Condition 2 for some neighborhood v ∈ V implies 2 for the same v. Indeed, suppose that there is b ∈ P such that b ≤ a + v holds for all elements a of some bounded below subset

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A ⊂ P. Then b ≤ a + ta for some ta ≤ v, hence b ≤ a + t for all a ∈ A, where t = supa∈A ta ≤ v. This yields b ≤ inf(A + t) = inf A + t ≤ inf A + v by 1 , hence our claim. 6.2 The Standard Full Extension of a Quasi-Full Cone. We shall construct a canonical embedding of a quasi-full locally convex cone (P, V) into a full locally convex cone in the following manner. Let PV = a ⊕ v | a ∈ P, v ∈ V ∪ {0} . We use the obvious algebraic operations on PV , that is (a ⊕ v) + (b ⊕ u) = (a + b) ⊕ (v + u)

and

α(a ⊕ v) = (αa ⊕ αv)

for a, b ∈ P, u, v ∈ V ∪ {0} and α ≥ 0. The order on PV is deﬁned as a⊕v ≤ b⊕u if c ≤ a + v implies that c ≤ b + u for all c ∈ P. This order relation is reﬂexive, and transitive, as for a, b, c ∈ P and u, v, w ∈ V ∪ {0} such that a ⊕ v ≤ b ⊕ u and b ⊕ u ≤ c ⊕ w, for every d ∈ P such that d ≤ a + v, we have d ≤ b + u, hence d ≤ c + w. Thus a ⊕ v ≤ c ⊕ w. Similarly, one veriﬁes compatibility with the algebraic operations: Compatibility with the multiplication by positive scalars is obvious; for compatibility with the addition, let (a ⊕ v), (b ⊕ u), (c ⊕ w) ∈ PV such that a ⊕ v ≤ b ⊕ u. If d ≤ (a + c) + (v + w), then d ≤ (a + c) + s for some s ≤ v + w by (QF1), and s ≤ s + s for some s ≤ v and s ≤ w by (QF2.) Hence d ≤ (a + s ) + (c + s ). Because a + s ≤ a + v implies that a + s ≤ b + u, we infer that d ≤ (b+c)+(u+w). This shows (a+c)⊕(v+w) ≤ (b+c)⊕(u+w). The embedding a → a ⊕ 0 : P → PV therefore preserves the algebraic operations and the order of P, since a ≤ b holds for elements a, b ∈ P if and only a ⊕ 0 ≤ b ⊕ 0 holds in PV . Moreover, for a neighborhood v ∈ V and a, b ∈ P we have a ≤ b + v in P if and only if a ⊕ 0 ≤ (b ⊕ 0) + (0 ⊕ v) = b ⊕ v holds in PV . We may therefore identify the neighborhoods v ∈ V with the elements 0 ⊕ v in PV . In this way V is embedded into PV as well, and (PV , V) becomes a full locally convex cone, containing (P, V) as a subcone. If a certain neighborhood v ∈ V is already contained in the given cone P, then the above deﬁnition of the order in PV yields that both v ⊕ 0 ≤ 0 ⊕ v and 0 ⊕ v ≤ v ⊕ 0. The elements v ⊕ 0 and 0 ⊕ v are therefore equivalent with respect to the canonical equivalence relation deﬁned by the order on PV . Thus for a full cone P, this extension PV yields only elements that in terms of the order relation are equivalent to existing ones in P. We shall call (PV , V) the standard full extension of the locally convex cone (P, V).

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Theorem 6.3. Let (P, V) be a quasi-full locally convex cone, and let (Q, W) be a locally convex complete lattice cone. Every continuous linear operator T : P → Q can be extended to a continuous linear operator T : PV → Q. Proof. Let (P, V) be quasi-full, (Q, W) a complete lattice cone, and let T : P → Q be a continuous linear operator. Recall from Section 3 that a continuous linear operator between locally convex cones is monotone with respect to the respective weak preorders. Because Q carries its weak preorder, this implies monotonicity with respect to the given orders of P and Q as well. For an element a ⊕ v ∈ PV we deﬁne T (a ⊕ v) = sup T (b) | b ∈ P, b ≤ a + v ∈ Q. Let us ﬁrst check linearity: Clearly T (αa ⊕ αv) = αT (a ⊕ v) for α ≥ 0. For additivity, let (a ⊕ v), (b ⊕ u) ∈ PV . Using Lemma 5.5(a), we infer T (a ⊕ v) + T (b ⊕ u) = sup T (c) | c ∈ P, c ≤ a + v + sup T (d) | d ∈ P, d ≤ b + u = sup T (c + d) | c, d ∈ P, c ≤ a + v, d ≤ b + u ≤ sup T (e) | e ∈ P, e ≤ (a + b) + (v + u) = T (a ⊕ v) + (b ⊕ u) . If on the other hand c ≤ (a + b) + (v + u) for c ∈ P, then c ≤ c + c for some c , c ∈ P such that c ≤ a + v and c ≤ b + v. by (QF1) and (QF2). Thus T (c) ≤ T (c ) + T (c ) ≤ T (a ⊕ v) + T (b ⊕ u). Taking the supremum over all such elements c ≤ (a + b) + (v + u) on the left-hand side yields T (a ⊕ v) + (b ⊕ u) ≤ T (a ⊕ v) + T (b ⊕ u). Next we observe that the operator T is monotone. Indeed, let a ⊕ v ≤ b ⊕ u for (a ⊕ v), (b ⊕ u) ∈ PV , and let c ∈ P such that c ≤ a + v. Then c ≤ b + u by our deﬁnition of the order in PV . This shows T (c) ≤ T (b ⊕ u). Taking the supremum over all such elements c ≤ a + v on the left-hand side yields T (a ⊕ v) ≤ T (b ⊕ u). Finally, for every w ∈ W there is v ∈ V such that a ≤ b + v implies that T (a) ≤ T (b) + w for all a, b ∈ P. Thus T (0 ⊕ v) = sup T (s) | s ∈ P, s ≤ v ≤ w. As (PV , V) is a full locally convex cone, this demonstrates the continuity of the monotone linear operator T : PV → Q. All left to verify is that T is indeed an extension of T if we consider P as a subcone of PV via its

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canonical embedding a → (a ⊕ 0). But this is obvious, as for a ∈ P we have T (a ⊕ 0) = sup T (b) | b ∈ P, b ≤ a = T (a).

Remarks 6.4. Let (P, V) be a (not necessarily quasi-full) locally convex cone satisfying the following condition: (QF*) For every a ∈ P and v ∈ V there is s ∈ P such that s ≤ v and λ ≥ 0 such that 0 ≤ a + λs. In this case we may remodel P into a quasi-full locally convex cone if we deﬁne an alternative neighborhood system V consisting of all families (rv )v∈V , where rv is a non-negative real and rv > 0 for at least one v ∈ V and rv = 0 else. Endowed with componentwise deﬁned algebraic operations and % = P ⊕V0 be the direct sum of P order V0 = V∪0 is an ordered cone. Let P % and V0 . We deﬁne the order on P in the following way: We set a ⊕ r ≤ b ⊕ s for elements a, b ∈ P and r, s ∈ V0 if r ≤ s and if there are elements c1 , . . . , cn ∈ P such that a ≤ b + (c1 + . . . + cn ) and ci ≤ (svi − rvi )vi % V) becomes a full for distinct elements v1 , . . . , vn ∈ V. In this way, (P, locally convex cone. Condition (QF*) in particular guarantees that its elements are bounded below. The neighborhoods u ∈ V may be identiﬁed with the elements r(u) ∈ V such that r(u)u = 1 and r(u)v = 0 else. As a % V), the locally convex cone (P, V) is seen to be quasi-full. subcone of (P, (Conditions (QF1) and (QF2) from 6.1 are implied by our deﬁnition of the neighborhoods in V.) Because a ≤ b ⊕ r(v) implies a ≤ b + v for a, b ∈ P and v ∈ V , the (upper, lower, symmetric) topologies induced on P by V are ∗ generally ﬁner than the given ones. The dual cone PV of P under this new locally convex topology is therefore larger than the given dual cone P ∗ . The polar r∗ of a neighborhood r ∈ V consists of all monotone R-valued linear functionals μ on P satisfying μ(c) ≤ 1 for all c ∈ P such that c ≤ r.

7. Cones of Linear Operators Endowed with the canonical (pointwise) algebraic operations, the linear operators between two cones N and M form again a cone L(N , M). We may introduce neighborhoods for L(N , M) in the following way (for a similar construction in the case of vector spaces see III.3 in [185]): Let W be a neighborhood system and let ≤ be an order for M such that (M, W) is a locally convex cone. Let Z be a family of subsets of N , directed upward by set inclusion. For every Z ∈ Z and w ∈ W we deﬁne a neighborhood V(Z,w) , setting S ≤ T + V(Z,w) for linear operators S, T ∈ L(N , M) if S(a) ≤ T (a) + w

for all a ∈ Z.

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The collection V(Z,W) = V(Z,w) | Z ∈ Z, w ∈ W of these neighborhoods deﬁnes a convex quasiuniform structure on a subcone H(N , M) of L(N , M) in the sense of I.5.3 in [100] provided that its elements are bounded below, that is if for each T ∈ H(N , M) and Z ∈ Z and w ∈ W there is λ ≥ 0 such that 0 ≤ T (a) + λw for all a ∈ Z. It is elaborated in I.5.3 [100] how such a convex quasiuniform structure can be used to construct an abstract neighborhood system V for the cone H(N , M), turning H(N , M), V into a locally convex cone in such a way that the neighborhoods in V(Z,W) form a basis for the neighborhood system V. In fact, all that needs to be done is to deﬁne suitable sums of the elements of V(Z,W) and thus create a cone that can be adjoined to H(N , M). The induced order for H(N , M) is given by S ≤ T for operators S, T ∈ H(N , M) if Z and w ∈ W. S(a) ≤ T (a) + w for all a∈ Z∈Z

Alternatively, if Condition (QF*) from 6.4 holds for the neighborhoods in VZ (with the order from above), then we may use the procedure from 6.4 in order % . As to turn H(N , M) into a quasi-full locally convex cone H(N , M), V % are generally ﬁner than those elaborated in 6.4, the topologies induced by V resulting from V. , M) of H(N , M) (see Remark 7.1. The standard lattice completion H(N 5.57) leads to a rather unwieldy setting in this case. It consists of R-valued functions deﬁned on the dual cone H(N , M)∗ which is diﬃcult to approach and depends on the particular topology of H(N , M), that is the choice for the family Z of subsets of N . It is therefore preferable to employ a simpliﬁed , M)Υ in the sense of 5.61 for which we shall use the lattice completion H(N ∗ ∗ subset Υ = Z∈Z Z ×M of H(N , M) , consisting of all continuous linear functionals whose elements (a, μ) act as linear functionals on H(N , M) as a, μ (T ) = μ T (a) for all T ∈ H(N , M). By our deﬁnition of the neighborhoods in H(N , M), this set Υ supports the separation property. The locally convex cone H(N , M) is therefore em , M)Υ , which in turn permits a more easily accessible rebedded into H(N alization of the lattice completion of H(N , M). In case that the subcone spanned by the sets Z ∈ Z is all of N , we may interpret the elements of , M) as linear the order closure of H(N , M) in its lattice completion H(N ∗∗ operators from N into M , the second dual of M. Indeed, we observed , M) in the order closure of H(N , M) in 5.60(a) that every element ϕ ∈ H(N is a linear functional on H(N , M)∗ . Since Υ = N × M∗ ⊂ H(N , M)∗ as elaborated above, the function ϕ : N × M∗ → R is linear in both arguments from N and from M∗ . Thus the mapping

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113

a → ϕa : N → M∗∗ , where ϕa (μ) = ϕ(a, μ) for μ ∈ M∗ is indeed a linear operator from N into M∗∗ . Moreover, if both N and M are in fact vector spaces over K = R or K = C and if all operators in H(N , M) are K-linear, then a similar argument shows the every function ϕ in the order closure of H(N , M) in , M) can be interpreted as a K-linear operator from N into M∗∗ . H(N Examples 7.2. (a) If both (N , U) and (M, W) are locally convex cones, and if all the sets Z ∈ Z are bounded below in N , then every continuous linear operator from N in to M is bounded below with respect to the neighborhoods in V(Z,W) . Thus, if H(N , M) is a cone of continuous linear operators from N into M, we may consider either of the following: (i) If Z is the family of all bounded below subset of N , we obtain the uniform operator topology for H(N , M). We may alternatively choose the families Z of all bounded or of all relatively bounded subsets of P (see 4.24) in this case. (ii) If Z is the family of all ﬁnite subsets of N , we obtain the strong operator topology for H(N , M). We shall also consider topologies on H(N , M) that arise if M is endowed with an alternative weak topology σ(M, L) generated by a third cone L and a bilinear form on M × L (see II.3 in [100]). In particular: (iii) If Z is the family of all ﬁnite subsets of N , and if M is endowed with the topology σ(M, M∗ ), we obtain the weak operator topology for H(N , M). (iv) If Z is the family of all ﬁnite subsets of N , and if M = L∗ is the dual cone of some locally convex cone (L, V), endowed with the topology σ(L∗ , L), we obtain the weak* operator topology for H(N , L∗ ). (b) If Z consists of the set Z = N , then V = 0 is the only resulting neighborhood for L(N , M), and boundedness from below requires that we consider linear operators that take only positive values on N for the cone H(N , M). The resulting order for operators S, T ∈ H(N , M) is S ≤ T if S(a) ≤ T (a) for all a ∈ N . If on the other hand, Z consists of the set Z = {0}, then V = ∞ is the only resulting neighborhood and the indiscrete topology arises for any subcone H(N , M) of L(N , M). (c) If N = M, then H(M, M) = R+ = {a ∈ R | a ≥ 0} is an example of a cone of linear operators on M, with the scalar multiplication as its operation. If (M, W) is a locally convex cone and if Z is an upward directed family of bounded below subsets of subsets of M, then the neighborhood V(Z,w) in R+ corresponding to some Z ∈ Z and w ∈ W according to the above is given by α ≤ β + V(Z,w) for α, β ∈ R+ if αa ≤ βa + w

for all

a ∈ Z.

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If all elements of the set Z are bounded in M, then this condition can be interpreted as follows: Let δ = inf{λ ≥ 0 | 0 ≤ a + λw for all a ∈ Z} and γ = inf{λ ≥ 0 | a ≤ λw for all a ∈ Z}. A simple argument using the cancellation rule I.4.2 in [100] then yields that the above is equivalent to β ≤α+

1 δ

and

α≤β+

1 . γ

1 (We set of course 10 = +∞ and +∞ = 0 is these expressions.) Thus depending on our choice for Z, one of the following can emerge as the upper neighborhoods V(Z,w) (α) for an element α ∈ R+ : The intervals (for ε > 0) (i) [α − ε, α + ε], yielding the Euclidean topology with equality as order; (ii) [0, α + ε], yielding the upper Euclidean topology with the natural order; (iii) [α − ε, +∞), yielding the lower Euclidean topology with reverse natural order; (iv) [α − ε, α], yielding the equality as order; (v) [0, α], yielding the natural order. Note that only in cases (ii) and (v) the resulting locally convex cone (R+ , V) is quasi-full. If (N , U) is indeed a locally convex topological vector space over K = R or K = C, endowed with its (modular) symmetric topology, then we may also consider H(N , N ) = K. Most useful choices for Z will yield the Euclidean neighborhoods Bε (α) = {β ∈ K | |β − α| ≤ ε} for elements α of K and the equality as order. Alternatively, there may be a subcone C of negative elements in K in this case, and the upper neighborhoods are the sets Bε (α) + C. (d) Every locally convex cone (P, V) can be represented as a locally convex cone of linear operators. Indeed, algebraically, P coincides with the cone H(R+ , P) of all linear operators from R+ = {a ∈ R | a ≥ 0} into P if we identify an element a ∈ P with the operator α → αa in H(R+ , P). The neighborhoods of P may be recovered for H(R+ , P) if we use the above procedure with Z containing only the singleton set {1} ⊂ R+ . We obtain a copy of the locally convex cone (P, V).

7.3 Cones of Linear Functionals. The Second Dual. Let (P, V) be a locally convex cone. In the general settings of this section we choose (N , U) = (P, V) and M = R with its usual neighborhood system W = {ε ∈ R | ε > 0}∗ see 1.4(a) . For the subcone H(P, R) of L(P, R) we choose the dual P of P. As in 7.2(a) let Z be a family of bounded below subsets of P. In this way, (P ∗ , V) becomes a locally convex cone. Its own dual cone, that is the second dual of P, then is well-deﬁned and depends on the choice for the topology of P ∗ , that is on the choice for the family Z of subsets of P. Considering the particular choices for Z as elaborated in 7.2(a) we shall use the following notations for the second dual of a locally convex cone (P, V) : (i) P ∗∗ denotes the cone of all R-valued linear functionals on P ∗ . ∗∗ ∗∗ ∗∗ , Psr and Psl denote the dual of (P ∗ , V) if Z consists of all (ii) Psl (bounded below, relatively bounded) or bounded subsets of P. These

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115

are referred to as the (lower strong, relative strong) or strong second dual of P. (iii) Pw∗∗ denotes the weak second dual of P, that is the dual of (P ∗ , V) if Z consists of all ﬁnite subsets of P. Since every element a ∈ P acts as an R-valued linear functional ϕa on P ∗ , and since this linear functional is obviously contained in the polar of the neighborhood V(Z,1) , where Z = {a} ⊂ P, the given cone P can be envisioned as a subcone of its second dual Pw∗∗ . Indeed, we have ∗∗ ∗∗ P ⊂ Pw∗∗ ⊂ Ps∗∗ ⊂ Psr ⊂ Psl ⊂ P ∗∗

in general. Now let us recall the construction of the standard lattice comple of P from Section 5.57. The elements of P were realized as R-valued tion P ∗ functions on P , and in Remark 5.60(a) we observed that the elements of the are linear on P ∗ . This order closure can therefore order closure of P in P also be considered as a subcone of P ∗∗ . Furthermore, we observe that for every choice of the family Z of bounded below subsets of P the linear functionals in the thus generated second dual PZ∗∗ of P, if considered as R-valued functions on P ∗ , are bounded below on the polars of all neighborhoods in V. Indeed, every functional ϕ in the second dual PZ∗∗ of P is contained in the polar of some neighborhood V(Z,ε) for Z ∈ Z and ε > 0. Given v ∈ V there is λ ≥ 0 such that 0 ≤ z + λv for all z ∈ Z. Then for every μ ∈ v ◦ we have 0 ≤ μ(z) + λ for all z ∈ Z, hence 0 ≤ μ + (λ/ε)V(Z,ε) by the deﬁnition of the neighborhood V(Z,ε) . ◦ , this yields ϕ(μ) ≥ −(λ/ε) for all μ ∈ v ◦ . Consequently, Since ϕ ∈ V(Z,ε) for every such choice of the family Z, the resulting second dual PZ∗∗ of P may be considered as a subcone of the locally convex complete lattice cone of P FVb (P ∗ , R) from 5.57. Recall that the standard lattice completion P had been introduced as the smallest locally convex complete lattice subcone of FVb (P ∗ , R) that contains P (see 5.57). We have P ⊂ PZ∗∗ ⊂ FVb (P ∗ , R) for any such choice of the family Z by the above. and the second dual P ∗∗ are Therefore both the order closure of P in P Z ∗∗ contained in the intersection of P and FVb (P ∗ , R), but it is in general not possible to identify one of these as a subcone of the other. We can, however, add the following often helpful observation: Let Z be is in a bounded below subset of P, and suppose that the element ϕ ∈ P the closure with respect to the order topology of (the embedding of) Z in This means that there is a net (ai )i∈I in Z converging pointwise as P. functions on P ∗ towards ϕ, that is ϕ(μ) = lim μ(ai ) i∈I

for all μ ∈ P ∗ . (Convergence is meant in the usual (order) topology of R.) Then the function ϕ is linear on P ∗ and μ ≤ ν + V(Z,1) for elements μ, ν ∈ P ∗ implies that μ(ai ) ≤ ν(ai ) + 1 holds for all i ∈ I, and therefore

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◦ ϕ(μ) ≤ ϕ(ν) + 1 as well. This shows that ϕ ∈ V(Z,1) . Hence the element ϕ ∈ ∗∗ ∗ is contained in the dual cone P of (P , V) whenever the neighborhood P Z generating family Z contains the bounded below set Z. For a locally convex vector space (P, V) over K = R or K = C the diﬀerent notions in (ii) for the strong second dual coincide, and according to 2.1(d) every real-valued (real) linear functional ϕ ∈ P ∗∗ corresponds canonically to a K-valued K-linear functional ϕK on PK∗ , that is an element of PK∗∗ , the (algebraic) second vector space dual of P. These ﬁnal observations will prove particularly useful in the subsequent chapters when we shall investigate integrals of cone-valued functions.

8. Notes and Remarks The theory of locally convex cones originated in a joint work [100] by the author and K. Keimel in 1992. We were then looking for a suitable setting for the formulation of Korovkin-type approximation theory which deals with certain restricted classes of continuous linear operators on locally convex vector spaces. These may be positive operators on ordered spaces, contractions on normed spaces, multiplicative operators on Banach algebras, etc. Approximation processes are modeled by sequences or nets of operators in such a class. The given restrictions then guarantee convergence towards the identity operator on a large subset of their domain if this property can be checked for a relatively small test set. The use of locally convex cones instead of locally convex vector spaces turns out to be very advantageous in this context, since it allows to formulate all those diﬀerent restriction properties for the operators in terms of the order structure alone, thus yielding a unifying approach. Subsequently the theory of locally convex cones has been expanded, mostly by the author of this book. Readers interested in further aspects of the subject should in particular familiarize themselves with the Hahn-Banach type extension and separation results that were laid out in [172] and form the foundations for the duality theory of locally convex cones. Ordered cones were earlier studied by various authors, in particular Fuchssteiner and Lusky [63] whose book contains a Hahn-Banach type sandwich theorem for R∪{−∞} valued linear functionals on an ordered cone, a non-topological predecessor to the results from [172]. An in-depth investigation for the relationship between order and topology can be found in the seminal work [135] by Nachbin. The compendia of continuous lattices [68] and [69] by Gierz, Hofmann, Keimel, Lawson, Mislove and Scott contain a detailed analysis of various ways to introduce topologies on lattices. The weak (global) preorder as deﬁned in Section 3 has an earlier analogue in the (global) preorder which was deﬁned in Section I.3 of [100] for elements a, b ∈ P as follows:

8. Notes and Remarks

117

ab

if

a≤b+v

for all v ∈ V. Clearly a ≤ b implies a b which in turn implies that a b. In some sense the preorder can be considered as a topological closure of the given order ≤ whereas the weak preorder signiﬁes a closure with respect to both topology and the linear structure. Like for the weak preorder there is also a local version v of the preorder see I.3 in [100] referring to a particular neighborhood v ∈ V rather than the whole neighborhood system. Relationships between the diﬀerent orders of a locally convex cone are investigated in detail in [175]. Since it provides the separation properties from Section 4, the weak preorder turns out to be the most suitable one for our purposes. An excellent historical account of the extensive literature on ordered topological vector spaces can be found in the classical book by Day [39]. The notions of order convergence and of order topology for complete locally convex lattice cones from Section 5 are also used in ordered vector spaces, but introduced in a slightly diﬀerent way which does not require a given topological or lattice structure see Chapter V.6 in [185] . However, on topological vector lattices this notion coincides with ours form Section 5. Topological vector lattices had ﬁrst been introduced as Banach lattices, and comprehensive treatments can for example be found in the books by Sch¨ afer [184] and [185] and by Meyer-Nieberg [132]. In locally convex vector spaces there are compatibility requirements between the algebraic and the lattice operations as well as the topology. These are reﬂected in the corresponding requirements of Section 5 for locally convex cones. The strong conditions for locally convex lattice cones mirror those for M-topologies in topological vector lattices. Since under circumstances the latter permit representations as function spaces see [94] , the result of Proposition 5.37 is not unexpected. Proposition 5.37 gives also the reason for using R-valued functions in the standard lattice completion of a locally convex cone. General lattices carrying diﬀerent orders leading to notions of order convergence and of approximation of elements are thoroughly investigated in [68] and [69].

Chapter II

Measures and Integrals. The General Theory

In this chapter we shall develop a general integration theory for cone-valued functions with respect to operator-valued measures. The structure of locally convex cones will allow the use of many of the main concepts of classical measure theory for (extended) real-valued functions. Section 1 introduces measurability for cone-valued functions on a set X with respect to a (weak) σ-ring of subsets of X. This notion does not involve any reference to a particular measure. Bounded operator-valued measures will be deﬁned in Section 3. The introduction of its modulus allows the extension of any given measure to a full locally convex cone containing the given cone and its neighborhood system, thus greatly facilitating the expansion of our concepts. This yields a new understanding of the variation of a measure, not as a separate positive real-valued measure associated with the given one, but as a component of its extension. The development of an integration theory for cone-valued functions with respect to an operator-valued measure follows in Section 4. Section 5 contains the general convergence theorems for sequences of functions and measures, that is variations and adaptations of the dominated convergence theorem. Chapter II concludes with a long list of special cases and examples in Section 6, demonstrating the generality of the approach. These examples include classical real-valued measure theory as well as settings with vector-, cone-, functional- and operator-valued measures and functions.

1. Measurable Cone-Valued Functions Throughout the following let X be a set, (P, V) a locally convex cone with dual P ∗ . Endowed with the pointwise algebraic operations and order, the P-valued functions on X form again a cone, denoted by F(X, P). As usual, we say that a function f ∈ F(X, P) is supported by a set E ⊂ X if f (x) = 0 for all x ∈ X \ E. For a positive real-valued function ϕ on X and f ∈ F(X, P) we denote by ϕ⊗f ∈ F(X, P) the mapping W. Roth, Operator-Valued Measures and Integrals for Cone-Valued Functions, Lecture Notes in Mathematics 1964, c Springer-Verlag Berlin Heidelberg 2009

119

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x → ϕ(x)f (x) : X → P. For an element a of P or of V we shall also use its symbol to denote the constant function x → a, hence ϕ⊗a for x → ϕ(x)a. 1.1 Weak σ-Rings. We shall develop our measure and integration theory with respect to a family R of subsets of X with the following properties: (R1) ∅ ∈ R. (R2) If E1 , E2 ∈ R, then E1 E2 ∈ R and E1 \ E2 ∈ R. En ∈ R. (R3) If En ∈ R for n ∈ N and En ⊂ E for some E ∈ R, then n∈N

We shall call a family R with these properties a (weak) σ-ring. (Condition (R3) is weaker then the usual one for σ-rings.) As E1 ∩ E2 = E1 \ (E1 \ E2 ), Condition (R2) implies that E1 ∩ E2 ∈ R whenever E1 , E2 ∈ R. Of course, any σ-algebra is a σ-ring in this sense, and a σ-ring R is a σ-algebra if and only if X ∈ R. However, because we shall require boundedness for measures deﬁned on R, using σ-algebras from the beginning would impose undue limitations. We may, however, associate with R in a canonical way the σ-algebra AR = {A ⊂ X | A ∩ E ∈ R

for all E ∈ R}

of measurable subsets of X. As usual, χE stands for the characteristic (or indicator) function on X of a subset E ⊂ X, and SR (X, P) is the subcone of F(X, P) of all P-valued step functions supported by R, that is functions h = ni=1 χEi⊗ai with Ei ∈ R and ai ∈ P. If the sets Ei are pairwise disjoint, then we shall call the above the standard representation for the step function h. Measurability for vector-valued functions has been introduced in various places (see for example Dunford & Schwartz [55], III.2.10). A suitable adaptation for cone-valued functions needs to consider the presence of unbounded elements in P and the absence of negatives. We shall therefore employ the relative topologies. 1.2 Measurable Functions. We shall say that a function f ∈ F(X, P) is measurable with respect to the σ-ring R if for every v ∈ V, with respect to the symmetric relative v-topology of P (M1) f −1 (O) ∩ E ∈ R for every open subset O of P and every E ∈ R. (M2) f (E) is separable in P for every E ∈ R. Note that Condition (M1) means that f −1 (O) ∈ AR for all open subsets O of P. Obviously the functions in SR (X, P) are measurable. Proposition 1.3. A function f ∈ F(X, R) is measurable if and only if it is measurable in the usual sense with respect to the σ-algebra AR . Proof. Let f ∈ F(X, R). The neighborhood system for R consists of the positive reals ε > 0. The symmetric relative topology therefore coincides

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121

with the usual topology on the elements of R, while +∞ is an isolated point. The range of f, hence f (E) for every E ∈ R, is separable in any case. Thus for measurability we require that f −1 (O) ∈ AR for every open subset of R and also that f −1 (+∞) ∈ AR . This coincides with the usual deﬁnition of measurability. Theorem 1.4. A function f ∈ F(X, P) is measurable if and only if for every E ∈ R, v ∈ V and ε > are sets En ∈ R, n ∈ N, such that 0 there n∈N En = E and f (x) ∈ vε f (y) whenever x, y ∈ En for some n ∈ N. Proof. First assume that the function f ∈ F(X, P) is measurable. For E ∈ R and v ∈ V let A = {an | n ∈ N} be a dense subset (with respect to the symmetric relative v -topology) of f (E). For a ∈ P and ε > 0 the sets ◦ ◦ v ε (a) = vε (a) and (a)v ε = (a)vε 0<ε <ε

0<ε <ε

are open in the upper and lower relative v -topologies, respectively, and their ◦ intersection v sε (a) is an open neighborhood of a in the symmetric relative v -topology. Set ◦ En = f −1 v s(ε/3) (an ) ∩ E ∈ R. Then Lemma I.4.1(a) shows that f (x) ∈ vε f (y) whenever x, y ∈ En . ◦ Furthermore, for every x ∈ E there is some an ∈ v sε/2 f (x) . Thus f (x) ∈ ◦ ◦ v sε/2 (an ) ⊂ v sε (an ), hence x ∈ En . This shows n∈N En = E, as required. For the converse, assume that the above condition holds for the function m f ∈ F(X, P) m ∈ N there are En m∈ R, and mlet E ∈ R Then for every such that n∈N En = E and f (x) ∈ v(1/m) f (y) whenever x, y ∈ En for m m m some n ∈ N. Choose am n = f (xn ) for some xn ∈ En . Then the set A = m {an | n, m ∈ N} is seen to be dense in f (E), which yields Condition (M1). For (M2), let O ⊂ P be open in the symmetric relative v -topology. With the sets Enm from above set F = Enm | Enm ⊂ f −1 (O }. Then F ∈ R by (R2) and F ⊂ f −1 (O) ∩ E. For x ∈ f −1 (O) ∩ E on the s f (x) ⊂ O. We ﬁnd x ∈ Enm other hand, there is m ∈ N such that v(1/m) m for some n ∈ N. Then for every y ∈ En we have f (y) ∈ v(1/m) f (x) ⊂ O, hence y ∈ f −1 (O). This shows x ∈ Enm ⊂ F, hence f −1 (O) ∩ E = F ∈ R. We can indeed assume that the sets En ∈ R from Theorem 1.4 are disjoint, since otherwise we may set G1 = E1

and

Gn = En \

n−1 i=1

Ei

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for n ≥ 2. Then Gn ∈ R and Gn ⊂ En . The sets Gn are disjoint and their union equals the union of the sets En , that is the given set E ∈ R. Corollary 1.5. The measurable functions form a subcone of F(X, P). Proof. Clearly, αf is measurable, whenever f ∈ F(X, P) is measurable and α ≥ 0. We proceed to show that for measurable functions f, g ∈ F(X, P) their sum f + g is also measurable. In a ﬁrst step, given E ∈ R and v ∈ V, using Theorem 1.4 and Lemma I.4.1(c) we can ﬁnd sets En ∈ R and λn ≥ 0 such that n∈N En = E and 0 ≤ f (x) + λn and 0 ≤ g(x) + λn whenever x ∈ En . Now, given ε > 0 we set δn = ε/(2 + 2λn ) and again using Theorem 1.4, for every n ∈ N we ﬁnd sets Enm ∈ R such that m m∈N En = En and f (x) ∈ vδn f (y) as well as g(x) ∈ vδn g(y) whenever x, y ∈ Enm . Following I.4.1(b) this yields f (x) ≤ (1 + δ)f (y) + δ(1 + λ)v = (1 + δ)f (z) + (ε/2)v and, likewise g(x) ≤ (1 + δ)f (y) + (ε/2)v. Thus f (x) + g(x) ≤ (1 + δ) f (y) + g(y) + εv and indeed f (x) + g(x) ∈ vε f (y) + g(y) . As m,n∈N Enm = E, this proves that the function f + g is also measurable. Theorem 1.6. For measurable functions f, g ∈ F(X, P) and v ∈ V the set {x ∈ X | f (x) v g(x)} is measurable, that is in AR . Proof. Let f, g ∈ F(X, P) be measurable, v ∈ V and E ∈ R. According to ∈ R such that Theorem 1.4, for 0 < ε≤ 1 there are disjoint sets E i∈N = i E and f (x) ∈ vε f (y) as well as g(x) ∈ vε g(y) whenever x, y ∈ Ei for some i ∈ N. Set Ei | f (xi ) ∈ vε g(xi ) for some xi ∈ Ei ∈ R. Fε = If x ∈ Fε , then f (x) ∈vε f (xi ) , f (xi ) ∈ vε g(xi ) and g(xi ) ∈ vε g(x) , hence f (x) ∈ v(7ε) g(x) by Lemma 2,1(a). This shows x ∈ E | f (x) ∈ vε g(x) ⊂ Fε ⊂ x ∈ E | f (x) ∈ v7ε g(x) . Then F =

F( n1 ) = {x ∈ E | f (x) v g(x)} ∈ R

n∈N

as well. As F = E

{x ∈ X | f (x) v g(x)}, our claim follows.

We shall use diﬀerent patterns of pointwise convergence for sequences of cone-valued functions. For functions (fn )n∈N and f in F(X, P), a subset F ⊂ X and a neighborhood v ∈ V we shall write v fn F f,

v fn F f,

or

v fn → f F

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123

if (fn )n∈N converges to f pointwise on F with respect to the upper, lower or symmetric relative v-topology of P, that is if for every x ∈ F and ε > 0 f (x) , f ∈ N such that f (x) ∈ v (x) ∈ f (x) vε or fn (x) ∈ there is n n ε n 0 s vε f (x) for all n ≥ n0 , respectively. Convergence in this sense for all v ∈ V means convergence in the (global) upper, lower or symmetric relative topology of P. We shall denote this by fn F f,

fn F f,

or

fn → f. F

All the above notions of convergence are compatible with the algebraic operations in P (see Lemma I.4.1(d)). Theorem 1.7. If for f ∈ F(X, P) and every E ∈ R and v ∈ V there v (f ), is a sequence of measurable functions fn ∈ F(X, P) such that fn → E then f is also measurable. Proof. Let f ∈ F(X, P), E ∈ R and v ∈ V. Suppose that there is a v sequence of measurable functions (fn )n∈N such that fn → (f ). If for all E i n ∈ N the sets An = {an | i ∈N} are dense (in the symmetric relative v topology) in fn (E), then A = n∈N An is obviously dense in f (E), which is thus seen to be separable. Now let O ⊂ P be open in the symmetric relative v -topology. For ε > 0 let Uε be the topological interior of the set Oε = {a ∈ O | vεs (a) ⊂ O}. For m, n ∈ N set (fk )−1 U(1/m) ∈ R Enm = E k≥n

and F = m,n∈N Enm ∈ R. If x ∈ f −1 (O) ∩ E, then there is ε > 0 such that s f (x) vεs f (x) ⊂ O. For m ≥ 7/ε there is n ∈ N such that fk (x) ∈ v(1/m) s for all k ≥ n. For any such k let a ∈ v(1/m) fk (x) . Then s s s v(1/m) fk (x) ⊂ v(7/m) f (x) ⊂ vεs f (x) ⊂ O (a) ⊂ v(3/m) s fk (x) ⊂ O(1/m) , thereby Lemma I.4.1(a). Thus a ∈ O(1/m) , hence v(1/m) fore fk (x) ∈ U(1/m) and x ∈ Enm ⊂ F. This shows f −1 (O) ⊂ F. For x ∈ F, on the other hand, there are m, n ∈ N such that fk (x) ∈ U(1/m) ⊂ O(1/m) for s fk (x) , hence f (x) ∈ O. all k ≥ n. There is such k such that f (x) ∈ v(1/m) −1 Thus f (O) ∩ E = F ∈ R. Theorem 1.8. Let f ∈ F(X, P) be measurable. (a) Let ϕ : X → R. If ϕ is positive and measurable with respect to AR , then ϕ⊗f ∈ F(X, P) is also measurable. (b) Let Φ : X → X. If Φ−1 (A) ∈ AR for all A ∈ AR , and Φ(E) ⊂ F for every E ∈ R with some F ∈ R, then the function then f ◦Φ ∈ F(X, P) is also measurable.

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(c) Let (N , U) be a locally convex cone and let Ψ : P → N . If for every u ∈ U there is v ∈ V such that the mapping Ψ is continuous with respect to the symmetric relative v - and u-topologies of P and N , then the function Ψ ◦ f ∈ F(X, N ) is also measurable. The assumption on Ψ holds in particular if Ψ : P → N is a continuous linear operator. Proof. (a) Our claim is obvious if ϕ is a real-valued step function (supported by AR ,) since the validity of the criterion from Theorem 1.4 for ϕ⊗f follows straight from its validity for f. Generally, there is a sequence (ψn )n∈N of positive real-valued step functions that converges pointwise from below to ϕ. All the functions fn = ψn ⊗f are measurable by the above. If ϕ(x) = 0 for x ∈ X, then ψn (x) = 0 for all n ∈ N. Otherwise, for v ∈ V and ε >0 ε . choose λ > 0 such that 0 ≤ f (x) + λv and set γ = min 1 + ε, 1 + 2λϕ(x) There is n0 ∈ N such that ψn (x) ≤ ϕ(x) ≤ γψn (x), hence ψn (x)(f (x) + λv) ≤ ϕ(x)(f (x) + λv) ≤ γψn (x)(f (x) + λv) for all n ≥ n0 . This shows fn (x) + λψn (x)v ≤ ϕ(x)f (x) + λϕ(x)v ≤ ϕ(x)f (x) + γλψn (x)v, hence fn (x) ≤ ϕ(x)f (x) + εv by the cancellation law for positive elements (see Lemma I.4.2 in [100]), as γλψn < λψn (x) + ε. Likewise, the above implies ϕ(x)f (x) + λϕ(x)v ≤ γfn (x) + γλψn (x)v ≤ γfn (x) + γλϕ(x)v, and ϕ(x)f (x) ≤ γfn (x) + εv as well. Thus fn (x) ∈ vεs ϕ(x)f (x) for all n ≥ n0 . This shows fn → ϕ⊗f, and by Theorem 1.7 the function ϕ⊗f is X seen to be measurable. For (b), let f and Φ : X → X be as stated, let g = f ◦Φ and v ∈ V. For E ∈ R we have Φ(E) ⊂ F for some F ∈ R, hence g(E) ⊂ f (F ) which is separable in the symmetric relative v -topology. Secondly, for an open subset O of P we have f −1 (O) ∈ AR , hence g −1 (O) = Φ−1 f −1 (O) ∈ AR , and the function g is seen to be measurable. For Part (c), let f and Ψ : P → N be as stated, let g = f ◦ Ψ and u ∈ U. Let v ∈ V be such that Ψ is continuous with respect to the symmetric relative v - and u-topologies of P and N . For every E ∈ R, the set f (E) is separable with respect to the relative v -topology of P, symmetric hence its continuous image g(E) = Ψ f (E) is separable with respect to the symmetric relative u-topology of N . Secondly, for an open subset O of N its inverse image Ψ−1 (O) is open in P, hence g −1 (O) = f −1 Ψ−1 (O) ∈ AR , and the function g is seen to be measurable. For the additional statement in (c), suppose that Ψ : P → N is a continuous linear operator. Given u ∈ U there is v ∈ V such that Ψ(a) ≤ Ψ(b) + u holds whenever a ≤ b + v for

1. Measurable Cone-Valued Functions

125

a, b ∈ P. Thus a ∈ vε (b), that is a ≤ γb + εv with some 1 ≤ γ ≤ 1 + ε implies Ψ(a) ≤ γΨ(b)+ ε, hence Ψ(a) ∈ uε Ψ(b) . Likewise, a ∈ vεs (b) implies that Ψ(a) ∈ usε Ψ(b) . The function Ψ is therefore continuous with respect to the symmetric relative v - and u-topologies of P and N . In the literature the terms weak measurability or scalar measurability are often used for a vector-valued function f if all the scalar-valued functions μ ◦ f for linear functionals μ in the dual of the range of f are measurable in the usual sense. The following theorem states that measurability in our sense implies scalar measurability. The converse holds true for functions with bounded values and separable ranges. Theorem 1.9. Let f ∈ F(X, P). (a) If f is measurable, then the R-valued functions μ ◦ f are measurable for all μ ∈ P ∗ . (b) If the values of f are bounded, f (E) is separable in the symmetric relative v-topology for all E ∈ R and v ∈ V, and the R-valued functions μ ◦ f are measurable for all μ ∈ P ∗ , then f is measurable. Proof. For (a), suppose that the function f ∈ F(X, P) is measurable, and let μ ∈ P ∗ , that is μ ∈ v ◦ for some v ∈ V. Recall from Proposition I.4.5 and Example I.4.37(a) that a continuous linear functional μ : P → R is also continuous, if we endow P with the symmetric relative v-topology and R with its given symmetric topology (which of course coincides with its symmetric relative topology). Following Theorem 1.8(c), this shows that the function μ◦f : X → R is measurable whenever the function f is measurable. Now suppose that the assumptions of Part (b) hold for the function f ∈ F(X, P). We shall verify the criterion of Theorem 1.4 for measurability. Recall from Proposition I.4.2(iv) that on the subcone B of bounded elements of P the corresponding given and relative topologies coincide. As the values of f are supposed to be bounded, this will greatly facilitate our arguments. In a ﬁrst step, let us consider an element a ∈ B and neighborhood v ∈ V. Then b ∈ / v(a), that is b a + v, for an element b ∈ B implies that b a + v/2, as indeed otherwise, for λ > 0 such that a ≤ λv and ε = min 1/9, 1/(3λ) there would be 1 ≤ γ ≤ 1 + ε such that b ≤ γ (a + v/2) + εv = γa +

2 + ε v ≤ γa + v. 2 3

γ

Because 1 γa = a + (γ − 1)a ≤ a + (γ − 1)λv ≤ a + ελv ≤ a + v, 3 this yields a ≤ b + v, contradicting our assumption. Consequently, following Theorem 3.2 in [175] (see also Corollary 4.34 in Chapter I), there is a linear functional μ ∈ v ◦ such that

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1 μ(b) > μ(a) + . 2 Now, in a second step of our argument, consider an element a ∈ B and for v ∈ V the symmetric neighborhood v s (a) = v(a) ∩ (a)v = {c ∈ P | c ≤ a + v

and a ≤ c + v}.

Given a set E ∈ R, let {bi }i∈N be a countable subset of f (E) \ v s (a) ⊂ B that is dense with respect to the (given) symmetric topology. Such a subset exists because on B the given and the relative topologies of P coincide. / v(a) or bi ∈ / (a)v. Accordingly, we may For each i ∈ N we have either bi ∈ choose linear functionals μi ∈ v ◦ corresponding to the elements bi such that either 1 1 or μi (a) > μi (b) + μi (b) > μi (a) + 2 2 if bi ∈ / v(a) or bi ∈ / (a)v, respectively. We denote 1 1 Oi = −∞ , μi (a) + or Oi = μi (a) − , +∞ 4 4

in these respective cases and set Ai = μ−1 i (Oi ) ⊂ P and A = i∈N Ai . For every c ∈ (v/4)s (a), that is c ≤ a + v/4 and a ≤ c + v/4 we have μ(c) ≤ μ(a) + 1/4 and μ(a) ≤ μ(c) + 1/4 for all μ ∈ v ◦ , hence c ∈ Ai for all i ∈ N. This shows v s (a) ⊂ A. We shall proceed to verify that A ∩ f (E) ⊂ (2v)s (a). For this, consider any element c ∈ f (E) \ (2v)s (a). First we observe that v s (c) ∩ v s (a) = ∅, because the existence of an element d ∈ v s (c) ∩ v s (a) would lead to c ∈ (2v)s (a), contradicting our choice of c. Thus f (E) ∩ (v/4)s (c) ⊂ f (E) \ v s (a) holds as well, and there is some bi ∈ (v/4)s (c). We have μi (bi ) ≤ μi (c) +

1 4

μi (c) ≤ μi (b) +

and

1 4

since μi ∈ v ◦ . Recall that either bi ∈ / v(a) or bi ∈ / (a)v. In the ﬁrst case, / Ai . this implies μi (bi ) > μi (a) + 1/2, hence μi (c) > μi (a) + 1/4, and c ∈ In the second case, we have μi (bi ) < μi (a) − 1/2, hence μi (c) < μi (a) − 1/4 / A. Summarizing, we veriﬁed that and, likewise c ∈ / Ai . Thus indeed c ∈ v s (a) ⊂ A

and

A ∩ f (E) ⊂ (2v)s (a).

Let us apply this to an element a = f (x) for some x ∈ E. By our assumption, all the R-valued functions ϕi = μi ◦ f are measurable, hence the sets Fi = ϕ−1 i (Oi ) are contained in AR . Likewise, −1 Fi = f −1 μ−1 f (Ai ) = f −1 (A) ∈ AR . F = i (Oi ) = i∈N

i∈N

i∈N

2. Inductive Limit Neighborhoods for Cone-Valued Functions

Thus

f −1 v s (a) ⊂ F

and

127

F ∩ E ⊂ f −1 (2v)s (a) .

Now in the third and ﬁnal step of our argument we shall verify the criterion of Theorem 1.4: For E ∈ R and v ∈ V let {an }n∈N be a subset of f (E) that is dense with respect to the symmetric relative v-topology, hence with respect to the given symmetric v-topology. For each element an and the neighborhood u = (ε/4)v ∈ V in place of v choose the set F = Fn ∈ AR as in the last part of the preceding step and set En = Fn ∩ E. Then f −1 us (an ) ∩ E ⊂ En and En ⊂ f −1 (2u)s (a) holds for all n ∈ N by the above. Thus, ﬁrstly, for x, y ∈ En we have s ⊂ f (x), f (y) ∈ (2u) (an ). But this obviously implies that f (x) ∈ (4u) f (y) s for any x ∈ E there is some a such that f (x) ∈ u (a ), vε f (y) . Secondly, n n that is x ∈ f −1 us (an ) ∩ E ⊂ En . This demonstrates n∈N En = E and completes our argument.

2. Inductive Limit Neighborhoods for Cone-Valued Functions Let (P, V) be a locally convex cone. In preparation of our integration theory for cone-valued functions with respect to an operator-valued measure, we shall introduce appropriate neighborhoods for the cone F(X, P) and corresponding subcones of measurable functions. Our integrals will constitute continuous linear operators on these cones. First, in order to allow greater generally, we shall extend the given neighborhood system of V. 2.1 Inﬁnity as a Neighborhood. We shall adjoin the maximal element ∞ to the neighborhood system V such that a ≤ b+∞ holds for all a, b ∈ P. The addition and multiplication by scalars involving this element is deﬁned in a canonical way: We set v + ∞ = ∞, 0 · ∞ = 0 and α · ∞ = ∞ for all v ∈ V and α > 0. The augmented neighborhood system which includes this inﬁnite element and 0 ∈ P will be denoted by V, that is V = V ∪ {0, ∞}. Obviously, (V, V) is a full locally convex cone. 2.2 Inductive Limit Neighborhoods. Let X and R be as before, and let F X, V be the family of V-valued functions on X, endowed with the pointwise operations and order. For functions f, g ∈ F(X, P) and s ∈ F X, V we write f ≤ g + s if f (x) ≤ g(x) + s(x) for all x ∈ X. The addition and multiplication by scalars for functions s, t ∈ F X, V is deﬁned pointwise, and s ≤ t means that f ≤ g + s implies f ≤ g + t for f, g ∈ F(X, P).

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An (R-compatible) inductive limit neighborhood for F(X, P) is a convex subset v of measurable functions in F X, V such that for every E ∈ R there is vE ∈ V and s ∈ v such that χE ⊗vE ≤ s. Measurability is meant with respect to R and the locally convex cone (V, V). For functions f, g ∈ F(X, P) and an inductive limit neighborhood v we denote f ≤g+v

if

f ≤ g + s,

for some

s ∈ v.

We deﬁne sums and multiples by positive scalars for inductive limit neighborhoods through the addition and multiplication of their elements. A canonical order relation is given by v≤u

if for every s ∈ v there is t ∈ u such that s ≤ t.

Inductive limit neighborhoods include uniform neighborhoods, consisting of a single constant function x → v; and if X ∈ R, that is if R is a σ-algebra, then the uniform neighborhoods form a base for the family of all inductive limit neighborhoods. 2.3 The Cone FR (X, P). We shall in the sequel deal with measurable functions in F(X, P) that can be reached from below by step functions; more precisely: We denote by FR (X, P) the subcone of all measurable functions f ∈ F(X, P) such that for every inductive limit neighborhood v there is h ∈ SR (X, P) satisfying h ≤ f + v. Lemma 2.4. Let f ∈ FR (X, P). (a) For every inductive limit neighborhood v there is λ ≥ 0 such thats 0 ≤ f + λv. (b) There is E ∈ R such that f (x) ≥ 0 for all x ∈ X \ E, and for every v ∈ V there is λ ≥ 0 such that 0 ≤ f + λ χE ⊗v. Proof. Let f ∈ FR (X, P). For (a), given an inductive limit neighborhood v, there is a step function h = ni=1 χEi⊗ai ∈ SR (X, P) such that h ≤ f + v, that is h ≤ f + s for some s ∈ v. We may assume that the sets Ei ∈ R are disjoint and E = ni=1 Ei ∈ R. There is v ∈ V such that χE ⊗v ≤ v, and in turn there is λ ≥ 0 such that 0 ≤ ai + λv for all i = 1, . . . , n. This shows 0 ≤ f (x) + s(x) + λv for all x ∈ E and 0 ≤ f (x) + s(x) for all x ∈ X \ E. Thus 0 ≤ f + (s + λ χE ⊗v), hence indeed 0 ≤ f + (1 + λ)v. For (b), let the inductive limit neighborhood v consist of all V-valued functions that are supported by some set in R. By (a) there is λ ≥ 0 and a function s ∈ v such that 0 ≤ f + λs. Because s is supported by some set E ∈ R, that is s(x) = 0 for all x ∈ X \ E, we have indeed f (x) ≥ 0 for all x ∈ X \ E. Now let v ∈ V and let v consist of the single function x → v. Then 0 ≤ f + λv with some λ ≥ 0 by (a). Thus 0 ≤ f + λ χE ⊗v, as claimed.

2. Inductive Limit Neighborhoods for Cone-Valued Functions

129

The following lemma provides a more straightforward characterization of the functions in FR (X, P), avoiding the use of inductive limit neighborhoods. Lemma 2.5. A measurable P-valued function f is in FR (X, P) if and only if (i) there is E ∈ R such that f (x) ≥ 0 for all x ∈ X \ E, and (ii) for every v ∈ V there is h ∈ SR (X, P) such that h ≤ f + χX ⊗v. Proof. If f ∈ FR (X, P), then (i) follows from Lemma 2.4(b). Statement (ii) follows from the deﬁnition of the cone FR (X, P) if we consider the singleton inductive limit neighborhood v = {χX ⊗v}. For the converse, suppose that (i) and (ii) hold for the measurable function f ∈ F(X, P), and let v be an inductive limit neighborhood. For the set E ∈ R from (i) there is v ∈ V such that χE ⊗v ≤ s for some s ∈ v. According to (ii), let h ∈ SR (X, P) such that h ≤ f + χX ⊗v and set h = χE ⊗h ∈ SR (X, P). Then h ≤ f + s, hence h ≤ f + v. This shows f ∈ FR (X, P). Note that a function χF ⊗a for F ∈ AR and a ∈ P is contained in FR (X, P) if and only if either a ≥ 0 or F ∈ R. Lemma 2.6. Let f ∈ FR (X, P) and let ϕ be a positive real-valued function on X, measurable with respect to AR . If either f is positive or if ϕ is bounded, then ϕ⊗f ∈ FR (X, P). Proof. Following Theorem 1.8(a), the function ϕ⊗f is measurable. If f is positive, then ϕ⊗f is also positive, hence in FR (X, P). Otherwise, there is ρ > 0 such that 0 ≤ ϕ(x) ≤ ρ for all x ∈ X. Given an inductive limit neighborhood v there is h ∈ SR (X, P) such that h ≤ f + (1/2ρ)v. Also, there is λ > 0 such that 0 ≤ f + λv. As ϕ is bounded and measurable, there is a real-valued positive step function ψ on X such that ψ(x) ≤ ϕ(x) ≤ ψ(x) +

1 2λ

for all x ∈ X. Then l = ψ ⊗h ∈ SR (X, P), and indeed l ≤ ψ ⊗f +

1 1 ψ ⊗v ≤ ψ ⊗f + v + (ϕ − ψ)⊗(f + λv) ≤ ϕ⊗f + v. 2ρ 2

Lemma 2.6 implies in particular that χF ⊗f ∈ FR (X, P) whenever f ∈ FR (X, P) and F ∈ AR . Also, if ϕ is a positive real-valued measurable function and a ∈ P, then ϕ⊗a ∈ FR (X, P) if a ≥ 0, and (χE ϕ)⊗a = ϕ⊗(χE ⊗a) ∈ FR (X, P) for every E ∈ R in general if the function ϕ is bounded. A sequence (fn )n∈N in FR (X, P) is said to be bounded below if for every inductive limit neighborhood v there is λ ≥ 0 such that 0 ≤ fn + λv for all n ∈ N.

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Theorem 2.7. Let f ∈ FR (X, P) and E ∈ R. For every inductive limit neighborhood v, every v ∈ V and ε > 0 there is 1 ≤ γ ≤ 1 + ε and a bounded below sequence (hn )n∈N of step functions in SR (X, P) such that (i) hn ≤ γf + v for all n ∈ N. (ii) For every x ∈ E there is n0 ∈ N such that f (x) ≤ hn (x) + v for all n ≥ n0 . Proof. Let f ∈ FR (X, P), E ∈ R, let v be an inductive limit neighborhood, let v ∈ V and ε > 0. Following Lemma 2.4(b) we may assume that f (x) ≥ 0 for all x ∈ X \ E. There is u ∈ V such that both u ≤ v and χE ⊗u ≤ v. Again using 2.4(b) we ﬁnd λ ≥ 0 such that 0 ≤ f + λχE ⊗u We set δ = 1 } and γ = (1+δ)2 ≤ 1+ε. By Theorem 1.4 there is a partition min{1, ε3 , 4(1+λ) of E into disjoint subsets Ei ∈ R, i ∈ N, such that f (x) ∈ uδ f (y) holds for all x, y ∈ Ei . Thus f (x) ≤ (1 + δ)f (y) + δ(1 + λ)u by Lemma I.4.1(b). We set ai = (1 + δ)f (xi ) ∈ P for some xi ∈ Ei . Thus for any x ∈ Ei we have f (x) ≤ ai + δ(1 + λ)u ≤ ai + v and

Thus

1 ai ≤ (1 + δ)2 f (x) + δ(1 + δ)(1 + λ)u ≤ γf (x) + u. 2 n ! i=1

1 1 χEi⊗ai ≤ χE ⊗(γf ) + χE ⊗u ≤ χE ⊗(γf ) + v. 2 2

Furthermore, there is h0 ∈ SR (X, P) such that h0 ≤ γf + 12 v, and therefore 1 χ(X\E) ⊗ho ≤ χ(X\E) ⊗(γf ) + v 2 holds as well. Now we choose the step functions hn =

n !

χEi⊗ai + χ(X\E) ⊗h0 .

i=1

Adding the above yields indeed hn ≤ γf + v for all n ∈ N, hence (i). Part (ii) of our claim follows directly from the above, as n for all x∈ Ei . f (x) ≤ hn (x) + v i=1

3. Operator-Valued Measures

131

Finally, given an inductive limit neighborhood u, there is λ ≥ 0 such that 0 ≤ h0 + λu, that is 0 ≤ h0 + λs for some s ≤ u. Also there is u ∈ V such that χE ⊗u ∈ u and ρ ≥ 0 such that 0 ≤ f + ρχE ⊗u. The latter implies 0 ≤ ai + ρu for all i ∈ N, hence 0 ≤ hn + λs + ρχE ⊗u ≤ hn + (λ + ρ)u for all n ∈ N. The sequence (hn )n∈N is therefore bounded below. Finally, if f ≥ 0, then we may choose h0 = 0, and as all the elements ai = (1 + δ)f (xi ) are also positive, we realize that hn ≥ 0 for all n ∈ N. If (P, V) is indeed a full locally convex cone, as will frequently occur in the subsequent sections, then the preceding result can obviously be simpliﬁed. We shall formulate this in a corollary. Corollary 2.8. Let (P, V) be a full locally convex cone. Let f ∈ FR (X, P) and E ∈ R. For every inductive limit neighborhood v and ε > 0 there is 1 ≤ γ ≤ 1 + ε and a bounded below sequence (hn )n∈N of step functions in SR (X, P) such that (i) hn ≤ γf + v for all n ∈ N. (ii) For every x ∈ E there is n0 ∈ N such that f (x) ≤ hn (x) for all n ≥ n0 . Proof. We choose v ∈ V such that χE ⊗v ≤ (1/2)vw and apply Theorem 2.7 with this v, the inductive limit neighborhood (1/2)vw and the given ε > 0. There is a sequence (hn )n∈N as in 2.7. The functions h n = hn + χE ⊗v then satisfy our claim.

3. Operator-Valued Measures Let (P, V) be a quasi-full locally convex cone and let (Q, W) be a locally convex complete lattice cone (see Sections 5 and 6 in Chapter I). Let L(P, Q) denote the cone of all (uniformly) continuous linear operators from P to Q. Recall from Section 3 in Chapter I that a continuous linear operator between locally convex cones is monotone with respect to the respective weak preorders. Because Q carries its weak preorder, this implies monotonicity with respect to the given orders of P and Q as well. Let X be a set, R a (weak) σ-ring of subsets of X. An L(P, Q)-valued measure θ on R is a set function E → θE : R → L(P, Q) such that θ(∅) = 0 and θ( i∈N

Ei )

=

∞ ! i=1

θEi

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II Measures and Integrals. The General Theory

holds whenever the sets Ei ∈ R are disjoint and ∞ i=1 Ei ∈ R. Convergence for the series on the right-hand side is meant in the following way: For ev ∞ θ (a) is order convergent in Q in the sense ery a ∈ P the series i=1 Ei of I.5.26. (Recall from Proposition I.5.42 that order convergence is implied by convergence in the symmetric relative topology.) Lemma 3.1. Let E ∈ R and a ∈ P. (a) If Ei ∈ R are such that Ei ⊂ Ei+1 for all i ∈ N and E = ∪∞ i=1 Ei , then θE (a) = lim θEi (a). i→∞

(b) If Ei ∈ R are such that E ⊃ Ei ⊃ Ei+1 for all i ∈ N, and ∩∞ i=1 Ei = ∅, then 0 ≤ lim θEi (a) + O θE (a) and lim θEi (a) ≤ O θE (a) . i→∞

i→∞

Proof. For Part (a), let F1 = E1 and Fi = Ei \ Fi−1 for i > 1. The sets Fi are disjoint, En = ∪ni=1 Fi and E = ∪∞ additivity of i=1 Fi . From the countable the measure θ we infer that θEn (a) = ni=1 θFi (a) and θE (a) = ∞ i=1 θFi (a), hence our claim. For Part (b), let Fi = E \ Ei for i ∈ N. Thus Fi ⊂ Fi+1 and ∪∞ i=1 Fi = E. This shows θE (a) = lim θEi (a) by Part (a). Furthermore, i→∞

θE (a) = θEi (a) + θFi (a) holds for all i ∈ N by Part (a). Using the limit rules in Lemma I.5.19 we infer that θE (a) ≤ lim θEi (a) + θE (a) ≤ lim θEi (a) + θE (a) ≤ θE (a), i→∞

i→∞

hence equality for these terms as Q carries the weak preorder which is supposed to be antisymmetric. Now the cancellation rule in Proposition I.5.10(a) yields our claim. For our upcoming integration theory for P-valued functions with respect to an L(P, Q)-valued measure θ (see Section 4 below) we shall also have to assign values of θ to the neighborhoods in P. This will be done by the introduction of its modulus |θ|. Recall that we require the locally convex cone (P, V) to be quasi-full. 3.2 The Modulus of a Measure. Throughout the following, let θ be a ﬁxed L(P, Q)-valued measure on R. For a neighborhood v ∈ V and a set E ∈ R, modulus (or semivariation) of θ is deﬁned as |θ|(E, v) = sup

n !

θEi (si ) si ∈ P, si ≤ v, Ei ∈ R disjoint subsets of E

.

i=1

The following is obvious from this deﬁnition. Lemma 3.3. Let v ∈ V and E ∈ R. If v ∈ P, then |θ|(E, v) = θE (v). Proof. Let E ∈ R and v ∈ V ∩ P. If si ∈ P such that si ≤ v and Ei ∈ R are disjoint subsets of E for i = 1, . . . , n, then

3. Operator-Valued Measures n !

133

θEi (si ) ≤

n !

i=1

θEi (v) = θ(∪ni=1 Ei ) (v) ≤ θE (v).

i=1

Thus |θ|(E, v) ≤ θE (v). The reverse inequality is obvious, as we may choose E1 = E and s1 = v in 3.2. Lemma 3.4. Let v ∈ V and E ∈ R. Then (a) 0 ≤ |θ|(E, v) and |θ|(∅, v) = 0. (b) θE (a) ≤ θE (b) + |θ|(E, v) whenever a ≤ b + v∞ for a, b ∈ P. sets such that E = (c) If Ei ∈ R are disjoint i=1 Ei , |θ|(E , v). then |θ|(E, v) = ∞ i i=1 Proof. Part (a) is obvious. Part (b) follows as the locally convex cone is supposed to be quasi-full. Indeed, for a ≤ b + v there is s ≤ v such that (b) + θE (s), and as θE (s) ≤ |θ|(E, v), a ≤ b + s. This implies θE (a) ≤ θE our claim follows immediately from 1 . For Part (c), let E = ∪∞ i=1 Ei for disjoint sets Ei ∈ R. Let F1 , . . . , Fn ∈ R be disjoint subsets of E and sk ∈ P such that sk ≤ v for k = 1, . . . , n. Then θFk (sk ) =

∞ !

θ(Fk ∩Ei ) (sk )

i=1

for every k = 1, . . . , n by the countable additivity of θ, hence # "∞ n n ! ! ! θFk (sk ) = θ(Fk ∩Ei ) (sk ) k=1

= ≤

k=1

i=1

i=1 ∞ !

k=1

" n ∞ ! !

# θ(Fk ∩Ei ) (sk )

|θ|(Ei , v)

i=1

by the limit rules established in Section 5 of Chapter I. For the converse inequality, let n ∈ N and for each i = 1, . . . , n, let F1i , . . . , Fni i ∈ R be disjoint subsets of Ei and si1 , . . . , sini ≤ v. Then "n # n i ! ! i θFki (sk ) ≤ |θ|(E, v), i=1

k=1

as the sets Fki ⊂ E are pairwise disjoint. Now taking the supremum over all such choices of sets Fik yields with ( 1) n !

|θ|(Ei , v) ≤ |θ|(E, v),

i=1

as n ∈ N was arbitrary.

hence

∞ ! i=1

|θ|(Ei , v) ≤ |θ|(E, v),

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II Measures and Integrals. The General Theory

Lemma 3.5. Let E ∈ R, α > 0 and u, v ∈ V. Then (a) |θ|(E, αv) = α|θ|(E, v). (b) |θ|(E, u + v) = |θ|(E, u) + |θ|(E, v). Proof. Part (a) is obvious. For Part (b), let E1 , . . . , En ∈ R be disjoint subsets of E and let ri ∈ P such that ri ≤ u+v for i = 1, . . . , n. According to (QF2) in I.6.1 there are elements si , ti ∈ P such that si ≤ u, ti ≤ v and si ≤ ri + ti . This shows n !

θEi (ri ) ≤

n !

i=1

θEi (si ) +

i=1

n !

θEi (ti ) ≤ |θ|(E, u) + |θ|(E, v).

i=1

As the sets Ei ∈ R and the elements ri ≤ u + v were chosen arbitrarily, this shows |θ|(E, u + v) ≤ |θ|(E, u) + |θ|(E, v). For the converse inequality, let . , Fm ∈ R be two collections of disjoint subsets of E1 , . . . , En ∈ R and F1 , . . E. We may assume that ni=1 Ei = m k=1 Fk = E. Let si ≤ u and tk ≤ v for si , tk ∈ P. Then n ! i=1

θEi (si ) +

m !

θFk (tk ) =

k=1

n ! m !

θ(Ei ∩Fk ) (si + tk ) ≤ |θ|(E, u + v)

i=1 k=1

by the above. Taking ﬁrst the supremum over all choices for the sets Ei ∈ R and the elements si ≤ u on the left-hand side of this inequality and using ( 1) yields |θ|(E, u) +

m !

θFk (tk ) ≤ |θ|(E, v + u).

k=1

In a second step, we obtain |θ|(E, u) + |θ|(E, v) ≤ |θ|(E, u + v) if we repeat this argument for the sets Fk ∈ R and the elements tk ≤ v. 3.6 Bounded Measures. Let (P, V) be a quasi-full locally convex cone and let (Q, W) be a locally convex complete lattice cone. We shall say that an L(P, Q)-valued measure θ on R is R-bounded or of bounded semivariation on R if (BV) For every w ∈ W and E ∈ R there is v ∈ V such that |θ|(E, v) ≤ w. In the sequel we shall always assume boundedness in this sense. Remarks 3.7. (a) If (P, V) is a full locally convex cone, then every L(P, Q)valued measure on R is bounded. Indeed, let E ∈ R and w ∈ W. Because the operator θE : P → Q is supposed to be continuous, there is v ∈ V such that θE (a) ≤ θE (b) + w whenever a ≤ b + v for a, b ∈ P. Following Lemma 3.3, this shows |θ|(E, v) = θE (v) ≤ w in particular.

3. Operator-Valued Measures

135

(b) Let P = K for K = R or K = C, endowed with the equality as order and the usual Euclidean topology; that is V = {εB | ε > 0}, where B is the unit ball in K and a ≤ b + εB means that a ∈ b + εB. Let Q = R. Then L(P, Q) can be identiﬁed with K, since every linear operator (functional) from K to R is given by an element z ∈ K via the evaluation a → e(za) for a ∈ K. This is therefore the case of a real- or complex-valued measure θ. According to 3.2, its modulus is computed as

n ! θEi · si si ∈ B, Ei ∈ R disjoint subsets of E |θ|(E, B) = sup i=1

= sup

n !

|θEi | Ei ∈ R disjoint subsets of E

,

i=1

that is the usual total variation of the real- or complex-valued measure θ (see II.1.4 in [55]). (c) If (P, V) is a locally convex topological vector space, and Q = R, that is the case of a functional-valued measure, our requirement of boundedness corresponds to Dieudonn´e’s notion of p-domination in [44] and to Prolla’s of ﬁnite p-semivariation in [155] (Ch. 5.5) for measures with values in the dual of a locally convex vector space. (d) If (N , ) is a normed space over K = R or K = C, then every N -valued measure θ may be considered to be an operator-valued measure in our sense. Indeed, the elements of N are linear operators from P = K, endowed with the Euclidean topology, into the standard lattice completion , W) $ of N as introduced in I.5.57. The notion of the semivariation of a (N vector-valued measure as given for example in IV.10.3 in [55] slightly diﬀers from our notion of the modulus, as there it is a real-valued expression (in fact, it is in some sense the norm in N of our modulus; see Section 8 below), which is however not countably additive in general. We shall consider this example in more detail in Section 6 below. (e) If (P, V) is a quasi-full locally convex cone and if we endow the sub % cone P+ of its positive elements with the neighborhood system V = {0}, for this, see also Example I.1.4(b) , then every L(P, Q)-valued measure θ can be canonically extended to an L(P+ , Q)-valued measure on the whole σ-algebra AR : For every set F ∈ AR we deﬁne the operator θF ∈ L(P+ , Q) by θF (a) = sup{θE (a) | E ⊂ F, E ∈ R} ∈ Q for a ∈ P+ . Linearity of this operator follows from I.5.22, and continuity is % = {0}. For trivial, since P+ is endowed with the neighborhood system V countableadditivity on AR let Fn ∈ AR , for n ∈ N, be disjoint sets and let F = n∈N Fn . Let a ∈ P+ . Then

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II Measures and Integrals. The General Theory

θF (a) = sup{θE (a) | E ⊂ F, E ∈ R}

∞ ∞ ! ! θE∩Fi (a) | E ⊂ F, E ∈ R ≤ θFi (a). = sup i=1

i=1

such that Ei ⊂ Fi for For every n ∈ N, on the other hand, and Ei ∈ R n i = 1, . . . , n, we set E = ni=1 Ei ∈ R and have i=1 θEi (a) = θE (a) ≤ over all such choices of sets Ei ∈ R yields θF (a). Taking the supremum n θ (a) ≤ θ (a). This holds for all n ∈ N with Lemma I.5.5(a) that F i=1 Fi ∞ θ (a) ≤ θF (a). and therefore yields the reverse inequality i=1 Fi 3.8 Extension of a Measure. We may use the modulus of an R-bounded L(P, Q)-valued measure θ to deﬁne an extension to an R-bounded L(PV , Q)valued measure, where (PV , V) denotes the standard full extension of the quasi-full locally convex cone (P, V) as constructed in Section 6.2 of Chapter I, that is PV = a ⊕ v | a ∈ P, v ∈ V ∪ {0} . This follows the extension of a continuous linear operator from P to Q into a continuous linear operator from PV to Q as elaborated in Theorem I.6.3. For E ∈ R and a ⊕ v ∈ PV we set θE (a ⊕ v) = θE (a) + |θ|(E, v). The required properties for a measure are readily checked. Indeed, for a ﬁxed set E ∈ R, Lemma 3.5 shows that θE is a linear operator on PV . In order to verify that this operator is monotone, let a ⊕ v ≤ b ⊕ u for a ⊕ v, b ⊕ u ∈ PV . Let E1 , . .. , En ∈ R be disjoint subsets of E and s1 , . . . , sn ≤ v. We set E0 = E \ ni=1 Ei and s0 = 0. Then ni=0 Ei = E and a + si ≤ b + u for all i = 0, . . . , n by our deﬁnition of the order in PV , hence a + si ≤ b + ti for some ti ≤ u by Condition (QF1) from I.6.1. Thus θEi (a + si ) ≤ θEi (b + ti ) for all i = 0, . . . , n by the monotonicity of the operators θEi , hence θE (a) +

n !

θEi (si ) =

i=1

≤

n ! i=0 n !

θEi (a + si ) θEi (b + ti )

i=0

= θE (b) +

n !

θEi (ti )

i=1

≤ θE (b) + |θ|(E, u). Taking the supremum over all such choices of sets Ei ∈ R and elements si ≤ v on the left-hand side of this inequality yields θE (a ⊕ v) = θE (a) + |θ|(E, v) ≤ θE (b) + |θ|(E, u) = θE (b ⊕ u).

3. Operator-Valued Measures

137

Furthermore, given w ∈ W, by the R-boundedness of the given measure θ, there is v ∈ V such that θE (0 ⊕ v) = |θ|(E, v) ≤ w. This implies that the linear operator θE : PV → R is indeed continuous. The countable additivity of the extended measure follows from Lemma 3.4(c). Furthermore, as (PV , V) is a full cone, the extension of θ remains R-bounded (see 3.7(a)), that is, |θ|(E, 0 ⊕ v) = θE (0 ⊕ v) = |θ|(E, v) holds for all E ∈ R and v ∈ V. If on the other hand, θ is an R-bounded L(PV , Q)-valued measure on R, and if θ0 denotes its restriction to an L(P, Q)-valued measure, then we have |θ0 |(E, v) ≤ θE (0 ⊕ v). This procedure of extending a given R-bounded measure from a quasi-full to a full cone yields an interesting new understanding of the (total) variation of a given measure, not as a separate positive real-valued measure associated with the given one, but as an integral part of its extension. Because this extension, evaluated at the neighborhoods is also Q- and not necessarily positive real-valued, its countable additivity is preserved, thus removing a major inconvenience that arises in the classical approach (see IV.10.3 in [55]). This therefore avoids the need to introduce the separate terms of variation and semivariation for a measure (see I.2 in [43]). The extension of a given measure as carried out in 3.8 will turn out to be invaluable in our upcoming integration theory for cone-valued functions with respect to an operator-valued measure. It does in fact justify the use of a full cone for P, that is the range of the concerned functions and the domain of the linear operators resulting from our measures. 3.9 Composition of Measures and Continuous Linear Operators. % V) % be quasi-full locally convex cones, and let (Q, W) Let (P, V) and (P, % & and (Q, W) be locally convex complete lattice cones. For continuous linear % P), T ∈ L(P, Q) and U ∈ L(Q, Q) % let U ◦ T ◦ S ∈ operators S ∈ L(P, % % L(P, Q) denote their composition, that is the continuous linear operator % → Q. % l → U T S(l) : P It is straightforward to verify that this operator is indeed linear and continuous. We shall use this in the following way: If θ is an L(P, Q)-valued % P) and if the operator U ∈ L(Q, Q) % is order measure on R, if S ∈ L(P, continuous (see I.5.29), then the set function % Q) % E → (U ◦ θE ◦ S) : R → L(P, % Q)-valued % is an L(P, measure, called the composition of θ with U and S and denoted as (U ◦ θ ◦ S). Countable additivity follows from the order sets such continuity of the operator U. Indeed, let Ei ∈ R be disjoint that ∞ % E= ∞ i=1 Ei ∈ R. Then for every l ∈ P we have θE S(l) = i=1 θEi S(l) by the countable additivity of θ, hence

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II Measures and Integrals. The General Theory

(U ◦ θ ◦ S)E (l) = U θE S(l) # "∞ ! =U θEi S(l) i=1 ∞ ! = U θEi S(l) i=1

=

∞ !

(U ◦ θ ◦ S)Ei (l)

i=1

by the order continuity of U. The modulus of the measure (U ◦ θ ◦ S) can % such that be estimated as follows: Let E ∈ R, and for v ∈ V let v˜ ∈ V % If E1 , . . . , En ∈ R are S(l) ≤ S(m) + v whenever l ≤ m + v˜ for l, m ∈ P. % such that li ≤ v˜ for i = 1, . . . , n, then disjoint subsets of E and if li ∈ P " n # n ! ! U ◦ θ ◦ S Ei (li ) = U θEi S(li ) ≤ U |θ|(E, v) . i=1

i=1

Taking the supremum over all such choices for sets Ei ∈ R and elements li ≤ v˜ yields |U ◦ θ ◦ S|(E, v˜) ≤ U |θ|(E, v) . % Q)-valued % The L(P, measure (U ◦θ ◦S) is therefore R-bounded whenever the L(P, Q)-valued measure θ is R-bounded. Indeed, for E ∈ R and w ˜∈ & there is w ∈ W such that U (s) ≤ U (t) + w W ˜ whenever s ≤ t + w ˜ for s, t ∈ Q. There is v ∈ V such that |θ|(E, v) ≤ w, hence |U ◦θ ◦S|(E, v˜) ≤ w ˜ % if v˜ ∈ V is chosen as above. We shall in particular make use of the combination of an L(P, Q)-valued measure θ with an order continuous linear functional μ ∈ Q∗ . (We choose % = P and the identity operator for S.) The resulting measure (μ ◦ θ) is P L(P, R) -, that is P ∗ -valued in this case. 3.10 Strong Additivity. Countable additivity of an L(P, Q)-valued measure θ is meant with respect to order convergence in the locally convex complete lattice cone (Q, W). Order convergence does in general not imply convergence in the weak or indeed convergence in the symmetric relative topology of Q (see I.5.42). However, the following result based on a wellknown theorem by Pettis (see Theorem IV.10.1 in Dunford & Schwartz, [55]) will show that in special cases some stronger type of convergence is implied. Theorem 3.11. Let θ be an L(P, Q)-valued measure, let a be a bounded element of P and let Q0 be the subcone of Q spanned by the set {θE (a) | E ∈ R}. If every continuous linear functional on Q0 can be extended to an order continuous linear functional on Q, then for disjoint sets Ei ∈ R such that ∪∞ i=1 Ei ∈ R the series

3. Operator-Valued Measures

139

θ(

Ei ) (a) =

i∈N

∞ !

θEi (a)

i=1

converges in the symmetric topology of Q. Proof. We shall follow the main lines of the arguments in the proof of Pettis’ Theorem as presented in [55]. Let a ∈ P be a bounded element, and let Q0 be the subcone of Q spanned by the set {θE (a) ∈ Q0 | E ∈ R}. As all the operators θE are continuous, the elements of Q0 are bounded in Q. We may therefore consider Q0 as a locally convex cone endowed with the symmetric topology generated by the neighborhood system W. Let Qs∗ 0 be the dual of Q0 under this topology. According to Proposition II.2.21 in [100], the linear functionals μ ∈ Qs∗ 0 can be expressed as the diﬀerence of two elements of the given dual cone (with respect to the given topology) Q∗0 of Q0 , that is ∗ ∗ ∗ Qs∗ 0 = Q0 −Q0 . As the elements of Q0 were supposed to be order continuous s∗ on Q0 , so are the elements of Q0 . Now let us consider a sequence of disjoint sets Ei ∈ R such that E = ∪∞ i=1 Ei ∈ R. Let Z0 ⊂ R be the set algebra in E generated by the sets Ei , and let Z ⊂ R be the σ-algebra in E generated by Z0 . The algebra Z0 is known to be countable (see III.8.4 in [55]). Let Q1 be the closure (with respect to the symmetric topology) in Q0 of the subcone that is spanned by the countable set {θE (a) | E ∈ Z0 }. In a ﬁrst step, an argument using the separation result from Corollary 4.6 in [172] will demonstrate that θE (a) ∈ Q1 for all E ∈ Z. For this, assume / Q1 for some E ∈ Z. Then according to the to the contrary that θE (a) ∈ separation result 4.6 in [172] there is a linear functional μ ∈ Qs∗ 0 such that such that μ θE (a) ≤ −1 ≤ μ(l) for all l ∈ Q1 . As Q1 is a cone, this implies indeed that μ(l) ≥ 0 holds for all l ∈ Q1. As the linear functional was seen to be order continuous, G → μ θ (a) : Z → R deﬁnes a μ ∈ Qs∗ G 0 countably additive real-valued measure (μ◦θ◦a) on Z. This measure, taking non-negative values on Z0 and a negative value on E ∈ R contradicts the uniqueness part of Hahn’s extension theorem for measures from an algebra Z0 to the σ-algebra Z generated by Z0 (see III.5.9 in [55] or 12.2.8 in [178]). Thus θE (a) ∈ Q1 as claimed. Now set Fn = ni=1 Ei for n ∈ N, and let us assume that, contrary to our claim, there exists a neighborhood w ∈ W, and a subsequence (Fm )m∈N of (Fn )n∈N such that either θE (a) θFm (a) + w

or

θFm (a) θE (a) + w

for all m ∈ N. Then according to Theorem 3.11 in [175] (see also Corollary 4.34 in Chapter I) there are linear functionals μm ∈ Qs∗ 0 , contained in the polar of the symmetric neighborhood w, such that μm θE (a) > μm θFm (a) + 1 for all m ∈ N. Let {lk | k ∈ N} be a countable dense (with respect to the topology) subset of Q1 . Since for every k ∈ N the sequence symmetric μm (lk ) m∈N is bounded in R, we may use a Cantor diagonal procedure to ﬁnd a subsequence (μmj )j∈N of (μm )m∈N such that the limit lim μmj (lk ) j→∞

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II Measures and Integrals. The General Theory

exists in R for all k ∈ N : Indeed, there is a subsequence (μm(1,j) )j∈N of (μm )m∈N such that lim μm(1,j) (l1 ) exists in R. Then there is a subsequence j→∞

(μm(2,j) )m∈N of (μm(1,j) )m∈N such that lim μm(2,j) (l2 ) exists, etc. We set j→∞

μmj = μm(j,j) for all j ∈ N. Then (μmj )j∈N is a subsequence of each of the sequences (μm(k,j) )m∈N for k ∈ N, thus satisfying our requirement. Now a simple argument will show that the limit lim μmj (l) exists indeed for all j→∞

l ∈ Q1 . In fact, given l ∈ Q1 and ε > 0 there is some lk such that both l ≤ lk + εw and lk ≤ l + εw, hence |μmj (l) − μmj (lk )| ≤ ε for all j ∈ N. Moreover, there is j0 ∈ N such that |μmj1 (lk ) − μmj2 (lk )| ≤ ε whenever j1 , j2 ≥ j0 . This implies |μmj1 (lk ) − μmj2 (l)| ≤ 3ε. Thus the sequence μmj (l) j∈N is a Cauchy sequence, hence convergent in R. For every j ∈ N let (μmj ◦ θ ◦ a) denote the real-valued measure G → μmj θG (a) : Z → R. Then lim (μmj ◦ θ ◦ a)(G) exists for every G ∈ Z by the above, hence j→∞

following Nikod´ ym’s theorem (see Corollary III.7.4 in [55]) the countable additivity of these measures is uniform in j. As E = ∞ j=1 Fj , this contradicts our assumption that (μmj ◦ θ ◦ a)(E) > (μmj ◦ θ ◦ a)(Fj ) + 1 holds for all j ∈ N. This result applies in particular if (Q, W) is the standard lattice completion of some subcone Q0 of Q and if the measure θ is L(P, Q0 )-valued. In this case, all continuous linear functionals on Q0 extend to order continuous linear functionals on Q, as required in Theorem 3.11. If all elements of P are bounded, then countable additivity of an L(P, Q0 )-valued measure implies convergence of the concerned operators with respect to the strong operator topology of L(P, Q0 ) see I.7.2(ii) . We shall provide a simple example of a measure that is countably additive with respect to order convergence but not with respect the symmetric topology of Q. Example 3.12. Let P = R with its usual (Euclidean) topology, and let Q be the cone of all R-valued bounded below functions on the interval [0, 1], endowed with the pointwise algebraic operations and order, and the constant functions w > 0 as neighborhoods. Then (Q, W) is a locally convex complete lattice cone. Let R be the σ-algebra of Borel sets on X = [0, 1]. For every E ∈ R let θE be the linear operator in L(P, Q) that maps ρ ∈ R into ρχE ∈ Q, where χE denotes the characteristic function of the set E. Clearly θ is countably additive with respect to order convergence, but not with respect to uniform convergence, that is convergence with respect to the symmetric topology in Q. If (Q0 , W0 ) is locally convex topological vector space over K = R or K = C, then the elements of Q0 may be considered as continuous linear operators from P = K, endowed with the Euclidean topology, into the standard lattice completion (Q, W) of Q0 . (This situation will be explored in greater detail in Example 6.23 below.) A Q0 -valued measure is required to be countably

4. Integrals for Cone -Valued Functions

141

additive with respect to order convergence in Q, that is weak convergence in Q0 . According to Theorem 3.11 (use a = 1 ∈ P), this implies convergence with respect to the symmetric topology of Q, that is the given topology of Q0 . This result is commonly known as Pettis’ theorem. Corollary 3.13. Let (P, V) be a locally convex topological vector space over R or C. For a P-valued measure countable additivity with respect to the weak topology implies countable additivity with respect to the given topology of P. 3.14 Weak Compactness. A well-known result due to Bartle, Dunford and Schwartz see Corollary I.2.7 in [43] or Theorem VI.7.3 in [55] about the relative weak compactness of the range of a vector-valued measure implies the following for operator-valued measures: Theorem 3.15. Suppose that (Q, W) is the standard lattice completion of a Banach space (Q0 , ) over R or C and that θ is a bounded L(P, Q0 )valued measure. Then for every a ∈ P and every E ∈ R the set {θG (a) | F ∈ R, G ⊂ E} is relatively compact in Q0 with respect to the weak topology σ(Q0 , Q∗0 ). Proof. Let a ∈ P and E ∈ R. The family RE = {G ∈ R, G ⊂ E} is a σalgebra on E, and the set function G → θG (a) : RE → Q0 is a countably additive Q0 -valued, that is a Banach space-valued measure on RE . Our claim then follows directly from Corollary I.2.7 in [43].

4. Integrals for Cone -Valued Functions Throughout this section, let (P, V) be a full locally convex cone and let (Q, W) be a locally convex complete lattice cone. Let R be a (weak) σring of subsets of X and θ an L(P, Q)-valued measure on R. The requirement that the locally convex cone (P, V) is full does in fact accommodate quasi-full cones as well. Indeed, in this case we may take advantage of the embedding of a quasi-full cone (P, V) into the full locally convex cone (PV , V), that is its standard full extension, as elaborated in I.6, and make use of the corresponding extension of an R-bounded L(P, Q)-valued measure θ to an L(PV , Q)-valued measure as constructed in Section 3.8; that is, we may set θE (v) = |θ|(E, v) for every set E ∈ R and every neighborhood v ∈ V ⊂ PV . We proceed to deﬁne integrals for cone-valued functions with respect to θ. The values of these integrals will be elements of Q. We shall use the cone FR (X, P) of all P-valued measurable functions on X that can be reached from below by P-valued step functions in the sense of Section 2.3. Similarly, FR (X, V) denotes the cone of all measurable V ∪{0} -valued functions on X.

142

II Measures and Integrals. The General Theory

In a ﬁrst step, we shall deﬁne integrals for P- and V-valued step functions n on X, that is functions s = i=1 χEi ⊗ai for Ei ∈ R and elements ai in P or V, respectively. We shall denote the corresponding subcones of in SR (X, V) F(X, P) by SR (X, P) and SR (X, V). Note that the functions n χ a ⊗ are V ∪ {0} -valued. Obviously, any representation i=1 Ei i for a given step function is not unique. To prepare our deﬁnition of the integral for functions in SR (X, P) we observe: Lemma 4.1. . , n and k = Let Ei , Fk ∈ R and ai , bk ∈ P nfor i = 1, . . m 1, . . . , m. If ni=1 χEi ⊗ai ≤ m k=1 χFk ⊗bk , then i=1 θEi (ai ) ≤ k=1 θFk (bk ). n Proof. First we shall verify that for any step function s = i=1 χEi ⊗ai there exists a representation m k=1 χFk ⊗bk such mthat the sets Fk ∈ R are pairwise n θ (a ) = disjoint and such that E i i i=1 k=1 θFk (bk ). We shall use induction with respect to n. For n = 1 there is nothing n+1to prove. Assume that our claim holds true for some n ≥ 1 and let s = i=1 χEi ⊗ai . There are disjoint n m ⊗bk satisfying the above. By χ χ ⊗ai = sets Fk ∈ R such that E F i k i=1 k=1 ⊗0 to the right-hand of the last equation, we may adding a suitable term χ F assume that En+1 ⊂ m k=1 Fk . Hence s= =

m ! k=1 m !

χFk ⊗bk + χEn+1 ⊗an+1 χ(Fk ∩En+1 ) ⊗(bk + an+1 ) +

k=1

m !

χ(Fk \En+1 ) ⊗bk .

k=1

The sets in the above representation for s are disjoint, and we have indeed n+1 !

θEi (ai ) =

i=1

=

m ! k=1 m !

θFk (bk ) + θEn+1 (an+1 ) θ(Fk ∩En+1 ) (bk + an+1 ) +

k=1

m !

θ(Fk \En+1 ) (bk )

k=1

as claimed. Thus, to prove our claim in Lemma 4.1, we may assume n that both and F are pairwise disjoint and that families of sets E i k i=1 χEi ⊗ai ≤ m χ b . By adding suitable terms χ ⊗ 0 and χ ⊗ 0 on the left- and ⊗ E F k=1 Fk k right-hand sides of the above inequality, we may assume in addition that F . Under these assumptions the sets E ∩ F form a E = ni=1 Ei = m i k k=1 k disjoint partition of E, and we have either Ei ∩ Fk = ∅ or ai ≤ bk . This yields n !

θEi (ai ) =

i=1

as claimed.

n ! m ! i=1 k=1

θ(Ei ∩Fk ) (ai ) ≤

m ! n ! k=1 i=1

θ(Ei ∩Fk ) (bk ) =

m ! k=1

θFk (bk ),

4. Integrals for Cone -Valued Functions

143

4.2 Integrals for P-Valued Step Functions. We are now in a position to deﬁne the integral for a P-valued step function h=

n !

χEi⊗ai ∈ SR (X, P)

i=1

over a measurable set F ∈ AR with respect to θ by ' h dθ = F

n !

θ(Ei ∩F ) (ai ).

i=1

Lemma 4.1 implies that the sum on the right-hand side is independent of the particular representation for h. The integral represents a monotone linear operator from SR (X, P) into Q. Lemma 4.3. Let F ∈ AR , let h, g ∈ SR (X, P) and α ≥ 0. Then ( ( (a) (F (αh) dθ = α (F h dθ. ( (b) (F (g + h) (dθ = F g dθ + F h dθ. (c) (F g dθ ≤ (F h dθ whenever g ≤ h. (d) F g dθ = X (χF ⊗g) dθ. All these properties are obvious from the deﬁnition of the integral and from Lemma 4.1. We shall demonstrate in the following lemma that, if the full cone (P, V) is in fact the standard full extension (P0 V , V) of a quasi-full cone (P0 , V), and if θ is the canonical extension of an R-bounded L(P0 , Q)-valued measure θ0 , as elaborated in I.6 and 3.8, then the way in which this extension was constructed, guarantees that the integral is already determined by its values on the subcone SR (X, P0 ) of SR (X, P), that is by P0 -valued step functions and the given measure θ0 . Lemma 4.4. Let F ∈ AR and g ∈ SR (X, P). If (P, V) is the standard full extension of the quasi-full cone (P0 , V), and if θ is the canonical extension of an R-bounded L(P0 , Q)-valued measure θ0 , then ' ' g dθ = sup h dθ h ∈ SR (X, P0 ), h ≤ g . F

F

Proof. Following 4.3(c), we may assume that F = X. Let us ﬁrst recall and reformulate from 3.2 the deﬁnition of the modulus of θ0 for a set E ∈ R and a neighborhood v ∈ V.

n ! θEi (si ) si ∈ P0 , si ≤ v, Ei ∈ R disjoint subsets of E |θ0 |(E, v) = sup i=1

' = sup

X

h dθ h ∈ SR (X, P0 ), h ≤ χE ⊗v .

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II Measures and Integrals. The General Theory

Recall that the extension of θ0 into θ was constructed by setting θE (v) = |θ0 |(E, v). Now we consider the case that n !

g=

χEi⊗vi ∈ SR (X, V)

i=1

is a V-valued function. We compute using Lemma I.5.5(a) ' g dθ = X

=

n ! i=1 n ! i=1

= sup

θEi (vi ) |θ0 |(Ei , vi )

n ' ! i=1

' = sup

hi dθ hi ∈ SR (X, P0 ), hi ≤ χEi ⊗v

X

h dθ h ∈ SR (X, P0 ),

h≤g ,

X

as claimed. Now for the general case, let g=

n !

χEi⊗(ai + vi ) ∈ SR (X, P),

i=1

n for ai ∈ P0 and vi ∈ V. Set g1 = i=1 χEi⊗ai ∈ SR (X, P0 ) and n χ ⊗vi ∈ SR (X, V). Then g = g1 + g2 , and the above yields g2 = i=1 Ei with property ( 1) ' ' ' g dθ = g1 dθ + g2 dθ X X X ' (g1 + h) dθ h ∈ SR (X, P0 ), h ≤ g2 = sup 'F ≤ sup h dθ h ∈ SR (X, P0 ), h ≤ g . F

The converse inequality is obvious from 4.3(c).

Subsequently, with every neighborhood w ∈ W we associate the inductive limit neighborhood vw , deﬁned as ' s dθ ≤ w . vw = s ∈ SR (X, V) X

(We shall write vw (θ) if diﬀerent measures are involved in our considerations.) The boundedness of θ guarantees that for every E ∈ R there is v ∈ V such that χE ⊗v ∈ vw . Convexity follows from Lemma 4.3. We have

4. Integrals for Cone -Valued Functions

145

v(λw) = λvw and vw + vw ≤ v(w+w ) for w, w ∈ W and λ > 0. We proceed to develop the integral over a measurable set F ∈ AR for a function f ∈ FR (X, P) in the following manner: First, for a neighborhood w ∈ W we set '

'

(w)

h dθ h ∈ SR (X, P), h ≤ f + vw .

f dθ = sup F

F

We note that in the situation of Lemma 4.4, the integral of a function in FR (X, P) is already determined by P0 -valued step functions alone: Lemma 4.5. Let F ∈ AR , f ∈ FR (X, P) and w ∈ W. If (P, V) is the standard full extension of a quasi-full cone (P0 , V), and if θ is the canonical extension of an L(P0 , Q)-valued measure θ0 , then '

'

(w)

h dθ h ∈ SR (X, P0 ), h ≤ f + vw .

f dθ = sup F

F

We proceed with a simple observation for step functions. Lemma 4.6. Let F ∈ AR , f ∈ SR (X, P) and w ∈ W. Then '

' f dθ ≤

'

(w)

f dθ ≤

F

F

f dθ + w. F

Proof. The ﬁrst part of the inequality is trivial. For the second part, let h ≤ f + vw for h ∈ SR (X, P), that is h ≤ f + s for some V-valued step function s ∈ vw . Following Lemma 4.3(b) and (c), this implies ' ' ' ' h dθ ≤ f dθ + s dθ ≤ f dθ + w F

F

F

F

for each such step function h ∈ SR (X, P), hence claimed.

( (w) ( F f dθ ≤ F f dθ + w as

Proposition 4.7. Let E ∈ R and f ∈ FR (X, P) and let (hn )n∈N be a bounded below sequence of step functions in SR (X, P) such that for every x ∈ E there is n0 ∈ N such that f (x) ≤ hn (x) for all n ≥ n0 . Then '

'

(w)

f dθ ≤ lim

E

n→∞

hn dθ + w E

for every w ∈ W. Proof. Let E ∈ R, f ∈ FR (X, P), and let (hn )n∈N be a sequence of step functions satisfying our assumptions. For w ∈ W let l ∈ SR (X, P) such that l ≤ f + vw , that is l ≤ f + s for some s ∈ vw . Now we set En = {x ∈ E | l(x) ≤ hm (x) + s(x) for all

m ≥ n}.

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II Measures and Integrals. The General Theory

All the sets En are measurable, En ⊂ En+1 and E = n∈N En by our assumption. Given any u ∈ W there is v ∈ V such that θE (v) ≤ u, and as the sequence (hn )n∈N is bounded below, there is ρ ≥ 0 such that 0 ≤ hn + ρχX ⊗v for all n ∈ N. Thus χEn ⊗l ≤ χEn ⊗(hn + s) + χ(E\En ) (hn + ρχX ⊗v) ≤ χE ⊗hn + χEn ⊗s + ρχ(E\En ) ⊗v. Hence by Lemma 4.3, and because ' ' χEn s dθ ≤ s dθ ≤ w, and X

' χ(E\En ) ⊗v = θ(E\En ) (v),

X

X

when taking the integrals over X in the above inequality, we obtain ' ' l dθ ≤ hn dθ + ρθ(E\En ) (v) + w. En

E

Because En ⊂ En+1 and

n∈N

En = E, Lemma 3.1(a) yields

θ(F ∩E) (a) = lim θ(F ∩En ) (a) n→∞

for all F ∈ R and a ∈ P. Considering the deﬁnition of the integral for a step function in 4.2, this renders ' ' lim l dθ = l dθ, n→∞

En

E

and Lemma 3.1(b) yields lim θ(E\En ) (v) ≤ O θE (v) ≤ ε u

n→∞

for all ε ≥ 0. Thus, using the limit rules from Lemma I.5.19, we obtain ' ' l dθ ≤ lim hn dθ + w + ε u. n→∞

E

E

Because u ∈ W and ε > 0 were arbitrary, and because Q carries the weak preorder, this shows ' ' l dθ ≤ lim hn dθ + w. E

n→∞

E

Our claim follows, since the above inequality holds true for all step functions l ∈ SR (X, P) such that l ≤ f + vw . Corollary 4.8. Let F ∈ AR , f ∈ FR (X, P) and u, w ∈ W. Then

4. Integrals for Cone -Valued Functions

'

147

'

(w)

(u)

f dθ ≤

f dθ + w.

F

F

Proof. Let F ∈ AR , f ∈ FR (X, P) and u, w ∈ W. Let l ∈ SR (X, P) such that l ≤ f + vw , and according to Lemma 2.4, we choose E ∈ R such that both h is supported by E and such that f (x) ≥ 0 for all x ∈ X \ E. For the set E ∩ F ∈ R, the inductive limit neighborhood v = vu and ε ≥ 0, let (hn )n∈N be a sequence of step functions in SR (X, P) approaching f as in Corollary 2.8. We may assume that the functions hn are supported by the set E ∩ F, since we may otherwise replace them by their product with the characteristic function of this set. Proposition 4.7 yields '

(w)

' f dθ ≤ lim

hn dθ + w.

n→∞

(E∩F )

(E∩F )

On the other hand, we have '

'

'

F

(w)

l dθ ≤

l dθ =

f dθ, (E∩F )

(E∩F )

since the function l is supported by E. Similarly, for the functions hn we observe that ' ' ' (u) hn dθ = hn dθ ≤ γ f dθ, (E∩F )

F

F

since hn ≤ γf + vw for all n ∈ N. Combining all of the above then yields '

' l dθ ≤ γ

(u)

f dθ + w

F

F

with some 1 ≤ γ ≤ 1 + ε, and indeed '

' l dθ ≤ F

(u)

f dθ + w, F

since ε > 0 was chosen independently. Finally, because this last inequality holds true for all l ∈ SR (X, P) such that h ≤ f + vw , our claim follows. 4.9 Integrals for Functions in FR (X, P). We may now deﬁne the integral over a set F ∈ AR for a function f ∈ FR (X, P) as '

'

(w)

f dθ = inf F

w∈W

f dθ. F

The above inﬁmum is well-deﬁned and yields an element of the locally convex complete lattice cone Q. Indeed, given any neighborhood u ∈ W there ( (λu) is λ ≥ 0 such that 0 ≤ f + λvu . Thus 0 ≤ F f dθ. According to Corollary 4.8, this yields

148

II Measures and Integrals. The General Theory

'

'

(λu)

0≤

(w)

f dθ ≤

f dθ + λu.

F

F

( (w) for all w ∈ V. This demonstrates that theset is F f dθ | w ∈ W bounded below, and its inﬁmum exists by ( 1). Moreover, our earlier observation in Lemma 4.5 justiﬁes that the above deﬁnition of the integral is consistent with the preceding one for step functions. Obviously, the integral is monotone, and we shall proceed to verify that it determines a continuous linear operator from FR (X, P) into Q. For Part (a) of the following lemma, recall from Lemma 2.6 that χF ⊗f ∈ FR (X, P) whenever f ∈ FR (X, P) and F ∈ AR . In Part (b) we consider R as the index set of a net, directed upward by set inclusion. Lemma 4.10. Let f ∈ FR (X, P) and F ∈ AR . Then ( ( (a) (F f dθ = X (χ(F ⊗f ) dθ. (b) F f dθ = lim (E∩F ) f dθ. E∈R

Proof. For Part (a) we ﬁrst note that χF ⊗f ∈ FR (X, P) (see Lemma 2.6). Let w ∈ W and h0 ∈ SR (X, P) such that h0 ≤ f + vw . We have '

'

(w)

h dθ h ∈ SR (X, P), h ≤ f + vw

f dθ = sup F

F

and '

'

(w)

χF ⊗f dθ = sup

h dθ h ∈ SR (X, P), h ≤ χF ⊗f + vw

X

X

First,(let h ∈ SR((X, P) such that h ≤ f +vw . Then h = χF ⊗h ≤ χF ⊗f +vw , and X h dθ = F h dθ by 4.3(d). This shows '

(w)

'

f dθ ≤

F

(w)

χF ⊗f dθ. X

For the converse inequality, let h ∈ SR (X, P) such that h ≤ χF ⊗f + vw . Then χF ⊗h ≤ χF ⊗f + vw and χ(X\F ) ⊗f + vw , hence h = ( ) ⊗h0 ≤( χ(X\F h dθ = h dθ. As χ(X\F ) ⊗h ≤ vw , χF ⊗h + χ((X\F ) ⊗h0 ≤ f + 2vw , and ( F ( F we have (X\F ) h dθ ≤ w, hence X h dθ ≤ F h dθ + w. This shows '

(w)

X

' χF ⊗f dθ ≤

(2w)

f dθ + w. F

Taking the inﬁma over all w ∈ W in the above inequality yields Part (a).

4. Integrals for Cone -Valued Functions

149

For Part (b) it is therefore suﬃcient to consider the case F = X, because the function f may be replaced by its product with the characteristic function χF . Let E0 ∈ R such that f (x) ≥ 0 for all x ∈ X \ E0 . (Then χE ⊗(f ≤ χE ⊗f whenever E0 ⊂ E ⊂ E for E, E ∈ R, hence f dθ ≤ f dθ by Part (a) and the monotony of the integral. This shows E E ' E∈R

'

'

lim

f dθ = E

f dθ ≤

sup

E0 ⊂E∈R

f dθ.

E

X

h is For the converse, let w ∈ W and h ≤ f +vw for h ∈ SR (X, ( P). Because ( supported by a set in R, there is E0 ⊂ E ∈ R such that X h dθ = E h dθ ≤ ( (w) ( (w) ( E f dθ. Moreover, Corollary 4.8 shows that E f dθ ≤ E f dθ + w. Thus '

' h dθ ≤

X

This shows

'

' f dθ ≤

X

(w)

X

sup

E0 ⊂E∈R

f dθ + w. E

' f dθ ≤

sup

E0 ⊂E∈R

f dθ + w, E

hence our claim, since w ∈ W was arbitrary and Q carries the weak preorder. 4.11 Sets of Measure Zero and Properties Holding Almost Everywhere. A set Z ∈ AR is said to be of measure zero (with respect to θ) if θ(E∩Z) = 0 for all E ∈ R. The family Z(θ) of all sets of measure zero is obviously closed for set complements and for countable unions. For a subset F of X we shall say that a pointwise deﬁned property of functions on X holds θ-almost everywhere on F if it holds on F \ Z with some Z ∈ Z(θ). ≤ or a.e.F = if the relations ≤ or In particular, we shall use the symbols a.e.F = hold θ-almost everywhere on the set F, respectively; that is for example, ≤ g + v for functions f, g ∈ F(X, P) and an inductive limit neighborf a.e.F hood v means that χ(F \Z) ⊗f ≤ χ(F \Z) ⊗g + v holds with some Z ∈ Z(θ). These relations are of course transitive and compatible with the algebraic operations. As θ(E∩Z) = 0 holds for all E ∈ R and ( Z ∈ Z(θ), ( we infer that θE = θ(E\Z) . Now Deﬁnition 4.2 yields that F h dθ = (F \Z) h dθ for all step functions h ∈ SR (X, P), F ∈ AR and Z ∈ Z(θ).( Considering ( our deﬁnition of the integral in 4.9 we observe that this yields F f dθ = (F \Z) f dθ for all ≤ g for functions f, g ∈ FR (X, P) f ∈ FR (X, P) as well. Consequently, f a.e.F implies that χ(F \Z) ⊗f ≤ χ(F \Z) ⊗g for some Z ∈ Z(θ), hence ' ' ' ' f dθ = f dθ ≤ g dθ = g dθ. F

(F \Z)

(F \Z)

F

150

II Measures and Integrals. The General Theory

In particular, any two functions in FR (X, P) that coincide θ-almost everywhere on a set F ∈ AR have the same integrals over F with respect to θ. 4.12 Integrability over a Set E ∈ R. We may now deﬁne integrability for cone-valued functions over measurable sets with respect to an operatorvalued measure. First, for a set E ∈ R we shall say that a function f ∈ F(X, P) is integrable over E with respect to θ if for every w ∈ W and ε > 0 there are functions f(w,ε) ∈ FR (X, P) and s(w,ε) ∈ FR (X, V) such that ' ≤ ≤ and s(w,ε) dθ ≤ εw f a.e.E f(w,ε) a.e.E γf + s(w,ε) E

for some 1 ≤ γ ≤ 1 + ε. Recall that the functions in FR (X, V) are actually V ∪ {0} -valued. However, in the case of Deﬁnition 4.12, without loss of generality we may assume that the function s(w,ε) ∈ FR (X, V) is indeed V-valued, as we can otherwise replace it by a function s˜((w,ε) = s(w,(ε/2)) + χX ⊗v, where v ∈ V is such that θE (v) ≤ (ε/2)w, hence E s˜(w,ε) dθ ≤ εw. Consequently, for an integrable function f ∈ F(X, P) and a net (f(w,ε) )ε>0 w∈W a of functions in FR (X, P) satisfying the above, we shall show that the limit ' ' f dθ = lim f(w,ε) dθ E

ε>0 w∈W

E

exists and is independent of the particular choice for the net (f(w,ε) )ε>0 w∈W . (The index set for this net is W × {ε > 0} with the reverse componentwise order.) Indeed, given w ∈ W and ε > 0, for all w1 , w2 ∈ W such that w1 , w2 ≤ w and 0 < ε1 , ε2 ≤ ε we have f(w1 ,ε1 ) ≤ γf + s(w1 ,ε1 ) ≤ γ1 f(w2 ,ε) + s(w1 ,ε1 ) for some 1 ≤ γ ≤ 1 + ε, hence ' ' f(w1 ,ε1 ) ≤ γ1 f(w2 ,ε2 ) + εw. E

(

E

ε>0

Thus E f(w,ε) w∈W forms a Cauchy net in the symmetric relative topology of Q, hence is convergent by Proposition I.5.41. The preceding argument together with Lemma I.5.20(c) also shows that this limit is independent of the particular choice for the net (f(w,ε) )ε>0 w∈W . 4.13 Integrability over a Set F ∈ AR . Obviously, integrability in the sense of 4.12 for a function f ∈ FR (X, P) over a set E ∈ R implies integrability over all subsets G ∈ R of E. This observation, together with Lemma 4.10 shows that we may consistently deﬁne integrability over sets in the σ-algebra AR in the following way: We shall say that a function f ∈ F(X, P) is integrable over F ∈ AR with respect to θ if f is integrable over the sets E ∩ F for all E ∈ R and if the limit

4. Integrals for Cone -Valued Functions

151

'

' f dθ = lim

f dθ

E∈R

F

(E∩F )

exists in Q. The set of all functions in F(X, P) that are integrable over F shall be denoted by F(F,θ) (X, P). Lemma 4.10(b) implies that FR (X, P) ⊂ F(F,θ) (X, P) for every F ∈ AR and every L(P, Q)-valued measure θ on R. We may use this deﬁnition of integrability also for functions that take the value ∞ ∈ V on a set of measure zero (see Section 2.1). Theorem 4.14. Let F ∈ AR . Then F(F,θ) (X, P) is a subcone of F(X, P) containing FR (X, P). More precisely, for f, g ∈ F(F,θ) (X, P) and 0 ≤ α ∈ R we have ( ( (a) (F (αf ) dθ = α (F f dθ ( (b) F (f + g) dθ = F f dθ + F g dθ ( ( ≤ g. (c) F f dθ ≤ F g dθ whenever f a.e.F Proof. In a ﬁrst case, let us assume that f, g ∈ FR (X, P) and that F = E ∈ R. Then Part (a) follows trivially from our deﬁnition of the integral. For (b), let w ∈ W, and h1 ≤ f + vw and h2 ≤ g + vw for h1 , h2 ∈ SR (X, P). Then h1 + h2 ≤ (f + g) + 2vw , hence '

'

(w)

(w)

f dθ + E

'

(f + g) dθ,

E

and therefore

E

'

'

(2w)

g dθ ≤ '

g dθ ≤

f dθ + E

(f + g) dθ.

F

E

For the converse inequality, let u ∈ W. For the set E ∈ R the inductive limit neighborhood vu and any ε > 0 choose sequences (hn )n∈N and (ln )n∈N of step functions in SR (X, P) approaching f and g as in Corollary 2.8, respectively. The sequence (kn )n∈N , where kn = hn + ln then approaches the function f + g with respect to F, the inductive limit neighborhood 2vu and ε. Thus by Proposition 4.7 we have '

'

(w)

(f + g) dθ ≤ lim

kn dθ + w

n→∞

E

'E

≤ lim

hn dθ + lim

n→∞

' ≤

'

E

'

(u)

n→∞

(u)

f dθ + E

ln dθ + w E

g dθ + w E

for all w ∈ W. This yields '

' (f + g) dθ ≤ E

'

(u)

(u)

f dθ + E

g dθ, E

152

II Measures and Integrals. The General Theory

since Q is a locally convex complete lattice cone, and indeed ' ' ' (f + g) dθ ≤ f dθ + f dθ E

E

E

after applying the inﬁma over all u ∈ W on the right-hand side and using the rules from Section I.5. Now in a second case, we still suppose that F = E ∈ R, and let ε>0 f, g ∈ F(E,θ) (X, P). Let (f(w,ε) )ε>0 w∈W and (g(w,ε) )w∈W be nets of functions in FR (X, P) approaching the functions f and g as in 4.12. Then the nets ε>0 (αf(w,ε) )ε>0 w∈W and (f(w,ε) +g(w,ε) )w∈W approach the functions αf and f +g, respectively, and the limit rules from Section 4 yield ' ' ' ' αf dθ = lim αf(w,ε) dθ = α f(w,ε) dθ = α f dθ ε>0 w∈W

E

and

E

E

'

E

' (f + g) dθ = lim ε>0 w∈W

E

(f(w,ε) + g(w,ε) ) dθ E

'

ε>0 w∈W

' f(w,ε) dθ + lim

= lim

ε>0 w∈W

E

'

'

f dθ +

=

g(w,ε) dθ E

g dθ.

E

E

≤ g and let (f(w,ε) )ε>0 and For Part (c) in this case, suppose that f a.e.E w∈W (g(w,ε) )ε>0 w∈W be nets in FR (X, P) as before. Then ≤ γf + s(w,ε)a.e.E ≤ γg + s(w,ε)a.e.E ≤ γg(w,ε) + s(w,ε) , f(w,ε) a.e.E hence

'

' f(w,ε) dθ ≤ γ

E

g(w,ε) dθ + εw E

with some 1 ≤ γ ≤ 1 + ε for all w ∈ W and ε > 0. According to the limit rules in Section I.5, this yields ' ' ' ' f dθ = lim f(w,ε) dθ ≤ lim g(w,ε) dθ = g dθ. E

ε>0 w∈W

E

ε>0 w∈W

E

E

For the ﬁnal and general case, let F ∈ AR and f, g ∈ F(F,θ) (X, P). Then the claims of Parts (a),(b) and (c) hold for integrals over all sets E ∩ F for E ∈ R. The deﬁnition of the respective integrals over F together with the limit rules from Lemma I.5.19 yield the validity of these claims for the integrals over F as well.

4. Integrals for Cone -Valued Functions

153

Simple examples can show that F ⊂ G for F, G ∈ AR does not necessarily imply that F(F,θ) (X, P) ⊂ F(G,θ) (X, P), but we have the following: Proposition 4.15. Let f ∈ F(X, P) and F, G ∈ AR (a) If F, G ∈ AR , then f is integrable over F ∩ G if and only if χG ⊗f is integrable(over F, if and ( only if χF ⊗(f is integrable over G. In this case we have (F ∩G) f dθ = F χG ⊗f dθ = G χF ⊗f dθ. (b) If F and G are disjoint and ( over F( and G, then f is ( f is integrable integrable over F ∪ G and (F ∪G) f dθ = F f dθ + G f dθ. ( (c) If ( F ⊂ G and f is integrable over F, G and G \ F, then O F f dθ ≤ O G f dθ . Proof. For Part (a), in a ﬁrst step let E ∈ R. First we observe from Deﬁnition 4.12 that a function f ∈ F(X, P) (is integrable ( over E if and only if χE ⊗f is integrable over E and that E f dθ = E χE ⊗f dθ. Thus, if for f ∈ F(X, P), the function χE ⊗f is integrable over X, then by Deﬁnition 4.13 the function χE ⊗f and therefore f is integrable over E. For the converse, assume that f ∈ F(X, P) is integrable over E. Let E ∈ R, w ∈ W and ε > 0. According to 4.12 there are f(w,ε) ∈ FR (X, P) ≤ ≤ and s(w,ε) ∈ FR (X, ( V) such that f a.e.E f(w,ε)a.e.E γf + s(w,ε) with some 1 ≤ γ ≤ 1 + ε and E s(w,ε) dθ ≤ εw. Then we have ≤ γf + χE ⊗s(w,ε) ≤ χE ⊗f(w,ε)a.e.E χE ⊗f a.e.E and

' E

χE ⊗s(w,ε) dθ ≤ εw

as well. Because the functions χE ⊗f(w,ε) and χE ⊗s(w,ε) are also contained in FR (X, P) and FR (X, V), respectively, we conclude that the function χE ⊗f is integrable over E and that ' ' ' χE ⊗f dθ = lim χE ⊗f(w,ε) dθ = lim χ(E ∩E) ⊗f(w,ε) dθ. E

ε>0 w∈W

E

ε>0 w∈W

X

The last equality follows from Lemma 4.10(a). The above holds for all sets E ∈ R, hence using Deﬁnition 4.13, we realize that the function χE ⊗f is indeed integrable over X and that ' ' ' χE ⊗f dθ = lim χ ⊗f dθ = lim χE ⊗f(w,ε) dθ E X

E ∈R

E

ε>0 w∈W

'

' f(w,ε) dθ =

= lim ε>0 w∈W

E

X

f dθ E

154

II Measures and Integrals. The General Theory

holds. Thus we have veriﬁed that a function f ∈ F(X, P) is integrable over ( ⊗ f is integrable over X and that f dθ = a set E ∈ R if and only if χ E E ( χ ⊗ f dθ in this case. Now in a second step, let F ∈ A and f ∈ F(X, P). R X E By the above, the function f is integrable over all sets E ∩ F for E ∈ R, if and only if all the functions χ(E∩F ) ⊗f = χE ⊗(χF ⊗f ) are integrable over X. In this case ' ' ' ' f dθ = χ(E∩F ) ⊗f dθ = χE ⊗(χF ⊗f ) dθ = χF ⊗f dθ (E∩F )

X

X

E

holds by our ﬁrst step. According to Deﬁnition 4.13 therefore f is integrable over F if and only χF ⊗f is integrable over X and ' ' ' ' f dθ = lim f dθ = lim χF ⊗f dθ = χF ⊗f dθ. F

E∈R

E∈R

(E∩F )

E

X

In a third and ﬁnal step for Part (a), let F, G ∈ AR . From the preceding we conclude that χG ⊗f is integrable over F if and only if χF ⊗(χG ⊗f ) = χ(F ∩G) ⊗f is integrable over X, that is f is integrable over F ∩ G, and all the integrals coincide. For Part (b), suppose that F ∩ G = ∅ and that f is integrable over both F and G. Then both functions χF ⊗f and χG ⊗f are integrable over X by Part (a), hence χ(F ∪G) ⊗f = χF ⊗f + χG ⊗f is also integrable over X by Theorem 4.14(b). Thus f is indeed integrable over F ∪ G and ' ' ' ' ' ' f dθ = χ(F ∪G) ⊗f = χF ⊗f + χG ⊗f = f dθ + f dθ (F ∪G)

X

X

X

F

G

by 4.14(b) For Part (c), suppose that F ⊂ G and that f is integrable over F, G and G \ F. Then ' ' ' f dθ = f dθ + f dθ G

F

(G\F )

by Part (b), and ' ' ' O f dθ ≤ O f dθ + O F

by Proposition I.5.11(a).

F

(G\F )

f dω

' =O

f dθ .

G

Proposition 4.16. Let f, g ∈ F(E,θ) (X, P) for E ∈ R( and let v( ∈ V. If f (x) v g(x) holds θ-almost everywhere on E, then E f dθ ≤ E g dθ + O θE (v) . Proof. Let E ∈ R, let v ∈ V and f, g ∈ F(E,θ) (X, P) such that f (x) v g(x) θ-almost everywhere on E. In a ﬁrst case, let us assume in addition that g ∈ FR (X, P). Lemma 2.4(b) implies that there is λ ≥ 0 such that

4. Integrals for Cone -Valued Functions

155

0 ≤ g(x) + λv for all x ∈ E. Recall from Section 2 that f (x) v g(x) means that f (x) ∈ vε g(x) for all ε > 0. In turn, f (x) ∈ vε g(x) and 0 ≤ g(x) + λv implies f (x) ≤ (1 + ε)g(x) + ε(1 + λ)v by Lemma I.4.1(b). Thus our assumption yields ≤ (1 + ε)g + ε(1 + λ)χE ⊗v f a.e.E for all ε > 0. By Theorem 4.14(c), this implies ' ' f dθ ≤ (1 + ε) g dθ + ε(1 + λ)θE (v). E

E

Now we let ε tend to 0 in the right-hand side of this expression. Lemma I.5.21 together with the deﬁnition of the zero component in I.5.8 leads to ' ' f dθ ≤ g dθ + O θE (v) . E

E

Now we may argue the general case: Suppose that f, g ∈ F(E,θ) (X, P), let w ∈ W and ε > 0, and for g choose the functions g(w,ε) ∈ FR (X, P) ≤ ≤ (and s(w,ε) ∈ FR (X, V) as in 4.13, that is g a.e.E g(w,ε) a.e.E γg + s(w,ε) and E s(w,ε) dθ ≤ εw for some 1 ≤ γ ≤ 1 + ε. Then f (x) v g(w,ε) (x) holds θ-almost everywhere on E, and our ﬁrst case together with 4.14(c) yields ' ' ' f dθ ≤ g(w,ε) dθ + O θE (v) ≤ γ g dθ + O θE (v) + εw. E

E

E

Because w ∈ W and ε > 0 were arbitrarily chosen, our claim follows.

The following Proposition 4.17 is an immediate consequence of 4.16 and strengthens Part (c) of Theorem 4.14(c). Proposition 4.17. Let f, g ∈ F(F,θ) (X,( P) for F ( ∈ AR . If f (x) g(x) holds θ-almost everywhere on F, then F f dθ ≤ F g dθ. Proof. Let F ∈ AR and f, g ∈ F(F,θ) (X, P) such that f (x) g(x) holds θ-almost everywhere on F. Let E ∈ R, w ∈ W, and choose v ∈ V such that θE (v) ≤ w. As f (x) g(x) implies f (x) v g(x), Proposition 4.16 yields ' ' f dθ ≤ g dθ + w, (

(

(E∩F )

(E∩F )

hence (E∩F ) f dθ ≤ (E∩F ) g dθ, since w ∈ W was arbitrarily chosen. Now our deﬁnition of the integral over a set F ∈ AR in 4.13 together with Lemma I.5.20(c) yields our claim. Proposition 4.18. Let f ∈ F(E,θ) (X, P) for E ∈ R. (a) If En( ∈ R such that ( En ⊂ En+1 for all n ∈ N, and E = then E f dθ = lim En f dθ. n→∞

n∈N

En ,

156

II Measures and Integrals. The General Theory

(b) If En ∈ R such that E ⊃ En ⊃ En+1 for all ( n ∈ N, and ( ( then 0 ≤ lim En f dθ ≤ lim En f dθ ≤ O E f dθ .

n∈N

En = ∅,

n→∞

n→∞

Proof. For Part (a), let En ∈ R such that En ⊂ En+1 for all n ∈ N, and E = n∈N En ∈ R. We shall ﬁrst assume that f ∈ FR (X, P). By Lemma 2.4, for w ∈ W there is a neighborhood v ∈ V and λ ≥ 0 such that θE (v) ≤ w and 0 ≤ f + λχE ⊗v. This implies χEn ⊗f ≤ χEn ⊗f + χ(E\En ) ⊗(f + λχE ⊗v) = χE ⊗f + λχ(E\En ) ⊗v. Thus

'

'

'

f dθ ≤ En

(f + λχ(E\En ) ⊗v) dθ = E

f dθ + λθ(E\En ) (v). E

Following Lemma 3.1(b), this yields ' ' lim f dθ ≤ f dθ + εw n→∞

En

E

for all ε ≥ 0. Now let h ∈ SR (X, P) be a step function such ( that h ≤ f +vw , (X, V) such that that is h ≤ f + s for some s ∈ S R X s dθ ≤ w. Then ( ( h dθ ≤ f dθ + w by 4.13(b) and (c), hence En En ' ' ' h dθ = lim h dθ ≤ lim f dθ + w. n→∞

E

n→∞

En

En

Taking the supremum over all such step functions h ≤ f + vw yields ' ' (w) f dθ ≤ lim f dθ + w. n→∞

F

En

Combining with the above we infer that ' ' ' f dθ ≤ lim f dθ + w ≤ lim n→∞

E

Thus indeed

( E

n→∞

En

f dθ = lim

(

n→∞ En

' f dθ + w ≤

En

f dθ + (1 + ε)w. E

f dθ, since w ∈ W and ε > 0 were arbi-

trary. Now for the general case, let f ∈ F(E,θ) (X, P). Given w ∈ W and ε > 0 choose the functions f(w,ε) ∈ FR (X, P) and s(w,ε) ∈ FR (X, V) as in Deﬁnition 4.12. Then the preceding yields ' ' ' ' f dθ ≤ f(w,ε) dθ = lim f(w,ε) dθ ≤ γ lim f dθ + εw, E

and

n→∞

E

' n→∞

En

'

' f dθ ≤ lim

lim

n→∞

n→∞

En

' f(w,ε) dθ ≤ γ

f(w,ε) dθ = En

En

E

f dθ + εw. E

Our claim from Part (a) follows, since both w ∈ W and ε > 0 were arbitrary.

4. Integrals for Cone -Valued Functions

157

For Part (b), let En ∈ R such that E ⊃ En ⊃ En+1 for all n ∈ N, and n∈N En = ∅. For the left-hand side of the inequality in (b) we shall again ﬁrst assume that f ∈ FR (X, P). Let w ∈ W. Following Lemma 2.4(b), there is v ∈ V and λ ≥ 0 such that θE (v) ≤ w and 0 ≤ f + λχE ⊗v, hence 0 ≤ χEn ⊗f + λχEn ⊗v. Then ' ' 0≤ (f + λχEn ⊗v) dθ = f dθ + λθEn (v). En

En

This yields

'

f dθ + λO θE (v)

0 ≤ lim

n→∞

En

by Lemma 3.1(b). Because O θE (v) ≤ εw for all ε > 0, because w ∈ W was ( arbitrary and Q carries the weak preorder, we infer that 0 ≤ lim En f dθ. For the general case, that is f ∈ F(E,θ) (X, P), given w ∈ W

n→∞

and ε >, 0 we choose functions f(w,ε) ∈ FR (X, P) and s(w,ε) ∈ FR (X, V) as in Deﬁnition 4.12. Then the preceding yields together with the limit rules from Lemma I.5.19 ' ' ' f(w,ε) dθ ≤ γ lim f dθ + lim s(w,ε) dθ 0 ≤ lim n→∞ E n→∞ En n→∞ En n ' ≤ γ lim f dθ + εw. n→∞

Thus indeed 0 ≤ lim

n→∞

( En

En

f dθ, since w ⊂ W and ε > 0 were arbitrarily

chosen. For the right-hand side of the inequality in (b), let Gn = E \ En . Then Gn ⊂ Gn+1 , E = ∪∞ n=1 Gn and E = Gn ∪ En . Thus ' ' ' f dθ + f dθ = f dθ Fn

En

E

( ( for all n ∈ N by 4.15(b). Part (a) of 4.18 yields E f dθ = lim Gn f dθ. n→∞ Again using the limit rules in Lemma I.5.19 we infer that ' ' ' ' lim f dθ + f dθ = lim f dθ + lim f dθ n→∞ E n→∞ E n→∞ G E n n n ' ' ' f dθ + f dθ = f dθ. ≤ lim n→∞

En

Gn

Now the cancellation rule in Proposition I.5.10(a) yields ' ' lim f dθ ≤ O f dθ . n→∞

En

E

E

158

II Measures and Integrals. The General Theory

Given a set F ∈ AA we shall denote by F(|F |,θ) (X, P) the subcone of all functions in F(X, P) that are integrable over all complements in F of sets in R, that is F(|F |,θ) (X, P) = F(F \E,θ) (X, P). E∈R

Using this notion, we obtain: Proposition 4.19. Let f ∈ F(|F |,θ) (X, P) for F ∈ AR . ( ( ( Then 0 ≤ lim (F \E) f dθ ≤ lim (F \E) f dθ ≤ O F f dθ . E∈R

E∈R

Proof. For every E ∈( R the function f is integrable over E ∩ F ∈ R and ( ( F \ E ∈ AR . Thus (F \E) f dθ + (E∩F ) f dθ = F f dθ by 4.15(b). Taking the limit over all E ∈ R and using the deﬁnition of the integral in 4.13 and Lemma I.5.19, we obtain ' ' ' lim f dθ + f dθ ≤ f dθ, E∈R

(F \E)

F

'

'

hence lim

E∈R

F

(F \E)

f dθ ≤ O

f dθ F

by the cancellation rule Proposition I.5.10(a). For the ﬁrst part of the inequality in 4.19, we ﬁx E0 ∈ R and let E0 ⊂ E ∈ R. Then ' ' ' f dθ = f dθ + f dθ. (F \E0 )

(F \E)

(F \E0 )∩E

Passing to the limits over E ∈ R in this equation and again using I.5.19 and the deﬁnition of the integral leads to ' ' ' f dθ ≤ lim f dθ + f dθ. (F \E0 )

E∈R

(F \E)

(F \E0 )

Now passing to the limit over E0 ∈ R, we obtain ' ' ' lim f dθ ≤ lim f dθ + lim E∈R

(F \E)

E∈R

(F \E)

E∈R

f dθ.

(F \E)

Following Proposition I.5.10(a) and Proposition I.5.14, the latter implies ' 0 ≤ lim f dθ, E∈R

as claimed.

(F \E)

5. The General Convergence Theorems

159

5. The General Convergence Theorems We shall proceed to establish a range of general convergence results for sequences of measures and functions and their respective integrals. These results are modeled after the dominated convergence theorem from classical measure theory. However, the presence of unbounded elements and the general absence of negatives will considerably complicate some technical aspects of the approach. First we shall extend some of the concepts of the preceding section from a single measure to families of measures. Subsequently, we shall set up suitable notions for convergence of sequences of measures and functions. Convergence for sequences of integrals will generally refer to order convergence in Q, though in some special cases we will be able to establish stronger convergence with respect to the symmetric topology. As in the preceding section, let (P, V) be a full locally convex cone and let (Q, V) be a locally convex complete lattice cone. R denotes a (weak) σring of subsets of X. We shall consider L(P, Q)-valued measures on R. 5.1 Families of Measures and Properties Holding Almost Everywhere. In the following we shall simultaneously deal with families of measures, and therefore need to extend our notion of properties holding almost everywhere from 4.11 to this situation: Given a (non-empty) family Θ of L(P, Q)-valued measures, we denote by Z(Θ) the collection of all sets Z ∈ AR such that θ(E∩Z) = 0 for all E ∈ R and θ ∈ Θ. This collection is obviously closed for set complements and for countable unions. Correspondingly, for a subset F of X we shall say that a pointwise deﬁned property of functions on X holds Θ-almost everywhere on F if it holds on F \ Z with some Z ∈ Z(Θ). If the concerned family Θ of measures is clearly identiﬁed, ≤ or a.e.F = if the relations for the sake of simplicity we may use the symbols a.e.F ≤ or = hold Θ-almost everywhere on the set F, respectively. 5.2 Equibounded Families of Measures. A family Θ of measures on R is called equibounded if for every E ∈ R and w ∈ W there is v ∈ V such that |θ|(E, v) = θE (v) ≤ w for all θ ∈ Θ. 5.3 Integrability with Respect to Equibounded Families of Measures. Likewise, we need to adapt our notation of integrability from Section 4.12 and 4.13. We shall say that a function f ∈ F(X, P) is integrable over a set E ∈ R with respect to a family Θ of L(P, Q)-valued measures if Θ is equibounded and if for every w ∈ W and ε > 0 there are functions f(w,ε) ∈ FR (X, P) and s(w,ε) ∈ FR (X, V) such that ≤ f(w,ε) a.e.E ≤ γf + s(w,ε) f a.e.E

' s(w,ε) dθ ≤ εw

and E

≤ for some 1 ≤ γ ≤ 1 + ε and all θ ∈ Θ. The almost-everywhere relation a.e.E is meant with respect to the family Θ. As in 4.12, we may again assume that

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the function s(w,ε) ∈ FR (X, V) is indeed V- rather than V ∪ {0} -valued. Integrability over a set F ∈ AR with respect to Θ then follows as in 4.13: The function f ∈ F(X, P) is integrable over F ∈ AR with respect to Θ if f is integrable over the sets E ∩ F with respect to Θ for all E ∈ R and all θ ∈ Θ the limit ' ' f dθ = lim F

E∈R

f dθ (E∩F )

exists. The subcone of all these functions f ∈ F(X, P) is denoted by F(F,Θ) (X, P). Likewise, F(|F |,Θ) (X, P) denotes the subcone of all functions in F(X, P) that are integrable with respect to Θ over all complements in F of sets in R, that is F(|F |,Θ) (X, P) = F(F \E,Θ) (X, P). E∈R

Repeating the argument from Proposition 4.15(a), one can verify that a function f ∈ F(X, P) is in F(F,Θ) (X, P) or in F(|F |,Θ) (X, P) if and only if the function χF ⊗f is contained in F(X,Θ) (X, P) or in F(|X|,Θ) (X, P), respectively. While integrability with respect to a family of measures obviously implies integrability with respect to every member of this family, the converse is not always true (see Example 5.15 below). The following results 5.4 to 5.7 are already of interest for integration with respect to a single measure and might therefore have been placed into the preceding section. We shall, however, also refer to the subsequent more general versions which refer to integration with respect to equibounded families of measures. Proposition 5.4. Let Θ be an equibounded family of measures on R. Let E ∈ R and f ∈ F(E,Θ) (X, P). For every w ∈ W there is s ∈ FR (X, V) ( ≤ f + s and and λ ≥ 0 such that 0a.e.E E s dθ ≤ λw for all θ ∈ Θ. Proof. Let E ∈ R, let f ∈ F(E,Θ) (X, P) and w ∈ W. According to the deﬁnition of integrability in 4.12, for ε = 1 there are g ∈ FR (X, P) and ≤ ≤ (s ∈ FR (X, V) such that f a.e.E g a.e.E γf + s for some 1 ≤ γ ≤ 2 and E s dθ ≤ w for all θ ∈ Θ. We choose v ∈ V such that θE (v) ≤ w for all θ ∈ Θ. Following Lemma 2.4(b) there is G ∈ R and λ ≥ 0 such that ≤ g + λχE ⊗v, hence 0 ≤ g + λχG ⊗v. The latter implies 0a.e.E ≤ f + 1 (s + λχE ⊗v) a.e.E ≤ f + (s + λχE ⊗v). ≤ 1 (g + λχE ⊗v) a.e.E 0 a.e.E γ γ ( As s + λχE ⊗v ∈ FR (X, V) and E (s + λχE ⊗v) dθ ≤ (1 + λ)w for all θ ∈ Θ, our claim follows. 5.5 The Locally Convex Cone F(F,Θ) (X, P), V(F, Θ) . Let Θ be ≤ , an equibounded family of measures on R. Endowed with the order a.e.F

5. The General Convergence Theorems

161

that is the given pointwise order Θ-almost everywhere on the set F ∈ AR , F(F,Θ) (X, P) is an ordered cone. We generate a canonical convex quasiuniform structure (see I.1.3) in the following way: With every w ∈ W ˘E and E ∈ R we associate the neighborhood v w (Θ), deﬁned for functions f, g ∈ F(F,Θ) (X, P) by ˘E f ≤g+v w (Θ)

if

≤ g+s f a.e.E

( for some s ∈ FR (X, V) such that E s dθ ≤ w for all θ ∈ Θ. Let V(F, Θ) ˘E denote the neighborhood system generated by the neighborhoods v w (Θ) for all w ⊂ W and E ∈ R such that E ⊂ F. As Θ is equibounded, according to Proposition 5.4, for every function f ∈ F(F,Θ) (X, P), every w ∈ W and E ∈ R such that( E ⊂ F there is s ∈ FR (X, V) and λ ≥ 0 such that E ≤ f + s and 0 a.e.E E s dθ ≤ λw holds for all θ ∈ Θ. Thus 0 ≤ f + λvw (Θ). All functions in F(F,Θ) (X, P) aretherefore bounded below with respect to these neighborhoods. In this way, F(F,Θ) (X, P), V(F, Θ) becomes a locally convex cone as elaborated in I.1.3. Theorem 4.14(c) implies that for every E ∈ R such that E ⊂ F and every θ ∈ Θ the mapping ' f → f dθ : F(F,Θ) (X, P) → Q E

is ( a continuous linear operator. EIndeed, for w ∈ W we have ˘ E g dθ + w whenever f ≤ g + vw (Θ) for f, g ∈ F(F,Θ) (X, P).

( E

f dθ ≤

5.6 Subcone-Based Integrability. The following deﬁnition of subconebased integrability is motivated by the fact that in many realizations (P, V) is indeed the standard full extension of some subcone of P, and we might be particularly interested in functions with values in this subcone. Given a subcone P0 of P and a neighborhood subsystem V0 of V, we shall say that a function f in F(X, P) is (P0 , V0 )-based integrable over a set E ∈ R with respect to an equibounded family Θ of measures if for every w ∈ W and ε > 0 there are functions f(w,ε) ∈ FR (X, P0 ) and s(w,ε) ∈ FR (X, V0 ) such that ' ≤ γf + s(w,ε) and ≤ f(w,ε) + s(w,ε) , f(w,ε) a.e.E s(w,ε) dθ ≤ εw f a.e.E E

≤ for some 1 ≤ γ ≤ 1 + ε and all θ ∈ Θ. The almost-everywhere relation a.e.E is meant with respect to the family Θ. In this context, FR (X, P0 ) is the subcone of FR (X, P) consisting of all measurable P0 -valued functions such that for every inductive limit neighborhood v for F(X, P) there is a P0 valued step function h ∈ SR (X, P0 ) satisfying h ≤ f(w,ε) + v. Measurability is still deﬁned with respect to the given neighborhood system V rather than the subsystem V0 . Similarly, FR (X, V0 ) consists of all V0 -valued functions in FR (X, P).

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Because (P0 , V0 )-based integrability over a set E ∈ R implies (P0 , V0 )based integrability over all subsets G ∈ R of E, (P0 , V0 )-based integrability over a set F ∈ AR may be deﬁned as in 5.3. Note that a (P0 , V0 )-based integrable function is not required to be P0 valued. Obviously, this notion of subcone-based integrability implies integrability based on the given cone (P, V) in the sense of 4.13 and 5.3, and the (P0 , V0 )-based integrable functions form a subcone of F(F,Θ) (X, P). For P0 = P and V0 = V the deﬁnition of (P0 , V0 )-based integrability coincides of course with the deﬁnition of integrability from 5.3: Clearly, integrability in the sense of 5.3 implies (P, V)-based integrability. For the converse, use f˜(w,ε) = f(w,ε) + s(w,ε) instead of f(w,ε) in 5.3. Other than in the classical scenario (see for example [25], [55], [178] and [179]), our deﬁnition of integrability does not generally guarantee that an integrable cone-valued function f ∈ F(X, P) can be approximated (even with respect to pointwise convergence) by a sequence of step functions whose integrals then converge towards the integral of f. However, a combination of Theorem 2.7 with Proposition 4.7 yields some corresponding results. Theorem 5.7. Let Θ be an equibounded family of measures on R. Let E ∈ R and let f ∈ F(X, P) be (P0 , V0 )-based integrable over E with respect to Θ for a subcone P0 of P and a subsystem V0 of V. For every w ∈ W such that θE (v) ≤ w for some ( v ∈ V0 and all θ ∈ Θ, and every ε > 0 there is s ∈ FR (X, V0 ) such that E s dθ ≤ w for all θ ∈ Θ, 1 ≤ γ ≤ 1 + ε and a bounded below sequence (hn )n∈N of P0 -valued step functions such that: ≤ γf + s holds for all n ∈ N. (i) hna.e.E

(ii) Θ-almost everywhere on E, for x ∈ E there is n0 ∈ N such that f (x) ≤ hn (x) + s(x) for all n ≥ n0 . ( ( ( ( (iii) G f dθ ≤ lim G hn dθ + w and G hn dθ ≤ γ G f dθ + w n→∞

for all n ∈ N, all G ∈ R such that G ⊂ E, and all θ ∈ Θ. Proof. Let f ∈ F(X, P) be (P0 , V0 )-based integrable over E ∈ R. Given w ∈ W and 0 < ε ≤ 1, following our assumption there are f(w,ε) ∈ ( FR (X, P0 ) and s(w,ε) ∈ FR (X, V0 ) such that E s(w,ε) dθ ≤ w/4 for all ≤ γf + s(w,ε) with some 1 ≤ γ ≤ ≤ f(w,ε) + s and f(w,ε) a.e.E θ ∈ Θ and f a.e.E 1 + ε/3. By our assumption there is v ∈ V0 such that θE (v) ≤ w/2 for all θ ∈ Θ. We shall apply Theorem 2.7 to the locally convex cone (P0 , V0 ), the function f(w,ε) ∈ FR (X, P0 ), the neighborhood v ∈ V0 , ε/3 in place of ε, and the inductive limit neighborhood v = {χX ⊗v}. For this we observe that the measurability conditions (M1) and (M2) in Section 1 with respect to the neighborhood system V imply those with respect to the subsystem V0 ⊂ V. There is 1 ≤ γ ≤ 1 + ε/3 and a bounded below sequence (hn )n∈N of P0 -valued step functions such that (i), hn (x) ≤ γ f(w,ε) (x) + v for all x ∈ E and n ∈ N, and (ii), for every x ∈ E there is n0 ∈ N such that f(w,ε) (x) ≤ hn (x) + v for all n ≥ n0 . This yields

5. The General Convergence Theorems

163

≤ (γγ )f + (γ s(w,ε) + χE ⊗v) hna.e.E for all n ∈ N. We set s = γ s(w,ε) + χE ⊗v ∈ FR (X, V0 ) ( and observe that E s dθ ≤ γ w/4 + w/2 ≤ w for all θ ∈ Θ. Because 1 ≤ γγ ≤ (1 + ε/3)2 ≤ 1 + ε, this yields Part (i) of our claim with s in place of s. Part (ii) also follows from the above, since f(w,ε) (x) ≤ hn (x) + v for x ∈ E implies that f (x) ≤ f(w,ε) (x) + s(w,ε) (x) ≤ hn (x) + s(w,ε) (x) + v ≤ hn (x) + s (x). For Part (iii) let G ∈ R such that G ⊂ E and let (θ ∈ Θ. The second part ( ≤ γf + s implies that of (iii) is obvious, since hna.e.E h dθ ≤ γ f dθ +w n G G for all n ∈ N. For the ﬁrst part of (iii), consider the full cone P and let h n = hn + χE ⊗v ∈ SR (X, P). The sequence (h n )n∈N of step functions approaches f(w,ε) ∈ FR (X, P) as required in Proposition 4.7, which therefore yields '

' f(w,ε) dθ ≤

f(w,ε) dθ ≤ lim

n→∞

G

G

'

(w )

h n dθ + w

G

for every θ ∈ Θ and all w ∈ W, hence ' ' ' f(w,ε) dθ ≤ lim h n dθ = lim hn dθ + θG (v) n→∞

G

since

( G

h n dθ = '

G

n→∞

G

(

hn dθ + θG (v). Thus ' ' f dθ ≤ f(w,ε) dθ + s(w,ε) dθ G G G ' ' ≤ lim hn dθ + s(w,ε) dθ + θG (v) n→∞ G G ' ≤ lim hn + w G

n→∞

G

(

since G s(w,ε) dθ ≤ w/4 and θG (v) ≤ w/2. This yields the ﬁrst inequality in (iii). If the family Θ of measures is equibounded relative to the subsystem V0 of V, that is if for every E ∈ R and every w ∈ W there is v ∈ V0 such that θE (v) ≤ w for all θ ∈ Θ, then the condition on the neighborhood v ∈ V0 in Theorem 5.7 is obviously superﬂuous. Indeed, given w ∈ V and any v ∈ V0 there is v ∈ V0 as above. Because the neighborhood system V0 is supposed to be directed downward, there is v ∈ V0 such that both v ≤ v and v ≤ v . Thus θE (v ) ≤ w for all θ ∈ Θ, and we may apply Theorem 5.7 with v in place of v.

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II Measures and Integrals. The General Theory

For future use, it is worthwhile to formulate as a corollary the simpliﬁcations that occur in Theorem 5.7 if the subcone (P0 , V0 ) of (P, V) is indeed a full cone, that is if V0 ⊂ P0 . Corollary 5.8. Let Θ be an equibounded family of measures on R. Let E ∈ R and let f ∈ F(X, P) be (P0 , V0 )-based integrable over E with respect to Θ for a subcone P0 of P and a subsystem V0 ⊂ P0 of V. For every w ∈ W such that θE (v) ≤ w for some v ∈ V ( 0 and all θ ∈ Θ, and every ε > 0, there is s ∈ FR (X, V0 ) such that E s dθ ≤ w for all θ ∈ Θ, 1 ≤ γ ≤ 1 + ε and a bounded below sequence (hn )n∈N of P0 -valued step functions such that: ≤ γf + s holds for all n ∈ N. (i) hna.e.E (ii) Θ-almost everywhere on E, for x ∈ E there is n0 ∈ N such that ( all n ≥ n0 . ( ( (f (x) ≤ hn (x) for (iii) G f dθ ≤ lim G hn dθ and G hn dθ ≤ γ G f dθ + w n→∞

for all n ∈ N, all G ∈ R such that G ⊂ E, and all θ ∈ Θ. Proof. Given a neighborhood w ∈ W satisfying the requirement of the corollary, and 0 < ε ≤ 1 we apply Theorem 5.7 with the neighborhood w/4 ∈ W instead of w. As in the proof of 5.7 we choose v ∈ V such that θE (v) ≤ w/8. Let s ∈ FR (X, V0 ) and the sequence (hn )n∈N of P0 -valued step functions as in 5.7. We apply Corollary 2.8 to the full cone (V0 , V0 ) for the function s ∈ FR (X, V0 ) with the inductive limit neighborhood v = {χX ⊗v} : There is a bounded below sequence (sn )n∈N of V0 -valued step functions satisfying (i) sn ≤ γ s + χX ⊗v with some 1 ≤ γ ≤ 1 + ε and (ii) for every x ∈ E there is n0 such that s(x) ≤ sn (x) for all n ≥ n0 . The latter implies ' ' s(x) dθ ≤ lim sn (x) dθ G

n→∞

G

for all G ∈ R such that G ⊂ E, and all θ ∈ Θ, by Proposition 4.7. Now we set h n = hn + sn + χE ⊗v ∈ SR (X, P0 )

and

s = 3s + 2χE ⊗v ∈ FR (X, V0 ).

These are the functions that we use for Corollary 5.8: We have ' s dθ ≤ 3(w/4) + 2(w/8) = w, E

and h n ≤ (γf + s) + sn + χE ⊗v aeE

≤ (γf + s) + (γ s + χX ⊗v) + χE ⊗v ≤ γf + s

5. The General Convergence Theorems

165

(

(

since 1 + γ ≤ 3. This implies G h n dθ ≤ γ G f dθ + w for all n ∈ N, all G ∈ R such that G ⊂ E, and all θ ∈ Θ. The ﬁrst part of (iii) follows from the last inequality in the proof of 5.7, that is ' ' ' f dθ ≤ lim hn dθ + s dθ + θG (v) n→∞ G G G ' ' ≤ lim hn dθ + lim sn dθ + θG (v) n→∞ G n→∞ G ' ≤ lim (hn + sn + χE ⊗v) dθ n→∞ G ' ≤ lim h n dθ, n→∞

G

hence our claim.

For the following recall the deﬁnition of the order topology of a locally complete lattice cone from Section I.5.43. We shall also consider integrals of measurable V-, that is V ∪ {0, ∞}-valued functions (see Section 2.1), if they take the value ∞ ∈ V only on a set of measure zero (see 4.12 and 4.13). Corollary 5.9. Let Θ be an equibounded family of measures on R. Let F ∈ AR and let f ∈ F(X, P) be (P0 , V)-based integrable over F with respect to Θ for a subcone P0 of P. Then there is a net (hi )i∈I of P0 valued step functions, a net (si )i∈I of measurable V-valued functions and a net (γi )i∈I in R such that for every θ ∈ Θ : (i) For every x ∈ F there is i0 ∈ I such that f (x) ≤ γi hi (x) + si (x) and hi (x) ( ≤ f (x) +(si (x) for all i ≥ i0 . (ii) lim F hi dθ = F f dθ in the order topology of Q. i∈I ( (iii) lim F si dθ = 0 in the symmetric topology of Q. i∈I

(iv) γi ≥ 1 for all i ∈ I and lim γi = 1. i∈I ( Consequently, F f dθ is contained in the closure with respect to the order topology of the subcone of Q spanned by the set {θ(E∩F ) (a) | E ∈ R, a ∈ P0 }. Proof. Suppose that the function f ∈ F(X, P) is (P0 , V)-based integrable over the set F ∈ AR with respect to Θ and in a ﬁrst step let E ∈ R be a subset of F. For every w ∈ W and ε > 0, let (sw,ε n )n∈N be a sequence of P0 -valued step functions, 1 ≤ γ w,ε ≤ 1 + ε and sw,ε ∈ SR (X, ( V) as in Theorem 5.7 with εw in place of w. According to 5.7 we have E sw,ε dθ ≤ εw for all θ ∈ Θ, and ' ' ' ' w,ε w,ε hw,ε dθ + εw and lim h dθ ≤ γ f dθ + εw f dθ ≤ lim n n E

n→∞

E

n→∞

E

follows from 5.7(iii). That is, for all θ ∈ Θ, both

E

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II Measures and Integrals. The General Theory

'

' hw,ε n dθ

lim n→∞

and

hw,ε n dθ

lim

n→∞

E

E

( are elements of the symmetric relative neighborhood wεs E f dθ . Let the index set J consist of all triples (w, ε, φ), where w ∈ W, ε > 0 and φ : W × {ε > 0} → N. The set J is ordered and directed upward by (w1 , ε1 , φ1 ) ≤ (w2 , ε2 , φ2 ) if w2 ≤ w1 , ε2 ≤ ε1 , and φ1 (w, ε) ≤ φ2 (w, ε) for all w ∈ W and ε > 0. Note that the index set J does not depend on the subset E ∈ R of F. We set hj = χE ⊗ hw,ε φ(w,ε) for j = (w, ε, φ) ∈ J , as well as sj = χE ⊗ sw,ε + χ(Z∪ZE ) ⊗ ∞

and

γj = γ w,ε ,

where ∞ is the inﬁnite element of the augmented neighborhood system V (see 2.1), and Z = X \ E∈R E . This is a set of Θ-measure zero. Likewise, ZE ∈ R is a subset of E of Θ-measure zero and such that the conclusions of 5.7(i) and (ii) hold for all x ∈ E \ ZE . (For this, recall that the union of countably many sets of measure zero is again of measure zero.) Therefore, 5.7(i) and (ii) hold for all x ∈ E, not just Θ-almost everywhere if we replace the function sw,ε by sj . Moreover, since the function sj takes the value ∞ ∈ V only on a zero set, we infer ' ' sj dθ = sw,ε dθ ≤ εw F

E

for all θ ∈ Θ. We have 1 ≤ γj ≤ 1 + ε. Thus ' sj dθ = 0 and lim γj = 1. lim j∈J

j∈J

F

The ﬁrst of these limits is taken in the symmetric topology of Q. Next we shall verify that ' ' f dθ = lim hj dθ E

j∈J

F

holds for every θ ∈ Θ in the order topology of Q. Indeed, let θ ∈ Θ and ( let U be a convex and order convex neighborhood of E f dθ ∈ Q in the order topology. As the order topology is coarser than the symmetric relative topology of Q (see Proposition I.5.44), there are w0 ∈ W and ε0 > 0 such that U is a neighborhood for every point in the symmetric relative ( neighborhood w0 sε0 ( E f dθ). Then for each choice of w ≤ w0 and ε ≤ ε0 we have both ' ' ' ' w,ε s s lim hw,ε dθ ∈ w f dθ and lim h dθ ∈ w f dθ 0 0 n n ε0 ε0 n→∞

E

E

n→∞

E

E

5. The General Convergence Theorems

167

by ( the above. As U is an order topology neighborhood of every element in vεs0 E f dθ , there is an integer φ0 (w, ε) ∈ N such that both ' ' w,ε w,ε inf hn dθ ∈ U and sup hn dθ ∈ U. n≥φ0 (w,ε)

n≥φ0 (w,ε)

E

E

Now the order convexity of the neighborhood U guarantees that ' hw,ε n dθ ∈ U E

whenever w ≤ w0 , ε ≤ ε0 and n ≥ φ0 (w, ε). We set j0 = (w0 , ε0 , φ0 ) ∈ J and φ0 (w, ε) = 1 if either w w0 or ε ε0 . Then the above yields indeed that ' ' hj dθ = hw,ε φ(w,ε) dθ ∈ U F

E

for all j = (w, ε, φ) ∈ J such that j ≥ j0 , that is w ≤ w0 , ε ≤ ε0 and φ(w, ε) ≥ ψ0 (w, ε), thus demonstrating our claim. Finally, given x ∈ E, for every w ∈ W and ε > 0 there is φ0 (w, ε) ∈ N such that f (x) ≤ hj (x) + sj (x)

hj (x) ≤ γj f (x) + sj (x)

and

for all j = (w, ε, φ) ∈ J such that φ(w, ε) ≥ φ0 (w, ε). If we set j0 = (w0 , 1, φ0 ) for any choice of w0 ∈ W, then the above inequalities hold whenever j ≥ j0 . Now in the second step of our construction, for every set E ∈ R we shall construct a net (hE j )j∈J of (P0 , V)-valued step functions as in our ﬁrst step with respect to the set E ∩ F ∈ R, that is in particular ' ' f dθ = lim hE j dθ (E∩F )

j∈J

F

for every θ ∈ Θ. Similarly, we select the corresponding nets (sE j )j∈J and (γjE )j∈J . Now we choose another index set I consisting of all pairs (E, ψ), where E ∈ R and ψ : R → J , ordered and directed upward by (E1 , ψ1 ) ≤ (E2 , ψ2 ) if E1 ≤ E2 , and ψ1 (E) ≤ ψ2 (E) for all E ∈ R. We set hi = hE ψ(E) for i = (E, ψ) ∈ I and realize that the net (hi )i∈I of (P0 , V)-valued step functions satisﬁes the properties stated in our Corollary. A straightforward diagonal argument similar to the preceding one shows that ' ' f dθ = lim hi dθ F

i∈I

F

holds for every θ ∈ Θ in the order topology of Q, as claimed in (ii). Because the integrals of all step functions hi involved are linear combinations of elements θ(E∩F ) (a) for E ∈ R and a ∈ P0 , (ii) does indeed imply that

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II Measures and Integrals. The General Theory

(

F f dθ is contained in the closure with respect to the order topology of the subcone of Q spanned by these elements. Similarly, as claimed in (iii) and in (iv), we verify that ' lim si dθ = 0 and lim γi = 1, i∈I

i∈I

F

where the ﬁrst of these limits is taken in the symmetric topology of Q. For Part (i), let x ∈ F. If x ∈ / E∈R E, then our claim is trivial, as we have si (x) = ∞ ∈ V for all i ∈ I. Otherwise, there is E0 ∈ R such that x ∈ E0 . We ﬁx any w0 ∈ W and choose the index i0 = (E0 , ψ0 ) ∈ I, where ψ0 : R → J is the mapping E → (w0 , 1, φE ) ∈ J . The mapping φE : W × {ε > 0} → N is chosen as constant φE (w, ε) = 1 if E0 ⊂ E, and otherwise we chose φE (w, ε) ∈ N such that f (x) ≤ hj (x) + sj (x)

and

hj (x) ≤ γj f (x) + sj (x)

for every j = (w, ε, φ) ∈ J such that j ≥ (w0 , 1, φE ). This holds for all E ∈ R such that E0 ⊂ E, hence we infer that f (x) ≤ hi (x) + si (x) holds for all i ≥ i0 .

and

hi (x) ≤ γi f (x) + si (x)

It its important to keep in mind that the limit in 5.9(ii) refers to the order topology of Q, not necessarily to order convergence as deﬁned in I.5.18. Because in general the order topology is not known to be Hausdorﬀ, this limit need therefore not be unique. Corollary 5.9 is of particular interest in case that the locally convex complete lattice cone (Q, W) is indeed the standard completion of some locally convex cone (Q0 , W) (see I.5.57) and that the measure θ is indeed L(P, Q0 )-valued. The closure of Q0 in Q with respect to the order topology was seen to be a subcone of the second dual Q∗∗ 0 (see Remark I.5.60(a) and Section I.7.3) in this case, and integrals of functions in F(X, P) are therefore elements of Q∗∗ 0 . Moreover, if the full locally convex cone (P, V) is indeed the standard full extension of a quasi-full locally convex cone (P0 , V) (see I.6.2) and if for all E ∈ R the operator θE maps the elements of P0 into Q0 , then a similar statement holds for all functions f ∈ F(X, P) that are (P0 , V)-based integrable over F (see Proposition 6.7 below). Because the values of our measures, that is continuous linear operators from P into Q, may be restricted to linear operators on a subcone P0 of P, one may raise the question, if and how such a restriction does aﬀect the integrals of functions with values only in this subcone P0 . Let us be precise: Let P0 be a subcone of P, and let V0 ⊂ P0 be a neighborhood subsystem of V. If for a given L(P, Q)-valued measure θ, for all E ∈ R, the restrictions of the linear operators θE from the full cone (P, V) to (Q, W ) are continuous linear operators from the full cone (P0 , V0 ) to (Q, W ), then, obviously, θ

5. The General Convergence Theorems

169

may also be considered to be an L(P0 , Q)-valued measure. This situation requires that for every E ∈ R and w ∈ W there is v ∈ V0 such that θE (v) ≤ w. To avoid confusion, we shall denote this restriction of the measure θ to (P0 , V0 ) by θ0 , and by F(F, θ0 ) (X, P0 ) the cone of all P0 -valued functions that are integrable over a set F ∈ AR with respect to θ0 . Similarly, we shall use F(F, Θ0 ) (X, P0 ) for functions that are integrable with respect to a family of restricted measures. Because our notions of measurability, of being reached from below by step functions and, consequently, of integrability depend on the given neighborhood system as well as on the cone, we shall have to clarify our notions for this situation. For a P0 -valued function, measurability with respect to (P, V) obviously implies measurability with respect to (P0 , V0 ), since V0 ⊂ V (see Conditions (M1) and (M2) in Section 1.2). The cone FR (X, P0 ) is however not necessarily a subcone of FR (X, P) since the condition for the elements of FR (X, P0 ) of being reached from below (see Section 2.3) involves only inductive neighborhoods that use the neighborhoods in V0 ⊂ V. Positive functions in FR (X, P0 ) are however contained in FR (X, P), since they can be trivially reached from below by the step function h = 0. Conversely, every P0 valued function in FR (X, P) that can be reached from below by P0 -valued step functions is contained in FR (X, P0 ). This implies in particular that FR (X, V0 ) consists of the V0 -valued elements of FR (X, V), that is FR (X, V0 ) = FR (X, V) ∩ F(X, V0 ). Furthermore, we note that every set Z ∈ AR of measure zero with respect to a measure θ is also of measure zero with respect to its restriction θ0 . The almost everywhere notion with respect to θ therefore implies the almost everywhere notion with respect to θ0 . The converse does not necessarily hold true. Proposition 5.10. Let P0 be a subcone of P, and let V0 ⊂ P0 be a neighborhood subsystem of V. Let Θ be an equibounded family of L(P, Q)-valued measures on R such that the family Θ0 of all restrictions of the measures in Θ to (P0 , V0 ) is an equibounded family of L(P0 , Q)-valued measures. Let F ∈ AR . If a P0 -valued function f is (P0 , V0 )-based integrable over F with respect to Θ, then f ∈ F(F, Θ0 ) (X, P0 ), and ' ' f dθ = f dθ0 F

F

holds for all θ ∈ Θ. Proof. Let P0 be a subcone of P, and let V0 ⊂ P0 be a neighborhood subsystem of V. Let Θ be an equibounded family of L(P, Q)-valued measures on R. The family Θ0 of all restrictions of the measures in Θ to (P0 , V0 ) is an equibounded family of L(P0 , Q)-valued measures if and only if for every E ∈ R and w ∈ W there is v ∈ V0 such that θE (v) ≤ w for all θ ∈ Θ. By our assumption, Θ satisﬁes this requirement. Given a set F ∈ AR , we

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II Measures and Integrals. The General Theory

shall consider P0 -valued functions as elements of the cones F(F,Θ) (X, P) or F(F, Θ0 ) (X, P0 ), respectively. We shall proceed in several steps: First we observe that every P0 -valued step function h = ni=1 χEi ⊗ai , for Ei ∈ R and a(i ∈ P0 , is(contained in both F(F,Θ) (X, P) and F(F,Θ0 ) (X, P0 ), and we have F h dθ = F h dθ0 for all θ ∈ Θ. In a second step we consider a neighborhood-valued function s ∈ FR (X, V0 ). As we remarked before, positivity implies that s is also contained in FR (X, V). Given a neighborhood w ∈ W, the inductive limit neighborhood vw formed by the neighborhoods in V contains the corresponding neighborhood v0w formed by the neighborhoods in V0 as a subset (see Section 4). Thus ' ' (w) s dθ0 = sup s dθ h ∈ SR (X, P0 ), h ≤ s + v0w F

F

' ≤ sup

' h dθ h ∈ SR (X, P), h ≤ s + vw =

F

(w)

s dθ.

F

Taking the inﬁma over all w ∈ W on both sides yields ' ' s dθ0 ≤ s dθ. F

F

Now in a third step, let E ∈ R, and let us consider a P0 -valued function that is (P0 , V0 )-based integrable over E with respect to Θ. We shall ﬁrst verify that f ∈ F(E, Θ0 ) (X, P0 ). Indeed, the former property requires that for w ∈ W and ε > 0 there is a V0 -valued function s(w,ε) ∈ FR (X, V), and a P0 -valued function f(w,ε) ∈ FR (X, P) that can be reached from below by P0 -valued step functions, such that ≤ f(w,ε) a.e.E ≤ γf + s(w,ε) f a.e.E ( for some 1 ≤ γ ≤ 1 + ε and such that E s(w,ε) dθ ≤ εw holds for all θ ∈ Θ. As FR (X, V0 ) = FR (X, V) ∩ F(X, V0 ), we have ' ' s(w,ε) dθ0 ≤ s(w,ε) dθ ≤ εw E

E

for all θ0 ∈ Θ0 by our ﬁrst step. As we mentioned before, the almost every≤ with respect to θ implies the same relation with respect where relation a.e.E to θ0 . The function f is therefore indeed integrable over E with respect to the family Θ0 of the restricted measures, that is f ∈ F(E, Θ0 ) (X, P0 ). Now let w ∈ W and ε > 0. We shall apply Corollary 5.8 with the family Θ and the given subcone (P0 , V0 ) to ﬁnd a sequence (hn )n∈N of P0 -valued step functions as in 5.8. Statements (i) and (iii) refer to the measures θ ∈ Θ. However, all functions involved are also contained in F(E, Θ0 ) (X, P0 ), the integrals with respect to the measures in Θ and in Θ0 coincide for the step functions hn , and for the function s ∈ FR (X, V0 ) in (i) we have

5. The General Convergence Theorems

171

'

' s dθ0 ≤ F

s dθ ≤ w F

by our second step. Property (ii) therefore yields together with Proposition 4.7 that ' ' f dθ0 ≤ lim hn dθ0 n→∞

E

E

holds for all θ0 ∈ Θ0 . Using this together with our ﬁrst step and the second part of statement (iii) in 5.8, we obtain ' ' ' ' f dθ0 ≤ lim hn dθ0 = lim hn dθ ≤ γ f dθ0 + w n→∞

E

n→∞

E

E

E

and, likewise, ' E

'

'

f dθ ≤ lim hn dθ = lim hn dθ0 n→∞ E ' ' n→∞ E ' ≤γ f dθ0 + s dθ0 ≤ γ f dθ0 + w E

E

E

with some 1 ≤ γ ≤ w ∈ W and ε > 0 were arbitrarily ( ( 1 + ε. Because chosen, this yields E f dθ = E f dθ0 . Now for the ﬁnal step of our argument, let F ∈ AR , and let f ∈ F(X, P) be P0 -valued and (P0 , V0 )-based integrable over F. Then f is (P0 , V0 )-based integrable over the sets E ∩ F, ( for all E ∈( R, hence f ∈ F(E∩F, Θ0 ) (X, P0 ) by our second step, and (E∩F ) f dθ = (E∩F ) f dθ0 by our ﬁrst step. This shows ' ' ' ' f dθ = lim f dθ = lim f dθ0 = f dθ0 . F

E∈R

E∈R

(E∩F )

(E∩F )

F

5.11 Sums, Multiples and Order for Measures. Let θ and ϑ be two R-bounded L(P, Q)-valued measures, and let α ≥ 0. We deﬁne the L(P, Q)valued measures θ + ϑ and αθ by and

(θ + ϑ)E (a) = θE (a) + ϑE (a) (αθ)E (a) = α θE (a)

for E ∈ R and a ∈ P. The properties of a measure are readily checked. Corresponding to a subcone P0 of P we deﬁne an order relation for measures θ and ϑ setting θ ≤P0 ϑ

if

θE (a) ≤ ϑE (a)

holds for all E ∈ R and a ∈ P0 . We write θ ≤ ϑ for the canonical choice of P0 = P+ = {a ∈ P | a ≥ 0}. In this case, for any family Θ of measures

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II Measures and Integrals. The General Theory

and every set F ∈ R, every P+ -valued function f ∈ F(F,Θ) (X, P) is seen to be (P0 , V)-based integrable over F with respect to Θ. Note that θ ≤ ϑ and ϑ ≤ θ in this sense implies that θ = ϑ, that is equality for the positive elements implies equality for all elements of P. Indeed, given E ∈ R, a ∈ P and w ∈ W there is v ∈ V such that θE (v) = ϑE (v) ≤ w. There is λ ≥ 0 such that 0 ≤ a+λv. Thus θE (a+λv) = ϑE (a+λv) and θE (a) ≤ ϑE (a)+w by the cancellation rules. This shows θE (a) ≤ ϑE (a) and likewise, ϑE (a) ≤ θE (a). Proposition 5.12. Let θ and ϑ be L(P, Q)-valued measures, let α ≥ 0, F ∈ AR , and let P0 be a subcone of P. (a) (If f ∈ F(F,{θ,ϑ}) (X, ( P), then ( f ∈ F(F,θ+ϑ) (X, P) and f d(θ + ϑ) = f dθ + F F F f dϑ. ( ( then f (b) If f ∈ F(F,θ) (X, P), ( (∈ F(F,αθ) (X, P) and F f d(αθ) = α F f dθ. (c) If θ ≤P0 ϑ, then F f dθ ≤ F f dϑ holds for every f ∈ F(X, P) that is (P0 , V)-based integrable over F with respect to Θ = {θ, ϑ}. Proof. Without loss of generality, we may assume that F = X. For Part (a), it is clear of the ( from our deﬁnition ( ( sum of two measures that our claim, namely X h d(θ + ϑ) = X h dθ + X h dϑ holds for all step functions h ∈ SR (X, P). Let Θ = {θ, ϑ}. Every zero set for Θ is obviously a zero set for θ + ϑ. We shall ﬁrst show that every function f ∈ F(X,Θ) (X, P) is integrable over every set E ∈ R with respect to θ + ϑ. Indeed, given w ∈ W and ε > 0, let w = w/2 and let the functions f(w ,ε) ∈ FR (X, P) and s(w ,ε) ∈ FR (X, V) be as in Deﬁnition 5.3, that is ≤ f(w ,ε) a.e.E ≤ γf + s(w ,ε) f a.e.E ( ( for some 1 ≤ γ ≤ 1 + ε and E s(w ,ε) dθ ≤ εw and E s(w ,ε) dϑ ≤ εw . The function s(w ,ε) ∈ FR (X, V) is integrable with respect to every measure on R, and for any u ∈ W we realize that '

'

(u)

s(w ,ε) d(θ + ϑ) = sup E

' ≤

h d(θ + ϑ) h ∈ SR (X, P), h ≤ s(w ,ε) + vw

E

'

(u)

(u)

s(w ,ε) dθ + E

s(w ,ε) dϑ. E

Taking the respective inﬁma over all neighborhoods u ∈ W and using Lemma I.5.20(c), we infer that ' ' ' s(w ,ε) d(θ + ϑ) ≤ s(w ,ε) dθ + s(w ,ε) dϑ ≤ εw. E

E

E

This shows integrability for f over E with respect to the family Θ = {θ, ϑ, θ + ϑ} of measures. Next for P0 = P and V0 = V, the family Θ from above, a set E ∈ R, a neighborhood w ∈ W and ε ≥ 0 let (hn )n∈N

5. The General Convergence Theorems

173

be a sequence of step functions in SR (X, P) approaching the function f ∈ F(X,Θ) (X, P) as in Corollary 5.8. Part (iii) of 5.8 then yields ' ' f d(θ + ϑ) ≤ lim hn d(θ + ϑ) n→∞ E E ' ' ≤ lim hn dθ + lim hn dϑ n→∞ E n→∞ E ' ' ≤γ f dθ + γ f dϑ + 2w. E

And similarly, '

E

'

'

'

f dϑ ≤ lim

f dθ + E

E

n→∞

hn dθ + lim n→∞

'E

hn dϑ E

≤ lim hn d(θ + ϑ) n→∞ E ' ≤γ f d(θ + ϑ) + w. E

This in turn shows

'

' f d(θ + ϑ) = E

f d(θ + ϑ), E

since w ∈ W and ε > 0 were arbitrarily chosen. The latter equality holds for all E ∈ R, hence or claim follows from the deﬁnition of the integral over X. Part (b) may be veriﬁed in a similar way. For Part (c), let θ ≤P0 ϑ for a subcone P0 of P, and let f ∈ F(X, P) be (P0 , V)-based integrable over X with respect to Θ = {θ, ϑ}. For E ∈ R, w ∈ W and ε > 0 we choose v ∈ V such that both θE (v) ≤ w and ϑE (v) ≤ w. Let (hn )n∈N be a sequence of P0 -valued step functions ( P) approaching the function ( in SR (X, f as in Theorem 5.7. We have E hn dθ ≤ E hn dϑ for all n ∈ N since θ ≤P0 ϑ, hence by Part (iii) of 5.7 ' ' ' ' f dθ ≤ lim hn dθ + w ≤ lim hn dϑ + w ≤ γ f dϑ + 2w. E

n→∞

(

(

E

n→∞

E

E

Thus E f dθ ≤ E f dϑ, since w ∈ W and ε > 0 were arbitrary. Our claim now follows from the deﬁnition of the integral over a set F ∈ R. By the restriction of a measure θ on R to a subset F ∈ AR we mean the measure θ|F on R, deﬁned as (θ|F )E = θE∩F for all E ∈ R. It is immediate from the deﬁnition of the integral in Section 4 that f ∈ F(X,θ|F ) (X, P) ( if f ∈ F(F,θ) (X, P) for a function ( if and only f ∈ F(X, P), and that X f dθ|F = F f dθ in this case.

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II Measures and Integrals. The General Theory

5.13 Convergence of Sequences of Measures. Let θ and (θn )n∈N be L(P, Q) -valued measures on R. We shall deﬁne lower and upper setwise convergence for measures and denote θn θ or θn θ if (i) θE (a) ≤ lim θn E (a) or lim θn E (a) ≤ θE (a) holds for all E ∈ R and n→∞

n→∞

a ∈ P, respectively. (ii) There is a set E0 ∈ R such that θ|(X\E0 ) ≤P θn |(X\E0 ) or θn |(X\E0 ) ≤P θ|(X\E0 ) holds for all n ∈ N, respectively. We shall denote θn −→ θ if both θn θ and θn θ. Lemma 5.14. Let Θ = {θn }n∈N be equibounded L(P, Q)-valued measures on R such that θn θ for a measure θ. Let E ∈ R. (a) (If f ∈ F(E,Θ) (X, ( P), then f ∈ F(E,Θ∪{θ}) (X, P). (b) E f dθ ≤ lim E f dθn for every f ∈ F(E,Θ) (X, P). n→∞ ( ( (c) E f dθ = lim E f dθn for every invertible function f such that both n→∞

f ∈ F(E,Θ) (X, P) and −f ∈ F(E,Θ) (X, P). Proof. Let Θ = {θn }n∈N be equibounded L(P, Q)-valued measures on such that θn θ for a measure θ. Let E ∈ R. We shall defer the proof of Part (a) since it will use elements of the statement of Part (b). For our proof of Part (b) we shall therefore assume that the function f is integrable over ¯ E with respect nto the family Θ = Θ ∪ {θ}. First, let us consider a step function h = i=1 χEi ⊗ai ∈ SR (X, P). Using Lemma I.5.19 we observe that θn θ implies ' h dθ = X

m !

θEi (ai ) ≤

i=1

m ! i=1

≤ lim

n→∞

lim θn Ei (ai )

n→∞

"m !

# θn Ei (ai )

' = lim n→∞

i=1

h dθn . X

Now let f ∈ F(E,Θ) ¯ (X, P). We shall use Corollary 5.8 with P0 = P and V0 = V in order to establish our claim. Given w ∈ W and ε > 0 there is a bounded below sequence (hk )k∈N of P-valued step functions( such that: (i) there is 1 ≤ γ ≤ 1 + ε and s ∈ FR (X, V) such that E s dϑ ≤ w for ≤ γf + s holds for all n ∈ N; ¯ and hna.e.E all ϑ ∈ Θ, ¯ (ii) Θ-almost everywhere on E, for x ∈ E there is n0 ∈ N such that f (x) ≤ (hn (x) for all n ≥ n(0 ; ( ( (iii) E f dϑ ≤ limn→∞ E hn dϑ and E hn dϑ ≤ γ E f dϑ + w for all ¯ n ∈ N and ϑ ∈ Θ. For every k ∈ N then we have by the above ' ' ' hk dθ ≤ lim hk dθn ≤ γ lim f dθn + w. E

n→∞

X

n→∞

E

5. The General Convergence Theorems

175

(

Note that this argument implies in particular that the sequence E f dθn is bounded below. Using (iii), we proceed from this and conclude that ' ' f dθ ≤ γ lim f dθn + w. n→∞

E

n∈N

E

Claim (b) follows, since this last inequality holds for all w ∈ W and ε > 0. In Part (c) we assume in addition that the negative −f of the function f is also contained in F(E,Θ) (X, P). Then Part (b) yields that both sequences ( ( ¯ f dθ and (−f ) dθn n∈N are bounded below, and n n∈N E E '

'

' f dθ ≤ lim

n→∞

E

Thus

f dθn

lim

n→∞

E

'

' f dθn = lim

E

(−f ) dθ ≤ lim

and

E

' n→∞

' (−f ) dθn . E

'

(−f ) dθ + f dθ E ' ' ≤ lim f dθn + lim (−f ) dθn + f dθ n→∞ E n→∞ E E ' ' (−f ) + f ) dθn + ≤ lim f dθ n→∞ E E ' ' ≤ f dθ ≤ lim f dθn , n→∞

E

f dθn +

'E

E

n→∞

E

and our claim (c) follows. We shall ﬁnally prove Part (a) of the lemma: Let Z ∈ AR be a zero-set for Θ = {θn }n∈N , that is θn (E∩Z) = 0 for all n ∈ N and E ∈ R. As θn θ, this implies θ(E∩Z) (a) ≤ 0 for all a ∈ P, hence θ(E∩Z) (a) = 0 for all 0 ≤ a ∈ P. However, for every a ∈ P there is v ∈ V such that 0 ≤ a + v. Hence θ(E∩Z) (a) = θ(E∩Z) (a) + θ(E∩Z) (v) = θ(E∩Z) (a + v) = 0. Thus θ(E∩Z) = 0. Every zero-set for Θ is therefore a zero-set for Θ ∪ {θ} as well. Now let f ∈ F(|X|,Θ) (X, P) and let E ∈ R. According to Deﬁnition 5.3, for every w ∈ W and ε > 0 there are functions f(w,ε) ∈ FR (X, P) and s(w,ε) ∈ FR (X, V) such that ' ≤ f(w,ε) a.e.E ≤ γf + s(w,ε) f a.e.E and s(w,ε) dϑ ≤ εw E

for some 1 ≤ γ ≤ 1 + ε and all ϑ ∈ Θ. The almost everywhere relations refer to the family Θ and by the above therefore also to Θ ∪ {θ}. Because the function s(w,ε) ∈ FR (X, V) is integrable over E with respect to every family of measures on R, we may use Part (b) of the lemma for

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II Measures and Integrals. The General Theory

'

' s(w,ε) dθ ≤ lim

E

n→∞

s(w,ε) dθn ≤ εw. E

This shows that the function f is indeed integrable over E with respect to the family Θ ∪ {θ}. Example 5.15. The following example will demonstrate that a result corresponding to 5.14(b) for upper convergence of measures is not available in general, (that is θn ( θ for measures θn , θ on R does not necessarily imply that lim E f dθn ≤ E f dθ holds for every integrable function f ∈ F(X, P). For

n→∞

this, let X = [0, 1], let R be the σ-algebra of all Borel sets on X, and let θ be the Lebesgue measure. This may be considered as an L(P, Q)-valued measure if we set√ P = Q = R (see Examples I.2.1). We deﬁne the measures θn as θn (E) = n θ E ∩ [0, n1 ] for E ∈ R. This yields θn (E) ≤ √1n for all E ∈ R and n ∈ N, hence θn −→ 0, that is the zero measure on R. Now consider the function f on X deﬁned as f (x) = √1x for x > 0 and f (0) = 0. As f is positive and measurable, it is contained in FR (X, P), hence in F(E,Θ) (X, P), where Θ is the equibounded family {θn }n∈N . We ( ( ( √ ( 1 calculate X f dθn = n 0n f dθ = 2, hence X f dθn → X f d 0 = 0, indeed. Note that Part (c) of Lemma 5.14 does not apply in this case. The function f is in fact invertible in F(X, P) and its inverse −f is integrable with respect to each of the measures θn . Indeed, given ε > 0 we may choose fε (x) = −f (x) for x ≥ ε and fε (x) = 0 else. Then fε is bounded below, hence in FR (X, P), and we have −f ≤ fε ≤ −f + sε , where sε (x)= 0 for x = 0 or x ≥ ε, and sε (x) = f (x) else. (This function sε is V ∪{0} -valued as required in Deﬁnition 4.12, since the neighborhood(system V of √ R consists of all strictly positive reals.) For ε ≤ n1 we calculate X sε dθn = 2 nε. Thus −f ∈ F(E,θn ) (X, P) for all n ∈ N, but −f is not contained in F(E,Θ) (X, P) as required in 5.14(c). 5.16 Residual Components. Let (θn )n∈N be an equibounded sequence of measures, let F ∈ AR and f ∈ F(F,{θn }) (X, P). We deﬁne the residual component of f on F with respect to (θn )n∈N as follows: Let F be the collection of

all sequences (En )n∈N of sets in R such that En ⊂ F, En ⊃ En+1 and n∈N En = ∅. Recall that integrability for a function f ∈ F(X, P) over F requires integrability over all subsets E ∈ R of F. Thus for f ∈ F(F,{θn }) (X, P) we deﬁne ' lim f dθn lim . Rs θn , F, f = sup (Em )∈F

m→∞

n→∞

Em

This appears to be a rather unwieldy expression. It will however turn out to be useful for our continuing investigations.

5. The General Convergence Theorems

177

Lemma 5.17. Let (θn )n∈N be an equibounded sequence of measures, and let F ∈ AR . Then (a) Rs θn , F, f ≥ 0 for all f ∈ F(F,{θn }) (X, P). ω (b) If θn ≤ ω for a measure ( and all n ∈ N, then Rs θn , F, f ≤ O F f dω for all f ∈ F(|F |,{θn ,ω}) (X, P). Proof. Part (a) is trivial, as we may choose the stationary sequence (Em )m∈N ∈ F, where Em = ∅ for all n ∈ N. For Part (b), suppose that θn ≤ ω holds for a measure ω on R and all n ∈ N, let Θ = {θn , ω}n∈N , let f ∈ F(F,Θ) (X, P) and (Em )m∈N ∈ F. For every w ∈ W there is by ≤ f +s Proposition 5.4 a function s ∈ FR (X, V) and λ ≥ 0 such that 0 a.e.E 1 ( and E1 s dϑ ≤ λw for all ϑ ∈ Θ. Because s ≥ 0, this yields for all m ∈ N ' n→∞

'

' f dθn ≤ lim

lim

n→∞

Em

(f + s) dθn ≤ Em

(f + s) dω. Em

Thus by Proposition 4.18(b) and Proposition I.5.11 ' ' lim f dθn ≤ lim (f + s) dω lim m→∞ n→∞ Em m→∞ Em ' (f + s) dω ≤O E ' 1 ' =O f dω + O s dω E E1 ' 1 ≤O f dω + w, E1

(

since O E1 s dω ≤ εw for all ε > 0. Because w ∈ W was arbitrarily chosen, and because Q carries the weak preorder, this yields ' ' lim f dθn ≤ O f dω . lim m→∞

n→∞

Em

E1

( ( Furthermore, Proposition 4.15(c) states that O E1 f dω ≤ O F f dω . Now combining all of the above, we have indeed ' ' Rs θn , F, f = sup lim f dθn f dω , lim ≤O (Em )∈F

as claimed.

m→∞

n→∞

Em

F

Lemma 5.17(b) implies in particular that for a stationary sequence (θn )n∈N of ( measures, that is θ = θ for all n ∈ N, we have Rs θ , F, f ≤ n n O F f dθ for all f ∈ F(|F |,θ) (X, P). This leads to the following notation:

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II Measures and Integrals. The General Theory

For a set F ∈ AR , and an equibounded sequence (θn )n∈N of measures, a measure θ and a family F of functions in F(F,{θn ,θ}) (X, P) we shall denote ' F if Rs θn , F, f ≤ O f dθ (θn ) F≺ θ F

holds for all f ∈ F. Setwise convergence of the measures θn towards θ, that θ holds for evis θn −→ θ, does however not necessarily imply that (θn ) {fF≺ } ery integrable function f ∈ F(|F |,{θn ,θ}) (X, P), as our preceding Example 5.15 can demonstrate. Indeed, let us calculate the residual component of the function f in 5.15 on the interval F = [0, 1] with respect to the given sequence (Em )m∈N be a sequence of Borel (θn )n∈N of measures on [0, 1]. First, let

sets in [0, 1] such that Em ⊃ Em+1 and m∈N Em = ∅. Then ' ' f dθn ≤ f dθn ≤ 2 Em

[0,1]

1 , on the for all k, l ∈ N. This shows Rs θn , F, f ≤ 2. For Em = (0, m other hand, we have Em ⊃ Em+1 and m∈N Em = ∅, and Em f dθn = 2 ( whenever n ≥ m. Thus lim Emf dθn ≥ 2 for all m ∈ N, and therefore n→∞ Rs θn , F, f ≥ 2. Together with the above, this yields Rs θn , F, f = 2. But we have θn −→ 0. A stronger requirement on the integrability of the function f will however avoid such cases. 5.18 Strongly Integrable Functions. Let Θ be a an equibounded family of measures, and let E ∈ R. We shall say that a function f ∈ F(X, P) is strongly integrable over E with respect to Θ if it is integrable over E in the sense of 5.3, and if in addition, for every w ∈ W there is a step ( ( ( such that f dθ ≤ h dθ + w and h function h ∈ SR (X, P) G G G dθ is w( bounded relative to G f dθ in Q, for all θ ∈ Θ and every subset G ∈ R of E. Note that this requirement strengthens the corresponding property from Theorem 5.7 which holds for integrable functions in general. Similarly, for a set F ∈ AR , a function f ∈ F(X, P) is strongly integrable over F with respect to Θ if it is integrable over F in the sense of 5.3 and strongly integrable over the sets E ∩ F for all E ∈ R. Because strong integrability over a set E ∈ R obviously implies strong integrability over every subset G ∈ R of E, this last part of our deﬁnition is consistent with the ﬁrst one. It is straightforward to verify that the strongly integrable functions form a subcone of F(F,Θ) (X, P). Lemma 5.19. Let Θ = {θ, θn }n∈N be equibounded measures on R such that θn θ. Let F ∈ AR , and suppose that the function f ∈ F(|F |,Θ) (X, P) is θ. strongly integrable over F with respect to Θ. Then (θn ) {fF≺ }

5. The General Convergence Theorems

179

Proof. Let Θ = {θ, θn }n∈N be equibounded measures such that θn θ. As in the ﬁrst step of the proof of Lemma 5.14, one easily veriﬁes that ' ' lim h dθn ≤ h dθ n→∞

G

G

holds for every step function h ∈ SR (X, P) and all G ∈ AR . Let F ∈ AR and suppose that the function f ∈ F(F,Θ) (X, P) is strongly integrable over F with respect to Θ. Let

Em ∈ R for m ∈ N be subsets of F such that Em ⊃ Em+1 and m∈N Em = ∅. Following 5.18, the function f is strongly integrable n over the set E = E1 . Given w ∈ W, we choose a step function h = ( i=1 χGi ⊗ai ∈ (SR (X, P)( as in the ﬁrst part of 5.18, that is ( h dθ ∈ B w G G f dθ , and G f dθ ≤ G h dθ + w holds for all θ ∈ Θ and every subset G ∈ R of E, in particular ' ' f dθn ≤ h dθn + w Em

Em

holds for all m, n ∈ N. Thus ' ' lim f dθn ≤ lim n→∞

n→∞

Em

' h dθn + w ≤

Em

for every m ∈ N, and consequently ' ' lim lim f dθn ≤ lim m→∞

n→∞

m→∞

Em

h dθ + w Em

' h dθ + w ≤ O h dθ + w

Em

E

( by Proposition ( 4.18(b). Because E h dθ is (w-bounded relative to E f dθ, to F f dθ by Proposition 4.15(c), and because ( E f dθ is bounded ( relative we have O E h dθ w O F f dθ by Proposition I.5.13(a). The latter implies ' ' h dθ ≤ O f dθ + w. O (

E

F

Thus, summarizing, lim m→∞

' lim

n→∞

f dθn

' f dθ + 2w. ≤O

Em

F

This holds for all w ∈ W and all sequences of sets Ek ∈ R such that Ek ⊂ F and k∈N Ek , and therefore demonstrates ' f dθ , Rs θn , F, f ≤ O F

our claim.

180

II Measures and Integrals. The General Theory

Lemma 5.20. Let Θ = {θ, θn }n∈N be equibounded L(P, Q)-valued measures on R such that θn θ. Let E ∈ R. (a) Let fn , f, f ∗ ∈ F(E,Θ) (X, P) and v ∈ V. If both fn (x) v f (x) and Θ-almost on fn (x)( v f ∗ (x) holds E for all n ∈ N, then everywhere ( lim E fn dθn ≤ E f dθ + Rs θn , E, f ∗ + O sup{θE (v) | θ ∈ Θ} . n→∞ ( ( (b) Let f ∈ F(E,Θ) (X, P). If (θn ) {fE≺ } θ, then lim E f dθn ≤ E f dθ. n→∞

Proof. Let Θ = {θ, θn }n∈N be equibounded measures such that θn θ. As seen before, this implies ' ' lim h dθn ≤ h dθ n→∞

G

G

for every step function h ∈ SR (X, P) and every G ∈ AR . Let E ∈ R, v ∈ V, and let fn , f, f ∗ ∈ F(E,Θ) (X, P) such that both fn v f and fn v f ∗ holds Θ-almost everywhere on E for all n ∈ N. Let us abbreviate d = O sup{θE (v) | θ ∈ Θ} ∈ Q. Recall from Proposition I.5.11(b) that αd = d for all α > 0. Given w ∈ W and ε > 0, we shall use Corollary 5.8 for the function f, with P0 = P and V0 = V, in order to obtain a sequence (hk )k∈N of step functions satisfying 5.8(i), (ii) and (iii). We set Gm = {x ∈ E | f (x) v hk (x)

for all k ≥ m}

for m ∈ N. Following Theorem 1.6, thesets Gm are contained in R, and we have Gm ⊂ Gm+1 . ( If we set( G = m∈N Gm , then 5.8(ii) implies that E \ G ∈ Z(Θ), that is E g dϑ = G g dϑ for the functions g = fn , f, f ∗ and all ϑ ∈ Θ. Because fn (x) v hm (x) holds Θ-almost everywhere on Gm , and fn (x) v f ∗ (x) holds Θ-almost everywhere on E for all n ∈ N, and because O ϑE (v) ≤ d for all θ ∈ Θ, Proposition 4.16 yields ' ' ' ' fn dϑ ≤ hm dϑ + d and fn dϑ ≤ f ∗ dϑ + d. Gm

Gm

(G\Gm )

for all ϑ ∈ Θ and n ∈ N. Thus ' ' ' fn dθn = fn dθn + E

Gm

fn dθn

(G\Gm )

' ≤

' hm dθn +

Gm

Gm

f ∗ dθn + d

(G\Gm )

holds for all m, n ∈ N. Let Em = G \ Gm . Then Em ⊃ Em+1 and

m∈N Em = ∅. Using this, we proceed with our argument. For a ﬁxed m ∈ N, we let n tend to inﬁnity in the preceding inequality, and obtain

5. The General Convergence Theorems

'

181

'

'

f dθn ≤ lim hm dθn + lim f ∗ dθn + d n→∞ n→∞ E Gm Em ' ' ∗ ≤ hm dθ + lim f dθn + d n→∞ E Gm m ' ' ≤γ f dθ + lim f ∗ dθn + d + w.

lim

n→∞

n→∞

Gm

Em

with some 1 ≤ γ ≤ 1 + ε. Finally, we let m tend to inﬁnity as well and use Proposition 4.18(a) for ' ' ' lim f dθ = f dθ = f dθ. m→∞

Gm

G

Thus n→∞

'

' f dθn ≤ γ

lim

E

E

f dθ + lim m→∞

E

'

'

f ∗ dθn

lim

n→∞

+d+w

Em

f dθ + Rs θn , E, f ∗ + d + w E ' ∗ ≤γ f dθ + Rs θn , E, f + d + w.

≤γ

E

The last inequality holds for all w ∈ W and ε > 0, hence ' ' lim f dθn ≤ f dθ + Rs θn , E, f ∗ + d, n→∞

E

E

since Q is endowed with the weak preorder. For Part (b), we set fn = f = f ∗ in Part (a). If (θn ) {fE}≺ θ, that is ( Rs θn , E, f ≤ O E f dθ holds in addition, then ' ' ' ' f dθ + Rs θn , E, f ≤ f dθ + O f dθ = f dθ E

E

E

E

follows from Proposition I.5.14. Given w ∈ W, we choose v ∈ V such that θE (v) ≤ w for all θ ∈ Θ. Then obviously O sup{θE (v) | θ ∈ Θ} ≤ w holds as well. Part (a) therefore yields ' ' ' lim f dθn ≤ f dθ + Rs θn , E, f + w ≤ f dθ + w n→∞

E

E

for all w ∈ W. Thus indeed

E

'

n→∞

' f dθn ≤

lim

E

since Q carries the weak preorder.

f dθ, E

182

II Measures and Integrals. The General Theory

Our upcoming convergence theorems will imply that the statements of Lemmas 5.14 and 5.20 do indeed extend to integrals over sets F ∈ AR , if the concerned functions are contained in F(|F |,{θn ,θ}) (X, P). Lemma 5.21. Let Θ = {θn }n∈N be equibounded L(P, Q)-valued measures on R such that θn −→ θ for a measure θ. Let F ∈ AR . If f ∈ F(F,Θ) (X, P), then f ∈ F(F,Θ∪{θ}) (X, P). Proof. We may assume that F = X, since for a function f ∈ F(X, P) integrability over F means equivalently that the function χF ⊗f is integrable over X. Let f ∈ F(X,Θ) (X, P). Then f ∈ F(E,Θ) (X, P) for every E ∈ R, hence f ∈ F(E,Θ∪{θ}) (X, P) by Lemma 5.14(a). Now let E0 ∈ R and v ∈ V be as in Deﬁnition 5.13(ii), that is θ|(X\E0 ) = θn |(X\E0 ) holds for all n ∈ N. Let E ∈ R such that E0 ⊂ E and ﬁx n0 ∈ N. Then ' ' ' ' ' f dθ = f dθ + f dθ = f dθ + f dθn0 . E

Hence

E\E0

'

' f dθ =

lim

E∈R

E0

E0

E∈R

E\E0

'

' f dθ + lim

E

E0

'

f dθn0 = E\E0

f dθ + E0

f dθn0 . X\E0

The function f is therefore indeed integrable over X with respect to θ, and we infer that f ∈ F(E,Θ∪{θ}) (X, P). 5.22 Convergence of Sequences in F (X, P). In Section 3 we introduced several notions of pointwise convergence for sequences of P-valued functions. They refer to the lower and upper relative topologies of P, that is for a subset F of X, a sequence (fn )n∈N and a function f in F(X, P) f if for every x ∈ F, v ∈ V and ε > 0 there we denote fn F f or fn F is n0 such that f (x) ∈ vε fn (x) or fn (x) ∈ vε f (x) for all n ≥ n0 , respectively. fn → f means that both fn f. F f and fn F F Correspondingly, if Θ is a family of measures on R, then we shall denote fn a.e.F f or fn −→ f if this convergence holds Θ-almost evfn a.e.F f, a.e.F erywhere on F, that is on a subset F \ Z with some Z ∈ Z(Θ). The following version of Fatou’s lemma is the ﬁrst of our main convergence theorems. It refers to lower convergence for both functions and measures. Theorem 5.23. Let Θ = {θ, θn }n∈N be equibounded L(P, Q)-valued measures on R such that θn θ. Let F ∈ AR , and let fn , f, f∗ , f∗∗ ∈ ≤ fn + f∗ for all F(|F |,Θ) (X, P) such that (θn ) {fF∗ }≺ θ. Suppose that f∗∗a.e.F n ∈ N, and that fn a.e.F f. Then ' ' ' f dθ ≤ lim fn dθn + O f∗ dθ . F

n→∞

F

F

5. The General Convergence Theorems

183

Proof. Without loss of generality, we may assume that F = X. Indeed, the respective integrals over F ∈ AR equal the integrals over X for the products of the concerned functions with χF , and these products satisfy the conditions of the theorem with X in place of F. Also, we may assume that the required convergence and boundedness properties hold everywhere on X instead of Θ-everywhere. Indeed, let Z(Θ) be the family of zero subsets of X. Then fn a.e.X f means that fn (X\Y ) f for some Y ∈ Z(θ). Using the fact that Z(Θ) contains countable unions of its members, we can ﬁnd Y ∈ Z(Θ) such that χ(X\Y ) ⊗f∗∗ ≤ χ(X\Y ) ⊗(fn + f∗ ) holds for all n ∈ N everywhere on X. Let Z = Y ∪ Y ∈ Z(Θ). The functions fn = χ(X\Z) ⊗fn , f = χ(X\Z) ⊗f and f∗ = χ(X\Z) ⊗f∗ , then fulﬁll everywhere all the assumptions of the theorem and their respective integrals coincide with those of the given functions. We shall proceed using these simpliﬁed assumptions of the theorem for the measures θn and θ and the functions fn , f, f∗∗ and f∗ . In a ﬁrst step of this proof we shall discuss the respective integrals of the functions involved over a set E ∈ R. Let w ∈ W be ﬁxed. ( Following Proposition 5.4, there is s ∈ FR (X, V) and λ > 0 such that E s dϑ ≤ λw for all ϑ ∈ Θ and both ≤ f + s. Using a similar argument as above, that is the replacement of the 0a.e.E functions fn and f by suitable functions fn and f which agree with the former ones Θ-almost everywhere, we may also assume that the last relation 1 }. holds indeed everywhere on E. Next we choose 0 < ε < min{1, 3λ According to Deﬁnition 5.3 (see also 4.12), we may assume that s(x) ∈ V for all x ∈ X. Thus, under the (now simpliﬁed) assumptions of the theorem, for every x ∈ E there is n0 ∈ N such that f (x) ∈ s(x) ε fn (x) that is f (x) ≤ γfn (x) + εs(x) for all n ≥ n0 with some 1 ≤ γ ≤ 1. According to Lemma I.4.1(c), the latter implies that f (x) ≤ (1 + ε)fn (x) + ε(1 + 1 + ε)s(x) ≤ (1 + ε)fn (x) + 3εs(x) for all n ≥ n0 . We choose a neighborhood v ∈ V such that ϑE (v) ≤ w for all ϑ ∈ Θ. Following Theorem 1.6, all the sets Em = {x ∈ E | f (x) v (1 + ε)fn (x) + 3εs(x) for all n ≥ m} are in R, we have Em ⊂ Em+1 , and m∈N Em = E by the above. Thus f (x) v (1 + ε)fn (x) + 3εs(x) for all x ∈ Em and n ≥ m. Now Proposition 4.16 yields that ' ' ' f dϑ ≤ (1 + ε) fn dϑ + 3ε s dϑ + O ϑ(Em , v) Em E Em ' m ≤ (1 + ε) fn dϑ + w Em

184

II Measures and Integrals. The General Theory

( holds for all ϑ ∈ Θ. The last part of the inequality follows, since Em s dϑ ≤ ( E s dϑ ≤ λw, 3ελ < 1 and O ϑ(Em , v) ≤ O ϑ(E, v) ≤ ε w for all ε > 0. ≤ fn + f∗ for Next we use f∗∗a.e.X ' ' ' f∗∗ dϑ ≤ fn dϑ + f∗ dϑ, E\Em

E\Em

E\Em

multiply the latter by (1 + ε) and add it to the preceding inequality for '

'

' Em

f∗ dϑ + w.

'

f∗∗ dϑ ≤ (1 + ε)

f dϑ + (1 + ε)

fn dϑ +

E\Em

E

E\Em

The latter holds true for all m ∈ N, n ≥ m and ϑ ∈ Θ. For ﬁxed m ∈ N, Lemma 5.14(b) yields together with I.5.19 '

'

'

'

f∗∗ dθ ≤ lim

f dθ + (1 + ε) Em

f dθn + (1 + ε) lim

n→∞

E\Em

n→∞

Em

'

≤ lim

'

f∗∗ dθn E\Em

f dθn + (1 + ε)

n→∞

Em

≤ (1+ε)

E\Em

'

lim

'

fn dθn + lim

n→∞

f∗∗ dθn

n→∞

E

f∗ dθn E\Em

+ w. Now we let m tend to inﬁnity and apply Proposition 4.18(a) for ' ' lim f dθ = f dθ m→∞

Em

and 4.18(b) for

E

' 0 ≤ lim

m→∞

f∗ dθ. (E\Em )

Moreover, the deﬁnition of the residual component in 5.16 together with our assumption (θn ) {fF∗≺ θ yields } lim n→∞

' lim

n→∞

f∗ dθn

≤ Rs θn , X, f∗ ≤ O

E\Em

m→∞

f∗ dθ .

X

The preceding inequality therefore leads to ' ' ' f dθ ≤ lim f dθ + (1 + ε) lim E

'

Em

'

≤ (1 + ε) lim

n→∞

m→∞

'

fn dθn + O E

f∗ dθ

(E\Em )

X

f∗ dθ + w.

5. The General Convergence Theorems

185

Because this last inequality holds for all w ∈ W and ε > 0, and as Q carries the weak preorder, we infer that '

'

' f dθ ≤ lim

n→∞

E

f∗ dθ .

fn dθn + O E

X

Now in the second step of our proof, we shall extend the preceding inequality from integrals over sets E ∈ R to the corresponding integrals over X. By our assumption there is E0 ∈ R such that θ|(X\E0 ) ≤P θn |(X\E0 ) holds for all( n ∈ N. Following Proposition 5.12(c), the latter implies that ( g dθ ≤ F g dθn for every F ∈ AR such that F ⊂ X \ E0 and F g ∈ F(F,Θ) (X, P). Recall that all functions involved in the theorem are in F(|X|,Θ) (X, P0 ), hence are integrable over complements of all sets in R. Using this, for every E ∈ R such that E0 ⊂ E we infer that '

'

'

f∗∗ dθ ≤

(X\E)

'

(X\E)

f∗ dθ,

(X\E)

f∗∗ dθ ≤

fn dθn + E

fn dθn +

(X\E)

'

'

'

f∗ dθ ≤

fn dθ +

(X\E)

hence

'

(X\E)

' fn dθn +

X

f∗ dθ (X\E)

for all n ∈ N. Thus using the above and the result of our ﬁrst step we obtain '

'

' E

fn dθn +

n→∞

(X\E)

E

(X\E)

≤ lim

X

'

'

' n→∞

' f∗∗ dθ + O f∗ dθ

'

f∗∗ dθ ≤ lim

f dθ +

fn dθn +

f∗ dθ .

f∗ dθ + O

X

(X\E)

X

Now we use the deﬁnition of the integral for ' ' f dθ = lim f dθ X

E∈R

E

and Proposition 4.19 for ' 0 ≤ lim E∈R

'

' f∗∗ dθ

(X\E)

and

lim

E∈R

f∗ dθ ≤ O (X\E)

f∗ dθ .

X

Finally, taking the limit over E ∈ R, and combining all of the above yields

186

II Measures and Integrals. The General Theory

'

'

'

f dθ ≤ lim

E∈R

X

f dθ + lim E∈R

E

'

'

≤ lim

E∈R

f dθ + '

E

≤ lim

n→∞

f∗∗ dθ (X\E)

f∗∗ dθ (X\E)

'

fn dθn + O X

f∗ dθ , X

( ( ( since O X f∗ dθ + O X f∗ dθ = O X f∗ dθ by Proposition I.5.11. This completes our proof. Because cone-valued functions do in general not have additive inverses, we require a result corresponding to Theorem 5.23 with respect to upper convergence for both measures and functions. Theorem 5.24. Let Θ = {θ, θn }n∈N be equibounded L(P, Q)-valued measures on R such that θn θ. Let F ∈ AR , and let fn , f, f ∗ ∈ F(|F |,Θ) (X, P) ≤ f ∗ for all n ∈ N, and that such that (θn ) {fF∗ }≺ θ. Suppose that fn a.e.F f. Then fn a.e.F ' ' ' ∗ lim fn dθn ≤ f dθ + O f dθ . n→∞

F

F

F

Proof. Our argument will follow the lines of the proof of Theorem 5.23, though some substantial adaptations will be required. For the reasons given in 5.23, without loss of generality, we may assume that F = X, and that the stated convergence and boundedness properties hold everywhere on X instead of Θ-everywhere. Suppose that the functions f, fn , f ∗ and the measures Θ = {θ, θn } fulﬁll these simpliﬁed assumptions of the theorem. Again, in a ﬁrst step we shall discuss the respective integrals of the functions involved over a set E ∈ R. For this, let w ∈ W be ﬁxed. Following Proposition 5.4, there is s ∈ ≤ ≤ f ∗ + s and 0 a.e.E (SR (X, V) and λ > 0 such that both 0 a.e.E f + s and ∗ ≤ s dϑ ≤ λw for all ϑ ∈ Θ. Moreover, we have f f for all n ∈ N by or n a.e.E E assumption. Using a similar argument as before, we may assume that all these 1 }. relations hold indeed everywhere on E. Next we choose 0 < ε < min{1, 2λ We may assume that s(x) ∈ V for all x ∈ X (see 5.3 and 4.12). Thus, under the (now simpliﬁed) assumptions theorem, for every x ∈ E there of the is n0 ∈ N such that fn (x) ∈ s(x) ε f (x) that is fn (x) ≤ γf (x) + εs(x) for all n ≥ n0 with some 1 ≤ γ ≤ 1. According to Lemma I.4.1(b), the latter implies that fn (x) ≤ (1 + ε)f (x) + 2εs(x) for all n ≥ n0 . We choose a neighborhood v ∈ V such that ϑE (v) ≤ w for all ϑ ∈ Θ. Following Theorem 1.6, all the sets Em = {x ∈ E | fn (x) v (1 + ε)f (x) + 2εs(x)

for all n ≥ m}

5. The General Convergence Theorems

187

are in R, we have Em ⊂ Em+1 , and

Em = E by the above. Thus

m∈N

fn (x) v (1 + ε)f (x) + 2εs(x)

fn (x) ≤ f ∗ (x)

and

holds Θ-almost everywhere on Em for all n ≥ m. We ﬁx m ∈ N, recall that δ(1 − 2ελ) > 0, and that O sup{θE (v) | θ ∈ Θ} ≤ δw for all δ > 0, and use Lemma 5.20(a) for ' ' (1 + ε)f + 2εs dθ + Rs θn , Em , f ∗ + (1 − 2ελ)w lim fn dθn ≤ n→∞ E Em m ' ' f dθ + O f ∗ dθ + w. ≤ (1 + ε) Em

X

( ( The last part of this inequality follows, since Em s dθ ≤ E s dθ ≤ λw, and since ' f ∗ dθ Rs θn , Em , f ∗ ≤ Rs θn , X, f ∗ ≤ O X

≤ f ∗ for

∗

by on the function f . Next we use fn ( ( our assumption ∗ f dϑ ≤ n (E\Em ) (E\Em ) f dϑ, and Lemma 5.20(b) for '

'

n→∞

f ∗ dθn ≤

fn dθn ≤ lim

lim

n→∞

(E\Em )

'

(E\Em )

a.e.E

f ∗ dθ.

(E\Em )

Thus, using the limit rules from Lemma I.5.19, we obtain ' ' ' lim fn dθn ≤ lim fn dθn + lim fn dθn n→∞

E

n→∞

n→∞

Em

'

'

≤ (1 + ε)

(E\Em )

f ∗ dθ + O

f dθ + Em

(E\Em )

'

f ∗ dθ + w.

X

This holds true for all m ∈ N. Now we let m tend to inﬁnity and apply Proposition 4.18(a) for ' ' lim f dθ = f dθ m→∞

Em

E

and 4.18(b) and Proposition I.5.11 for ' ' lim f ∗ dθ ≤ O f ∗ dθ m→∞

(E\Em )

E

' ≤O

E

' f ∗ dθ + O

f ∗ dθ

(X\E)

' =O X

f ∗ dθ .

188

II Measures and Integrals. The General Theory

Combining all of the above, we obtain ' ' ' ∗ lim fn dθn ≤ (1 + ε) f dθ + O f dθ + w. n→∞

E

E

X

Because this inequality holds for all w ∈ W and ε > 0, and as Q carries the weak preorder, we infer that ' ' ' lim fn dθn ≤ f dθ + O f ∗ dθ . n→∞

E

E

X

Now in a second step, we shall extend the preceding inequality from integrals over sets E ∈ R to the corresponding integrals over X. Following our deﬁnition of the convergence of measures in 5.13, there is E0 ∈ R such that θn |(X\E0 ) ≤P θ|(X\E0 ) holds ( ∈ N. Following Proposition 5.12(c), ( for all n the latter implies that F g dθn ≤ F g dθ for every F ∈ AR such that F ⊂ X \ E0 and g ∈ F(F,Θ) (X, P). Using this, for every E ∈ R such that E0 ⊂ E we infer that ' ' ' ' ' fn dθn = fn dθn + fn dθn ≤ fn dθn + f ∗ dθ X

E

(X\E)

E

(X\E)

for all n ∈ N. Thus using the above and the result of our ﬁrst step we obtain ' ' ' lim fn dθn ≤ lim fn dθn + f ∗ dθ m→∞

m→∞

X

E

' ≤

(X\E)

'

f dθ + E

'

∗

f dθ + O (X\E)

∗

f dθ . X

Now we use the deﬁnition of the integral for ' ' lim f dθ = f dθ E∈R

E

X

and Proposition 4.19 for ' lim E∈R

f ∗ dθ ≤ O

'

(X\E)

f ∗ dθ .

X

Combining all of these observations and taking the limit over E ∈ R in the above inequality yields ' ' ' lim fn dθn ≤ f dθ + O f ∗ dθ , m→∞

'

since

X

∗

X

'

f dθ + O

O X

X

∗

f dθ

' =O

X

by Proposition I.5.1. This completes our proof.

f dθ X

∗

5. The General Convergence Theorems

189

Note that for measures θn θ and a stationary sequence of functions, that is fn = f ∗ = f ∈ F(|F |,Θ) (X, P), such that (θn ) {fF}≺ θ, Theorem 5.24 yields ' ' f dθn ≤

lim

m→∞

f dθ,

F

F

( ( ( since F f dθ + O F f dθ = F f dθ by Proposition I.5.14. The combination of Theorems 5.23 and 5.24 leads to a version of Lebesgue’s theorem on dominated convergence (see Proposition 18 in Chapter 11 of [178]). It refers to symmetric convergence for both measures and functions. Theorem 5.25. Let Θ = {θn }n∈N be equibounded L(P, Q)-valued measures on R such that θn −→ θ for a measure θ. Let F ∈ AR , and let fn , f, f∗∗ , f∗ , f ∗∗ , f ∗ ∈ F(|F |,Θ) (X, P) such that (θn ) {fF∗ ,f ∗ }≺ θ. Suppose that ≤ fn + f∗ and fn + f ∗∗a.e.F ≤ f ∗ for all n ∈ N, and that fn −→ f. Then f∗∗a.e.F a.e.F ' ' ' f dθ ≤ lim fn dθn + O f∗ dθ n→∞ F F X ' ' ' and lim fn dθn ≤ f dθ + O f ∗ dθ . n→∞

F

F

F

Proof. Let the functions fn , f, f∗∗ , f∗ , f∗∗ f ∗ and the measures Θ = {θn }n∈N and θ be as in the assumptions of the theorem. Following Lemma 5.21, ¯ = integrability with respect to Θ implies integrability with respect to Θ Θ ∪ {θ}. Our assumptions therefore imply those of Theorem 5.23, and we conclude that ' ' ' f dθ ≤ lim fn dθn + O f∗ dθ . n→∞

F

F

F

In order to apply Theorem 5.24 we set gn = fn + f ∗∗ and g = f + f ∗∗ Then ≤ ∗ for all n ∈ N. Moreover, fn a.e.X f gn , g ∈ F(|F |,Θ) ¯ (X, P) and gn a.e.F f g, since the relative topologies were seen to be comimplies that gn a.e.X patible with the algebraic operations in P (see Section I.4). The functions gn , g therefore fulﬁll the assumptions of Theorem 5.24, and we infer that ' ' ' lim fn dθn + lim f ∗∗ dθn ≤ lim (fn + f ∗∗ ) dθn n→∞ F n→∞ F n→∞ F ' ' ' ∗∗ ∗ ≤ f dθ + f dθ + O f dθ . F

F

Lemma 5.14(b) yields '

' f dθ ≤ lim ∗∗

F

n→∞

f ∗∗ dθn . F

F

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II Measures and Integrals. The General Theory

Thus using the cancellation law in I.5.10(a), we obtain ' ' ' ' lim fn dθn ≤ f dθ + O f ∗ dθ + O f ∗∗ dθ . n→∞

F

F

F

F

∗∗ ≤ f ∗ and fn Finally, the relations fn +f ∗∗a.e.F a.e.F f imply that f (x)+f (x) ∗ f (x) holds θ-almost everywhere on F, and therefore ' ' ' ∗∗ f dθ + f dθ ≤ f ∗ dθ

F

F

F

(

by Proposition 4.17. element F f ∗∗ dθ of Q is therefore bounded relative ( The ∗ to the element F f dθ (see Proposition I.4.11(b)), and Proposition I.5.14 yields that ' ' ' f ∗∗ dθ + O f ∗ dθ = O f ∗ dθ , O F

F

F

thus completing our argument.

We may use the notions of boundedness from Chapter I.4.24 to formulate a special case of Theorem 2.25 that allows a stronger conclusion. Corresponding to I.4.24(iv) we shall say that a subset A of F(X, P) is bounded above relative to a function f ∈ F(X, P) if for every inductive limit neighborhood v there are λ, ρ ≥ 0 such that g ≤ ρf + λv holds for all g ∈ A. Similarly we deﬁne boundedness below and (relative) boundedness almost everywhere on a set F ∈ AR , as well as boundedness for nets and sequences in F(X, P). Recall the notations from I.4.24 and I.4.25. Corollary 5.26. Let θ be a bounded L(P, Q)-valued ( R. Let ( measure on F ∈ AR , and let fn , f ∗ ∈ F(|F |,θ) (X, P) such that F f ∗ dθ ∈ B F f dθ . Let (fn )n∈N be a sequence in F(|F |,θ) (X, P) that is θ-almost everywhere on f, then F bounded below and bounded above relative to f ∗ . If fn −→ a.e.F ' ' lim fn dθ = f dθ. n→∞

F

F

Proof. This is an immediately consequence of Theorem 5.25: We set Θ = {θ} and f∗∗ = f ∗∗ = 0. Given w ∈ W there are λ, ρ ≥ 0 and n0 ∈ N such that ≤ ρf ∗ + λvw holds for all n ≥ n0 . This means ≤ fn + λvw and fn a.e.F 0a.e.F ∗ ≤ ≤ 0a.e.F f(n + λs and fn a.e.F ( ρf + λt for functions s, t ∈ SR (X, V) such that both X s dθ ≤ w and X t dθ ≤ w. Now we apply Theorem ( 5.25 with λs in ∗ ∗ and ρf +λt in place of f from 5.25. Then O λs dθ place of f ∗ F ( (≤w and O F(ρf ∗( + λt) dθ ≤ O f ∗ + w by Proposition I.5.11. Because F f dθ + O f ∗ = F f dθ by Proposition I.5.14 and our assumption on the function f ∗ , and because the neighborhood w ∈ W was arbitrarily chosen, our claim follows.

5. The General Convergence Theorems

191

An elementary function is a function f = ϕ⊗a ∈ F(X, P), where ϕ is a bounded measurable non-negative real-valued function supported by a set E ∈ R, and a is an element of P. Note that elementary functions are contained in FR (X, P). Indeed, χE ⊗a ∈ FR (X, P) implies ϕ⊗a = ϕ⊗(χE ⊗a) ∈ FR (X, P) by Lemma 2.6. We make the following observations: Lemma 5.27. Let ϕ be a bounded measurable non-negative real-valued function supported by a set in R. There is a sequence (ϕn )n∈N of real-valued step functions converging uniformly on X to ϕ and such that 0 ≤ ϕn ≤ ϕ for all n ∈ N. Proof. Let the function ϕ be as stated, supported by the set E ∈ R. Without loss of generality, we may assume that 0 ≤ ϕ ≤ 1. For n ∈ N and i = 1, . . . n we set i−1 i i < ϕ(x) ≤ En = x ∈ E ∈R n n i and ϕn = ni=1 i−1 n χEn . Then 0 ≤ ϕn (x) ≤ ϕ(x) ≤ ϕn (x) + 1/n holds for all x ∈ X. Corollary 5.28. Let θ be a bounded L(P, Q)-valued measure. Let ϕ be a bounded non-negative real-valued function supported by a set in R, and let (ϕn )n∈N be a sequence of measurable real-valued functions such that ≤ ϕ for all n ∈ N, converging θ-almost everywhere to ϕ. Then 0 ≤ ϕn a.e.X for every a ∈ P the sequence (ϕn ⊗a)n∈N in F(X, P) is bounded beeverywhere bounded above relative to ϕ⊗a, the sequence low ( and θ-almost X ϕn ⊗a(dθ n∈N in Q is bounded below and bounded above relative to the element X ϕ⊗a dθ, and ' ' ϕ⊗a and lim ϕn ⊗a dθ = ϕ⊗a dθ. ϕn ⊗a −→ a.e.X n→∞

X

X

Proof. Let the function ϕ be as stated, supported by the set E ∈ R. We ≤ ϕ ≤ 1 holds for all n ∈ N. For a ∈ P we may assume that 0 ≤ ϕn a.e.X have ϕn ⊗a ∈ FR (X, P) for all n ∈ N by Lemma 2.6. There is a set Z ∈ AR of measure 0 such that the functions ϕ˜n = χ(X\Z) ⊗ϕn converge pointwise everywhere to ϕ˜ = χ(X\Z) ⊗ϕ and that ϕ˜n ≤ ϕ˜ ≤ 1 holds for all n ∈ N. (For this, recall that a countable union of zero sets is again a zero set.) Theorem 1.7 guarantees that ϕ˜ is measurable, hence ϕ˜⊗a ∈ FR (X, P), and the function ϕ⊗a is integrable by 4.12. Let x ∈ X \ Z. If ϕ(x) = 0, then that ϕn (x) = 0 for all n ∈ N as well. If ϕ(x) > 0, then, given v ∈ V and ε > 0, there is λ ≥ 1 such that 0 ≤ a + λv and n0 ∈ N such that ϕn (x) ≤ ϕ(x) ≤ (1 + ε)ϕn (x) for all n ≥ n0 . Thus ϕn (x)(a + λv) ≤ ϕ(x)(a + λv) ≤ ϕ(x)(a) + (1 + ε)λϕn (x)v

192

II Measures and Integrals. The General Theory

and ϕ(x)(a + λv) ≤ (1 + ε)ϕn (x)(a + λv) ≤ (1 + ε)ϕn (x)(a) + (1 + ε)λϕ(x)v. Now the cancellation law for positive elements (Lemma I.4.2 in [100]) yields ϕn (x)a ≤ ϕ(x)a + 2ελϕn (x)v ≤ ϕ(x)a + 2ελv and ϕ(x)a ≤ (1 + ε)ϕn (x)a + 2ελϕ(x)v ≤ ϕn (x)a + 2ελv. s This shows ϕn (x)a ∈ v2ελ ϕ(a) for all n ≥ n0 and demonstrates ϕn (x)a −→ ϕ(x)a in the symmetric relative topology of P. Thus ϕn ⊗a −→ ϕ⊗a holds as a.e.X claimed. Furthermore, given an inductive limit neighborhood v there is v ∈ V such that χE ⊗v ≤ v and λ ≥ 0 such that 0 ≤ a + λv. Then 0 ≤ ϕn ⊗(a + λv) ≤ ϕn ⊗a + λv and

≤ ϕ⊗(a + λv) ≤ ϕ⊗a + λv ϕn ⊗a ≤ ϕn ⊗(a + λv)a.e.X

for all n ∈ N. The sequence (ϕn ⊗a)n∈N in F(X, P) is therefore bounded below and θ-almost everywhere bounded above relative to the function the inductive limit ϕ⊗a. Furthermore, for any w ∈ W we may choose ( . Then the above yields 0 ≤ ϕ ⊗ neighborhood v w ( X n a dθ + λw as well ( ϕ ⊗ a dθ ≤ ϕ ⊗ a dθ + λw for all n ∈ N. Hence the sequence as X ( X n ϕ ⊗ a dθ in Q is seen to be bounded below and bounded above relX n n∈N ( ative to the element X ϕ⊗a dθ. The convergence statement for the sequence of integrals follows from Corollary 5.26. Corollary 5.28 in combination with Lemma 5.27 yields a strengthening of the result of Corollary 5.9, that is the approximation of integrable functions by a net of step functions, for elementary functions f = ϕ⊗a ∈ F(X, P) : There is a sequence (hn )n∈N of step functions that is bounded below and bounded above relative to f such that ' ' hn −→ f and lim hn dθ = f dθ. n→∞

X

X

5.29 Remarks. (a) If (Q, W) is the (simpliﬁed) standard lattice completion (see I.57) of some subcone (Q0 , W0 ), that is if (Q, W) is a cone of R-valued functions on P ∗ , then for elements l, m, n ∈ Q the statement l ≤ m + O(n) means that l(μ) ≤ m(μ) holds for all μ ∈ P ∗ such that n(μ) < +∞. The convergence statements of Theorems 5.23 to 5.25 can then be read in this light. The conclusion of Theorem 5.25 means for example that

5. The General Convergence Theorems

193

'

'

f dθ (μ) ≤ lim

n→∞

F

fn dθn (μ)

F

( holds for all μ ∈ P ∗ such that F f∗ dθ (μ) < +∞ , and ' ' lim fn dθn (μ) ≤ f dθ (μ) n→∞

F

(

F

for all μ ∈ P∗ such that F f ∗ dθ (μ) < +∞ . (b) The convergence statements in the preceding Theorems 5.23 to 5.25 refer to order convergence in Q for the concerned sequences of integrals. Stronger claims than those might state convergence in the lower, upper and symmetric topologies of Q, respectively. In the context of our approach, such claims are however not valid in general, even for stationary sequences of measures, as the following simple example can show: Let R be the σ-algebra of all Borel sets in X = [0, 1], let P = R with its usual order and locally convex cone topology. Let Q be the cone of all bounded below R-valued functions on X, endowed with the pointwise operations and order and the strictly positive constant functions w as neighborhoods. Clearly (Q, W) is a locally convex complete lattice cone, and order convergence in Q means pointwise convergence for the concerned functions. We deﬁne an L(P, Q)-valued measure θ on R, setting θE (α) = (αχE ∈ Q for E ∈ R and α ∈ P. It is then straightforward to check that X h dθ = h holds for every P-valued step function h on X, that is the integral over θ yields the identity operator from F(|X|,θ) (X, P) into Q. Now, if we consider the stationary sequences ϑn = θn = θ in Theorems 5.23 to 5.25, a review of the assumptions there reveals that only pointwise convergence is required for the sequences of functions (fn )n∈N and (gn )n∈N in F(|X|,θ) (X, P). Thus only pointwise, that is order convergence will result for their integrals in general. Note that in this example the measure θ is countably additive only with respect to order convergence in Q, not with respect to the weak (see Section I.4.6) or indeed the symmetric relative topology of Q. We shall demonstrate below (Theorem 5.36) that countable additivity for a measure with respect to the symmetric relative topology of Q in this situation would indeed imply the above stronger statement of convergence for the corresponding sequence of integrals. This shows in particular that no such measure can represent the identity operator from F(|X|,θ) (X, P) into Q. We shall in the following discuss some special cases where convergence with respect to the symmetric topology does indeed result from Theorems 5.23 to 5.25. For the sake of simplicity we shall restrict ourselves to stationary sequences of measures θn = θ in this context. The preceding Remark 5.29(b) suggests that we shall need to impose further conditions for this purpose. One of these conditions will refer to the countable additivity of the measure θ, another one will require the availability of suﬃciently many order continuous linear functionals on Q.

194

II Measures and Integrals. The General Theory

5.30 Strong Additivity. Countable additivity of an L(P, Q)-valued measure θ as introduced in Section 3 is meant with respect to order convergence in the locally convex complete lattice cone (Q, W). In Theorem 3.11 we veriﬁed that in special cases this implies convergence in a stronger sense. In this context, we shall say that an L(P, Q)-valued measure θ is strongly additive if for every decreasing sequence (En )n∈N of sets in R such that

n∈N En = ∅, for a ∈ P and w ∈ W there is n0 ∈ N such that θEn (a) ≤ O θE1 (a) + w holds for all n ≥ n0 . Similarly, we shall say that a family Θ of L(P, Q)valued measures is uniformly strongly additive if it is equibounded and if the above property holds with the same n0 for all θ ∈ Θ. Note that for strong additivity we do not require that a measure is countably additive with respect to the symmetric topology of Q, since this would be overly restrictive. For Q = R, for example, the element +∞ is isolated, that is both open and closed in the symmetric topology of R. Thus, for a disjoint union E = i∈N Ei of sets in R such that θE (a) = +∞ for a ∈ P, countable additivity with respect to the symmetric topology would require that θ(∪ni=1 Ei ) (a) = +∞ for all n greater than some n0 ∈ N. Lemma 5.31(b) will however imply that for a uniformly strongly additive family Θ of L(P, Q)-valued measures a requirement corresponding to 5.30 holds indeed with respect to the symmetric topology of Q; more

precisely: Given a decreasing sequence (En )n∈N of sets in R such that n∈N En = ∅, a ∈ P and w ∈ W, there is n0 ∈ N such that 0 ≤ θEn (a) + w and θEn (a) ≤ O θE1 (a) + w holds for all θ ∈ Θ and n ≥ n0 . There are several well-known results about strong additivity. Our version of Pettis’ theorem, that is Theorem 3.11, (see Theorem IV.10.1 in [55]), states that in case that (P, V) is a locally convex topological vector space and (Q, W) is the standard lattice completion of some subcone (Q0 , W), every L(P, Q0 )-valued measure is also strongly additive. The Vitali-HahnSaks theorem see Theorem III.7.2 in [55]) implies a theorem by Nikod´ ym which states that every setwise convergent sequence of real- or Banach spacevalued measures is in fact uniformly strongly additive (see Corollary III.7.4 and Theorem IV.10.6 in [55]). We shall investigate a few implications of strong additivity. Lemmas 5.31 and 5.32 will strengthen the corresponding statements from Proposition 4.18. Lemma 5.31. Suppose that the family Θ of L(P, Q)-valued measures is uniformly strongly additive. Let E ∈ R and f ∈ F(E,Θ) (X, P). (a) If En ∈ R such that En ⊂ En+1 for all n ∈ N, and E = n∈N En , ( w ∈ W there is n0 ∈ N such that (then for every f dθ ≤ En E f dθ + w for all θ ∈ Θ and n ≥ n0 .

5. The General Convergence Theorems

195

(b) If En ∈ R such that E ⊃ En ⊃ En+1 for all n ∈ N, and then(for every w ∈ W there is n0 ∈ N such that 0 ≤ En f dθ + w for all θ ∈ Θ and n ≥ n0 .

n∈N

En = ∅,

Proof. We shall ﬁrst prove Part (b) of the lemma.

Let En ∈ R for n ∈ N be subsets of E ∈ R such that En ⊃ En+1 and n∈N En = ∅. In a ﬁrst step, we shall consider a function f ∈ FR (X, P). Let w ∈ W. Because the family Θ is supposed to be equibounded, there is v ∈ V such that θE (v) ≤ w for all θ ∈ Θ. This implies O θE (v) ≤ εw for all θ ∈ Θ and ε > 0. Following Lemma 2.4(b), there is λ ≥ 0 such that 0 ≤ χE ⊗f + λχE ⊗v. By 5.30 there is n0 ∈ N such that 1 1 w≤ w θEn (v) ≤ O θE (v) + 2λ λ for all θ ∈ Θ and n ≥ n0 . This yields ' ' ' 0≤ (χE ⊗f + λχE ⊗v) dθ = f dθ + λθEn (v) ≤ En

En

f dθ + w

En

for all θ ∈ Θ and n ≥ n0 . Now in the second and general step, let f ∈ F(E,Θ) (X, P). Given w ∈ W and 0 < ε ≤ 1/2, let the functions f(w,ε) ∈ FR (X, P) and s(w,ε) ∈ FR (X, V) be as in the deﬁnition of integrability in 5.3, that is ' ≤ ≤ f a.e.E f(w,ε) a.e.E γf + s(w,ε) and s(w,ε) dθ ≤ εw E

for some 1 ≤ γ ≤ 1 +( ε and all θ ∈ Θ. Following our ( ﬁrst step, there is n0 ∈ N such that 0 ≤ En f(w,ε) dθ + w/2, hence 0 ≤ γ En f dθ + 12 + ε w for all θ ∈ Θ and n ≥ n0 . Because γ ≥ 1 and ε ≤ 1/2 this yields ' 0≤ f dθ + w En

for all θ ∈ Θ and n ≥ n0 , our claim in Part (b). For Part (a), let En ∈ R E = such that En ⊂ En+1 for all n ∈ N, and n∈N En . We set Fn =

E \ En ∈ R and have Fn ⊃ Fn+1 and n∈N Fn = ∅. For a function f ∈ F(E,Θ) (X, P) we may now ( use Part (b) of the lemma: Given w ∈ W, there is n0 ∈ N such that 0 ≤ Fn f dθ + w holds for all θ ∈ Θ and n ≥ n0 . This yields ' ' ' ' f dθ ≤ En

f dθ + En

f dθ + w

=

Fn

for all θ ∈ Θ and n ≥ n0 , our claim in Part (a).

f dθ + w E

Lemma 5.32. Suppose that the family Θ of L(P, Q)-valued measures is uniformly strongly additive and that the function f ∈ F(X, P) is strongly integrable over E ∈ R with respect to Θ.

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II Measures and Integrals. The General Theory

(a) If En ∈ R are such that En ⊂ En+1 for all n ∈ N, and E = n∈N En , w ∈ W ( there is n0 ∈ N such that ( (then for every f dθ ≤ f dθ + O n0 . E En E f dθ + w for all θ ∈ Θ and n ≥

(b) If En ∈ R are such that E ⊃ En ⊃ En+1 for all n ∈ N, and n∈N En = ( w ∈ W there is n0 ∈ N such that (∅, then for every f dθ ≤ O En E f dθ + w for all θ ∈ Θ and n ≥ n0 . Proof. Again, we shall ﬁrst prove Part (b) of the Lemma. Let f ∈ F(X, P) be strongly integrable over E ∈ R with respect to Θ. For w ∈ W, according to 5.18 there is a step function h = m i=1 χFi ⊗ai ∈ SR (X, P) such that ' ' f dθ ≤ h dθ + w/3 G

G

( and such that G h dθ is w-bounded relative to G f dθ for all θ ∈ Θ and every subset G ∈ R of

E. Let En ∈ R for n ∈ N be subsets of E such that En ⊃ En+1 and n∈N En = ∅. For every n ∈ N and θ ∈ Θ, we calculate ' m ! h dθ = θ(En ∩Fi ) (ai ). (

En

i=1

The measures in Θ are supposed to be uniformly strongly additive. Thus there is n0 ∈ N such that 1 w θ(En ∩Fi ) (ai ) ≤ O θ(E∩Fi ) (ai ) + 3m for all n ≥ n0 , θ ∈ Θ and i = 1, . . . , m. Thus, using Proposition I.5.11 ' h dθ ≤ En

m !

1 O θ(E∩Fi ) (ai ) + w = O 3

i=1

'

'

and

h dθ ≤ O (

(

En

f dθ +

E

'

1 h dθ + w 3 E

2 w, 3

since w E f dθ , which by Proposition I.5.13(a) implies that ( E hdθ ∈ B( O E h dθ ≤ O E h dθ + εw for all ε > 0. Thus for all n ≥ n0 and θ ∈ Θ we infer that ' ' ' 1 f dθ ≤ h dθ + w ≤ O f dθ + w, 3 En En E that is Part (b) of our claim. For Part (a), let En ∈ R such that En ⊂ En+1 for all n ∈ N, and E = n∈N En . We set Fn = E \ En ∈ R and have

Fn ⊃ Fn+1 and n∈N Fn = ∅. For a function f ∈ F(X, P) that is strongly integrable over E ∈ R with respect to Θ we may ( (b) of the ( now use Part lemma: Given w ∈ W, there is n0 ∈ N such that Fn f dθ ≤ O E f dθ +w

5. The General Convergence Theorems

197

for all θ ∈ Θ and n ≥ n0 . This yields ' ' ' ' f dθ = f dθ + f dθ ≤ E

En

Fn

' f dθ + O

En

for all θ ∈ Θ and n ≥ n0 , our claim in Part (a).

f dθ + w

E

5.33 Weakly Sequentially Compact Sets of Measures. A family Θ of L(P, Q)-valued measures is said to be weakly sequentially compact if every sequence (θn )n∈N in Θ contains a setwise convergent subsequence (θnk )k∈N , that is θnk −→ θ for some measure θ on R (see Deﬁnition II.3.18 in [55]). Note that we do not require that θ ∈ Θ. As a consequence, every subset of a sequentially compact set is again sequentially compact. Theorem IV.9.1 in [55] provides a well-known criterion for weak sequential compactness of a family of ﬁnite real-valued measures deﬁned on a σ-algebra R : Such a family Θ is weakly sequentially compact if and only if (i) Θ is equibounded, that is the total variation of its elements is bounded on X, and (ii) Θ is uniformly (strongly) additive. We shall use this to establish a criterion for sequential compactness of a family of functional-valued measures, that is for the case Q = R. Lemma 5.34. Suppose that all elements of P are bounded and that P is separable in the symmetric relative v-topology for every v ∈ V. Suppose that X ∈ R. Then every uniformly strongly additive family of P ∗ -valued measures on R is weakly sequentially compact. Proof. Let Θ be a uniformly strongly additive family of P ∗ -valued measures on R. Because we assume that X ∈ R, and because uniform strong additivity includes equiboundedness (see 5.30), there is v ∈ V such that θX (v) ≤ 1 for all θ ∈ Θ. In a ﬁrst step of our argument we ﬁx an element a ∈ P and choose λ ≥ 0 such that both 0 ≤ a + λv and a ≤ λv. The latter is possible because all elements of P are supposed to be bounded, which implies in particular that ϑE (a) < +∞ for every P ∗ -valued measure ϑ and E ∈ R. For every θ ∈ Θ we may therefore deﬁne a real-valued measure θa on R, setting θa (E) = θE (a) for every E ∈ R. The above implies that θa (E) ≤ λθE (v) and 0 ≤ θa (E) + λθE (v), hence |θa (E)| ≤ λθE (v) < λθX (v) ≤ λ. Using this, we can estimate the usual (total) variation var (θa , X) of this measure (see Deﬁnition III.1.4 in [55]) as follows: For disjoint sets E1 , . . . , En ∈ R we have n n ! ! |θEi (a)| ≤ λ θEi (v) = λθ(∪ni=1 Ei ) (v) ≤ λ, i=1

hence var (θa , X) = sup

i=1

n ! i=1

|θa (Ei )| | E1 , . . . , En ∈ R, disjoint

≤ λ.

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II Measures and Integrals. The General Theory

Thus for the family Θa = {θa | θ ∈ Θ} of real-valued measures, ﬁrstly the total variation of its elements is bounded by λ, and secondly, the countable additivity on R is uniform with respect to all measures in Θa . The latter follows from our requirement that the family Θ is uniformly

strongly countably additive. Indeed, let En ∈ R such that En ⊃ En+1 and n∈N En = ∅. Following 5.30, given ε > 0, there is n0 ∈ N such that θEn (a) ≤ O θE (a) + ε for all n ≥ n0 and θ ∈ Θ. Because θE (a) is ﬁnite, we have O θE (a) = 0. Thus θa (En ) = θEn (a) ≤ ε holds for all θa ∈ Θa and n ≥ n0 . Now the criterion from Theorem IV.9.1 in [55] (see the remark following 5.33) for weak sequential compactness of ﬁnite real-valued measures yields that the set Θa is indeed weakly sequentially compact. Now in the second step of our argument, following our assumption of the separability of P, we choose a countable subset {an }n∈N of P that is dense with respect to the symmetric relative v-topology. Let (θn )n∈N be a sequence in Θ. We shall apply a diagonal procedure in order to construct a weakly convergent subsequence of (θn )n∈N . For each n ∈ N, the set Θan of real-valued measures was seen to be weakly sequentially compact. Thus there is a subsequence (θn1 )n∈N of (θn )n∈N and a real-valued measure ϑ1 such that θn1 E (a1 ) → ϑ1 (E) for all E ∈ R. Likewise, there is a subsequence (θn2 )n∈N of (θn1 )n∈N and a real-valued measure ϑ2 such that θn2 E (a2 ) → ϑ2 (E) for all E ∈ R. And so on... We choose the subsequence (θnn )n∈N of (θn )n∈N and claim that this subsequence converges setwise towards some P ∗ -valued measure ϑ. Indeed, for every i ∈ N we have by our construction θnn E (ai ) → ϑi (E) for all E ∈ R. Let P0 be the subcone of P spanned by the elements {an }n∈N . By our assumption P0 is dense in P with respect to the symmetric relative v-topology. For a ﬁxed E ∈ R and a = ni=1 λi ai ∈ P0 for λi ≥ 0, set n ! n ϑE (a) = lim θn E (a) = λi ϑi (E) ∈ R. n→∞

i=1

Clearly, ϑE is a linear functional on P0 , and a ≤ b + v for a, b ∈ P0 implies that θE (a) ≤ θE (b) + θE (v) ≤ θE (b) + 1, for all θ ∈ Θ. Using the limit rules, this shows in turn that ϑE (a) ≤ ϑE (b)+1 holds as well. The linear functional ϑE : P0 → R is therefore continuous with respect to the locally convex topology on P generated by the single neighborhood v ∈ V, that is the neighborhood system Vv = {αv | α > 0}, and can therefore be uniquely extended to a continuous linear functional on the whole cone P (see Theorem I.5.56). Moreover, for every a ∈ P and 0 < ε ≤ 1 there is some b ∈ P0 such that both a∈ vε (b) and b ∈ vε (a). This implies by the above that θnn E (a) ∈ vε θnn E (b) and θnn E (b) ∈ vε θnn E (a) n for all n ∈ N. There is n0 ∈ N such that for all n ≥ n0 we have θn E (b) ∈ n vε ϑE (b) and ϑE (b) ∈ vε θn E (a) . Now combining all of the above yields with Lemma I.4.1(a)

5. The General Convergence Theorems

199

θnn E (a) ∈ vε θnn E (b) ⊂ v3ε ϑE (b) ⊂ v7ε ϑE (a) and likewise ϑE (a) ∈ vε ϑE (b) ⊂ v3ε θnn E (b) ⊂ v7ε θnn E (a) for all n ≥ n0 . This demonstrates that θnn E (a) → ϑE (a) for all a ∈ P. All left to show is that the mapping E → ϑE : R → P ∗ is countably additive, that is ϑ is indeed a P ∗ -valued measure on R. For this, let a ∈ P, and let Ei ∈ R, for i ∈ N, be disjoint sets. Using the additivity of the measures θnn and the limit rules, we have ϑ(∪i0

i=1

(a) = lim Ei )

n→∞

θnn (∪i0

i=1 Ei

i0 i0 ! ! n lim (a) = θ (a) = ϑEi (a). n E i ) n→∞

i=1

i=1

for every i0 ∈ N. This shows ﬁnite additivity in particular. Given ε > 0, it follows from the uniform strong additivity of the measures in Θ together with Lemma 5.31(b) that there is i0 ∈ N such that |θ(∪∞i=i +1 Ei ) (a)| ≤ ε holds 0 (a)| ≤ ε. This yields with the above for all θ ∈ Θ, hence also |ϑ(∪∞ i=i +1 Ei ) 0

i0 ! ∞ ϑEi (a) = ϑ(∪∞ (a) − ϑ(∪i0 ϑ(∪i=1 Ei ) (a) − i=1 Ei ) i=1

i=1

= ϑ(∪∞ i=i

0

(a) Ei )

(a) ≤ ε. E ) +1 i

Because ε > 0 was arbitrarily chosen, this yields ϑ(∪∞ (a) = i=1 Ei )

∞ !

ϑEi (a).

i=1

Summarizing, we have veriﬁed that the subsequence θnn n∈N of θn n∈N converges setwise towards the P ∗ -valued measure ϑ.

In Section 3.9 we introduced the composition of an operator-valued measure θ with two linear operators. We shall now investigate integrals with respect to this type of measures. Let us recall our notations: Let (P, V) and % V) % be full locally convex cones, and let (Q, W) and (Q, % W) & be locally (P, convex complete lattice cones. For an L(P, Q)-valued measure θ, a contin% P) and an order continuous linear operator uous linear operator S ∈ L(P, % % % U ∈ L(Q, Q), the L(P, Q)-valued measure (U ◦ θ ◦ S) was deﬁned as the set function % Q). % E → (U ◦ θE ◦ S) : R → L(P, % % and a linear operator S ∈ L(P, % P) For a P-valued function f ∈ F(X, P) we denote by S ◦ f ∈ F(X, P) the P-valued function x → S f (x) : X → P.

200

II Measures and Integrals. The General Theory

% V) % be full locally convex cones, and let Theorem 5.35. Let (P, V) and (P, % & (Q, W) and (Q, W) be locally convex complete lattice cones. Let Θ be an % be an equibounded family of L(P, Q)-valued measures, and let Υ ⊂ L(Q, Q) equicontinuous family of continuous and order continuous linear operators. % P) such that S is onto and S(V% ) ⊂ V. Let Let S ∈ L(P, % = (U ◦ θ) | U ∈ Υ, θ ∈ Θ} = (θ ◦ S) | θ ∈ Θ} Θ and Θ % and L(P, % Q)-valued composition be the corresponding families of L(P, Q)% is integrable measures on R. Let F ∈ AR . If the function f ∈ F(X, P) over F with respect to Θ, then the function S ◦ f ∈ F(X, P) is integrable % and over F with respect to Θ, ' ' (S ◦ f ) d(U ◦ θ) = U f d(θ ◦ S) F

F

holds for all θ ∈ Θ and U ∈ Υ. % Θ be as stated, and Proof. We may assume that F = X. Let Θ, Υ, S and Θ, n % ˜i ∈ SR (X, P) let θ ∈ Θ and U ∈ Υ. First, for a step function h = i=1 χEi ⊗a we have ' n ! U ◦ θ Ei S(˜ ai ) (S ◦ h) d(U ◦ θ) = X

i=1 n ! U θEi (S(˜ ai )) = i=1

=U

" n !

θ◦S

#

Ei

(˜ ai )

i=1

' =U

h d(θ ◦ S) .

X

% According to Theorem 1.8(c) the Next we consider a function f ∈ FR (X, P). function S ◦ f ∈ FR (X, P) is also measurable. Let v be an inductive limit neighborhood for F(X, P). Then for every E ∈ R there is vE ∈ V such % such that S(˜ vE ) ≤ vE that χE ⊗vE ≤ v. Correspondingly, there is v˜E ∈ V (see 2.2). Hence S ◦ (χE ⊗v˜E ) ≤ χE ⊗vE ≤ v. This shows that the convex % ˜ of all measurable V-valued set v functions s˜ such that S ◦ s˜ ≤ v is a % By 2.3 there is a corresponding inductive limit neighborhood for FR (X, P). % ˜. Then S ◦ h ∈ SR (X, P) step function h ∈ SR (X, P) such that h ≤ f + v and S ◦ h ≤ S ◦ f + v. This shows S ◦ f ∈ FR (X, P). Now let E ∈ R % Given w & and ε > 0 we choose w ∈ W such that and (U ◦ θ) ∈ Θ. ˜∈W U (s) ≤ U (t) + w ˜ whenever s ≤ t + w for s, t ∈ Q. Correspondingly, there % such that S(˜ v ) ≤ v. According is v ∈ V such that θE (v) ≤ w, and v˜ ∈ V ˜ = {χX ⊗v˜} there to Corollary 2.8, given the inductive limit neighborhood v is 1 ≤ γ ≤ 1 + ε and a bounded below sequence (hn )n∈N of step functions

5. The General Convergence Theorems

201

% such that: (i) hn ≤ γf + χX ⊗v˜ for all n ∈ N and (ii) for in SR (X, P) every x ∈ E there is n0 ∈ N such that f (x) ≤ hn (x) for all n ≥ n0 . Thus (i˜) S ◦hn ≤ γ(S ◦f )+χX ⊗v for all n ∈ N and (ii˜) for every x ∈ E there is that (S ◦ f )(x) ≤ (S ◦ hn )(x) for v) = n0 ∈ N such all n ≥ n0 . As (θ ◦ S)E (˜ v ) ≤ θE (v) ≤ w and (U ◦θ)E (v) = U θE (v) ≤ U (w) ≤ w, ˜ this yields θE S(˜ ' ' hn d(θ ◦ S) ≤ γ f d(θ ◦ S) + w E

and

E

'

' (S ◦ hn ) d(U ◦ θ) ≤ γ E

(S ◦ f ) d(U ◦ θ) + w ˜ E

for all n ∈ N, as well as ' ' f d(θ ◦ S) ≤ lim hn d(θ ◦ S) E

and

n→∞

'

E

' (S ◦ f ) d(U ◦ θ) ≤ lim

(S ◦ hn ) d(U ◦ θ)

n→∞

E

E

with Theorem 5.23. Using our observation for order continuous linear operators from I.5.29 and the latter we infer that ' ' f d(θ ◦ S) ≤ U lim hn d(θ ◦ S) U n→∞ E E ' ≤ lim U hn d(θ ◦ S) n→∞ E ' = lim (S ◦ hn ) d(U ◦ θ) n→∞ E ' ≤ γ (S ◦ f ) d(U ◦ θ) + w ˜ E

and

'

' (S ◦ f ) d(U ◦ θ) ≤ lim

(S ◦ hn ) d(U ◦ θ) ' hn d(θ ◦ S) = lim U n→∞ ' E ≤γ U f d(θ ◦ s) + w. ˜ n→∞

E

E

E

& and ε > 0 and therefore demonstrates This holds true for all w ˜∈W ' ' (S ◦ f ) d(U ◦ θ) = U f d(θ ◦ S) E

E

% and θ ∈ Θ and U ∈ Υ. for all f ∈ FR (X, P)

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II Measures and Integrals. The General Theory

Next we observe that any set in AR of measure zero with respect to Θ % Indeed, if (θ ◦ S)E = 0 for a set is also of measure zero with respect to Θ. % As the operator E ∈ R and all θ ∈ Θ, then θE S(˜ a) = 0 for all a ˜ ∈ P. S is supposed to be surjective, this yields θE (a) = 0 for all a ∈ P, hence % θE = 0 and (U ◦ θ)E = 0 for all U ∈ Υ. Now let f ∈ F(X,Θ) (X, P). Let & and ε > 0. Because the family Υ was supposed to be E ∈ R, let w ˜∈W equicontinuous, there is w ∈ W such that U (s) ≤ U (t) + w ˜ holds for all U ∈ Υ whenever s ≤ t + w for s, t ∈ Q. Our deﬁnition in 5.3 of integrability over the set E ∈ R requires that there are with respect to the family Θ % % such that functions f(w,ε) ∈ FR (X, P) and s(w,ε) ∈ FR (X, V) '

≤ f(w,ε) a.e.E ≤ γf + s(w,ε) f a.e.E

s(w,ε) d(θ ◦ S) ≤ εw

and E

for some 1 ≤ γ ≤ 1 + ε and all θ ∈ Θ. Then S ◦ f(w,ε) ∈ FR (X, P) and % ⊂ V. By the above we S ◦ s(w,ε) ∈ FR (X, V) by our assumption that S(V) have ≤ γ(S ◦ f ) + (S ◦ s(w,ε) ) ≤ S ◦ f(w,ε) a.e.E S ◦ f a.e.E and

'

'

s(w,ε) d(θ ◦ S) ≤ εw ˜

(S ◦ s(w,ε) ) d(U ◦ θ) = U E

E

% and θ ∈ Θ and U ∈ Υ. By Deﬁnition 5.3, the function for all f ∈ FR (X, P) % and S ◦ f is therefore also integrable over E with respect to the family Θ, we have '

' (S ◦ f ) d(U ◦ θ) = lim E

(S ◦ f(w,ε) ) d(U ◦ θ)

ε>0

E

w∈W

'

f(w,ε) d(θ ◦ S)

= lim U ε>0

E

w∈W

"

=U

#

'

f(w,ε) d(θ ◦ S)

lim ε>0

w∈W

' =U

E

f d(θ ◦ S)

E

for every θ ∈ Θ and U ∈ Υ. Finally, we verify the second part of Deﬁnition 5.3, that is integrability over F = X. We have

5. The General Convergence Theorems

203

'

' (S ◦ f ) d(U ◦ θ) = lim

(S ◦ f ) d(U ◦ θ) ' f d(θ ◦ S) = lim U E∈R E ' = U lim f d(θ ◦ S) E∈R E ' =U f d(θ ◦ S) E∈R

X

E

X

for all θ ∈ Θ and U ∈ Υ. Thus S ◦ f ∈ F(X,Θ) % (X, P), hence our claim.

%=P We shall in the following mainly use this result for the special case P % = R and an equicontinuous set Υ of and the identity operator for S, for Q order continuous linear functionals in P ∗ . Recall from Section I.5.32 that the order continuous linear functionals are said to support the separation property for a locally convex complete lattice cone (Q, W) if for every neighborhood w ∈ W we have l ≤ m + w for l, m ∈ Q whenever μ(l) ≤ μ(m) + 1 holds for all order continuous lattice homomorphisms μ ∈ w◦ . We are now prepared to formulate and prove a combined version of the Convergence Theorems 5.23, 5.24 and 5.25, that under additional assumptions yields convergence with respect to the upper, lower and symmetric topologies of Q, respectively, for the concerned sequence of integrals. Because we shall deal only with bounded elements of Q, we do not need to consider the relative topologies, since they coincide locally with the given topologies in this case (see Section I.4). Recall that for a sequence (an )n∈N in Q convergence towards a ∈ Q in the upper, or lower topology of Q means that for every w ∈ W there is n0 ∈ N such that an ≤ a + w,

or

a ≤ an + w

holds for all n ≥ n0 , respectively. Because these topologies are generally far from Hausdorﬀ, limits need not be unique. Convergence in the symmetric topology combines convergence in both the upper and lower topologies. Theorem 5.36. Suppose that the order continuous linear functionals support the separation property for Q. Let θ be a strongly additive L(P, Q)-valued measure on R, and let E ∈ R. Let fn , f, f∗∗ , f∗ , f ∗∗ , f ∗ ∈ F(X, P) be bounded-valued measurable functions, and suppose that for every w ∈ W there is v ∈ V such that θE (v) ≤ w and such that these functions are (P, V0 )-based integrable over E with respect to θ for the subsystem V0 = {ρv | ρ > 0} of V. Suppose that the functions f∗ and f ∗ are strongly integrable over E with respect to θ and that their respective integrals are bounded in Q.

204

II Measures and Integrals. The General Theory

≤ fn + f∗ for all n ∈ N, and fn (a) If f∗∗a.e.E a.e.E f, then ' ' f dθ = lim fn dθn n→∞

E

E

with respect to the lower topology of Q. ≤ f ∗ for all n ∈ N, and fn a.e.E f, then (b) If fna.e.E ' ' f dθ = lim fn dθn n→∞

E

E

with respect to the upper topology of Q. ≤ fn + f∗ and fn + f ∗∗a.e.E ≤ f ∗ for all n ∈ N, and fn −→ f, then (c) If f∗∗a.e.E a.e.E ' ' f dθ = lim fn dθn n→∞

E

E

with respect to the symmetric topology of Q. Proof. We shall deal with Parts (a), (b) and (c) simultaneously. By restricting the measure θ and all the functions involved to the set E, we may assume that X = E ∈ R. Let G = {fn , f, f∗∗ , f∗ , f ∗∗ f ∗ } be the family of the functions used in our statement. This family is countable. ( Suppose that contrary to our claim, the sequence E fn dθ n∈N does not ( converge towards E f dθ in the (a) lower, (b) upper or (c) symmetric topology of Q. Then there is w ∈ W and a subsequence (fnk )k∈N of (fn )n∈N such that either ' ' ' ' (a) f dθ ≤ fnk dθ + w or (b) fnk dθ ≤ f dθ + w, E

E

E

E

respectively, holds for all k ∈ N. In case (c), we can ﬁnd ( a subsequence (fnk )k∈N of (fn )n∈N either as in (a) or in (b). We have μ E f dθ < +∞ for all μ ∈ Q∗ since the integral of f is supposed to be bounded in Q. Let Υ be the family of all order continuous linear functionals in w◦ and Ω be the corresponding set {μ ◦ θ | μ ∈ Υ, } of P ∗ -valued measures on R. Theorem 5.35 yields that the functions ( in G are( integrable over E with respect to the family Ω and that μ E g dθ = E g d(μ ◦ θ) holds for all g ∈ G and μ ∈ Υ. By our assumption the order continuous linear functionals support the separation property for Q, thus there are functionals μk ∈ Υ ⊂ w◦ such that either ' ' ' ' f d(μk ◦θ) = μk f dθ > μk fnk dθ +1 = fnk d(μk ◦θ)+1 (a) X

or

X

X

fnk d(μk ◦θ) = μk

(b) X

'

'

fnk dθ X

X

' > μk X

' f dθ +1 = f d(μk ◦θ)+1 X

holds for all k ∈ N, respectively. We shall proceed as follows:

5. The General Convergence Theorems

205

There is v ∈ V such that θE (v) ≤ w and such that the functions in G are (P, V0 )-based integrable over E with respect to θ for the subsystem V0 = {ρv | ρ > 0} of V. All functions in G are supposed to be measurable, thus their ranges are separable with respect to the symmetric relative v-topology by (M2) in Section 1.2. For every g ∈ G, let A(g) be a countable dense subset in the range of g. Recall that by our assumption all elements of A(g) are bounded in P. Following Deﬁnition 5.6, that is the (P, V0 )-based integrability of the functions in G, for every g ∈ G and n ∈ N there is a function gn ∈ FR (X, P) and sn ∈ FR (X, V0 ) such that ' 1 ≤ ≤ gn a.e.E γn g + sn and sn dθ ≤ w g a.e.E gn + sn , n E ( for some 1 ≤ γn ≤ 1 + 1/n. The latter implies that E sn dω ≤ 1/n for all ω ∈ Ω. Again, measurability guarantees that there are countable dense subsets A(gn ) in the respective ranges of the functions gn . Now, recalling the deﬁnition of the cone FR (X, P) in Section 2.3, for every n ∈ N and m ∈ N m there is a step function hm gn ∈ SR (X, P) such that hgn (x) ≤ gn (x) + (1/m)v m m for all x ∈ E. Obviously, the range A(hgn ) of hgn is ﬁnite. We denote by B the union of all the sets A(g), A(gn ) and A(hm gn ), for g ∈ G and n, m ∈ N, and by

n ! C= ρi bi + δv | bi ∈ B, 0 ≤ ρi ∈ Q, 0 < δ ∈ Q . i=1

This set is also countable, and all its elements are v-bounded in P by our assumption on the functions g ∈ G. Finally, let P0 be the closure of C in P with respect to the symmetric relative v-topology. Then P0 is a subcone of P, separable, and all of its elements are v-bounded, that is bounded with respect to the neighborhood subsystem V0 , which itself is contained in P0 . Moreover, the above shows that all functions in G are indeed (P0 , V0 )based integrable (see 5.6) over E, hence over all subsets G ∈ R of E, with respect to the family Ω of P ∗ -valued measures. Proposition 5.10 now yields that all these functions are contained in F(E,Ω0 ) (X, P0 ), where Ω0 denotes the family of the restrictions to P0 of the measures in Ω, and that ' ' ' g d(μ ◦ θ)0 = g d(μ ◦ θ) = μ g dθ G

G

G

holds for all g ∈ G, μ ∈ Υ and subsets G ∈ R of E. We proceed to apply Lemma 5.34 to the cone (P0 , V0 ) in order to show that the family Ω0 of P0∗ -valued measures is weakly sequentially compact. As we mentioned before, the elements of P0 are bounded, and P0 is separable in the symmetric relative v-topology. For equiboundedness of the family Ω0 , let ε > 0 be a neighborhood for R. Correspondingly, we choose the neighborhood εv ∈ V0 and conclude that

206

II Measures and Integrals. The General Theory

(μ ◦ θ)0 E (v) = μ θE (v) ≤ μ(εw) ≤ ε for all (μ ◦ θ)0 ∈ Ω0 , since all functionals μ ∈ Υ involved are contained in w◦ . This shows that Ω0 is indeed equibounded. Likewise, Ω0 is seen to be uniformly strongly additive. Indeed, let En ∈ R such that En ⊃ En+1

and n∈N En = ∅. Following 5.30, given ε > 0, there is n0 ∈ N such that θEn (v) ≤ εw for all n ≥ n0 and θ ∈ Θ. Thus (μ ◦ θ)0 En (v) = (μ ◦ θ)En (v) = μ θEn (v) ≤ μ θEn (v) ≤ ε for all (μ ◦ θ)0 ∈ Ω0 and n ≥ n0 . Thus, following Lemma 5.34, the family Ω0 of P0∗ -valued measures is weakly sequentially compact. We may therefore assume that the sequence (μk ◦ θ)0 k∈N from the ﬁrst part of this proof converges setwise to some bounded P0∗ -valued measure ω. We abbreviate ωk for (μk ◦ θ)0 and recall that either ' ' ' ' (a) f dωk > fnk dωk + 1 or (b) fnk dωk > f dωk + 1 X

X

X

X

holds for all k ∈ N, respectively. Next we shall argue that (ωk ) {fE∗ ,f ∗ }≺ ω. In fact, we shall demonstrate that Rs ωk , E, g = 0 for every g ∈ {f∗ , f ∗ }. For this,

let Em ∈ R for m ∈ N be subsets of E such that Em ⊃ Em+1 and n∈N Em = ∅. Let ε > 0. Because the function g is supposed to be strongly integrable over E with respect yields that for ε > 0 there is m0 ∈ N ( 5.32(b) ( to θ, Lemma such that Em g dθ ≤ O E g dθ + εw for all m ≥ m0 . Because the element ( ( E g dθ is supposed to be bounded in Q, we infer that O E g dθ = 0 see Proposition I.5.10(c) . We have ' ' g dωk = μk g dθ ≤ ε Em

Em

for all m ≥ m0 and k ∈ N, since μk ∈ w◦ . Thus ' g dωk ≤ ε lim lim k→∞

m→∞

for all ε > 0, hence

lim m→∞

Em

' lim

g dωk

k→∞

This shows Rs ωk , E, g = sup (Em )∈F

≤ 0.

Em

lim

m→∞

' lim

k→∞

g dωk Em

= 0.

6. Examples and Special Cases

207

Now, ﬁnally, our preceding convergence theorems will yield a contradiction. We shall apply them to the cones P0 and R, the sequence of measures (ωk )k∈N and ω, and the given functions fn , f, f∗∗ , f∗ , f ∗∗ f ∗ . First, Lemma 5.14(a) states that all functions involved are in F(E,Ω0 ∪{ω}) (X, P0 ). Moreover, Lemmas 5.14(b) and 5.20(b) demonstrate that ' ' f dω = lim f dωk . k→∞

E

E

In case (a), Theorem 5.23 yields ' ' f dω ≤ lim fnk dωk k→∞

F

(

F

since O F f∗ dθ = 0, contradicting our assumption at the start of this argument. Similarly, in case (b), Theorem 5.24 leads to ' ' lim fnk dωk ≤ f dω, k→∞

F

F

contradicting the corresponding assumption for this case. In case (c), ﬁnally, Theorem 5.25 yields ' ' f dω = lim fnk dωk , F

k→∞

F

contradicting the assumptions of both cases (a) and (b). This completes our argument. As we established in I.5.57, every locally convex cone can be canonically embedded into a larger locally convex complete lattice cone whose order continuous lattice homomorphisms support the separation property. The corresponding requirement in Theorem 5.36 can therefore be met if we use this standard lattice completion for Q. In Section 6 below we shall identify several special cases where Theorem 5.36 can be applied.

6. Examples and Special Cases The generality of our approach to measures and integrals allows a wide range of settings, depending on the choices for the locally convex cones (P, V) and (Q, W). We shall present a selection of these special cases in this section. Throughout the following, we shall assume that (P, V) is a quasi-full locally convex cone and that (Q, V) is a locally convex complete lattice cone. (PV , V) shall denote the standard full extension of (P, V) into a full cone, as elaborated in Section 6 of Chapter I. (Q0 , W0 ), on the other hand, will stand for a locally convex cone whose standard lattice completion in the sense

208

II Measures and Integrals. The General Theory

of I.5.57 is (Q, W). We shall generally use the notations of the preceding sections. In particular, R stands for a weak σ-ring of subsets of a set X, and θ is a bounded measure on R. The concepts of the preceding Sections 4 and 5, in particular our notions of integrability, will be applied to the full cone (PV , V) instead of (P, V). Some of our general notions are considerably simpliﬁed in special cases. In the ﬁrst set of examples we shall discuss the speciﬁc insertions for P and Q that lead to classical integration theory. 6.1 The case Q = R. If we choose Q = R with the canonical order and the neighborhoods V = {ε ∈ R | ε > 0}, then the values of the measure θ are linear functionals in the dual cone P ∗ of P, and for each a ∈ P the mapping E → θE (a) : R → R is an extended real-valued measure on R. The modulus of the measure θ is given by

n ! |θ|(E, v) = sup θEi (si ) si ∈ P, si ≤ v, Ei ∈ R disjoint subsets of E , i=1

which is an element of R, for E ∈ R and v ∈ V. Boundedness therefore means that for every E ∈ R there is v ∈ V such that |θ|(E, v) < +∞. This coincides with Prolla’s notion of ﬁnite p-semivariation in [155] (Ch. 5.5). A bounded measure can be extended to the full cone (PV , V) as elaborated in Section 3.8. Integrals of P-valued functions with respect to an L(P, R)-, that is P ∗ -valued measure are also in R. For a meaningful statement ( in our Convergence Theorems 5.22, 5.24 and 5.25 we need to enforce that F f∗ dθ < ( +∞ and F f ∗ dθ < +∞ in this case. 6.2 Extended Positive-Valued Functions and Measures. We obtain classical integration theory for extended positive-valued functions with respect to extended positive-valued measures if we choose P = R+ , endowed with the singleton neighborhood system V = {0} (see Example 1.2(b) in ∗ Chapter I), and Q = R. The dual R+ of R+ consists of all elements of R+ (via the usual multiplication) and the singular functional ¯0 such that ∗ ¯ 0(+∞) = +∞. Every R+ -valued measure θ 0(α) = 0 for all α ∈ R+ and ¯ is therefore R-bounded and can be expressed as the sum of an R+ -valued measure θ1 in the usual sense and a measure θ0 that takes only the values 0 and ¯ 0. Because the symmetric relative topology renders the Euclidean topology on the interval (0, +∞), and the elements 0 and ∞ as isolated points see Example 4.18(a) in Chapter I , R+ is separable in this topology. Our notion of measurability from Section 1 for R+ -valued functions therefore coincides with the usual one in this case. Continuity for an R+ -valued function deﬁned

6. Examples and Special Cases

209

on a topological space X does however require that this function takes the values 0 and +∞ only on respective subsets of X that are both open and closed. Because v = 0 is the only neighborhood for R+ , according to Section 4, the integral of a measurable function f over a set F ∈ AR with respect to a measure θ is deﬁned as ' ' f dθ = sup h dθ h ∈ SR (X, P), h ≤ f , F

F

that is the classical deﬁnition of the integral. 6.3 Extended Real-Valued Functions and Positive-Valued Measures. We obtain classical integration theory for R-valued functions with ∗ respect to positive-valued measures if we choose P = Q = R. The dual R of R consists of all positive reals (via the usual multiplication) and the singular functional ¯ 0 such that ¯ 0(α) = 0 for all α ∈ R and ¯0(+∞) = +∞. ∗ Every R -valued measure θ is therefore R-bounded and can be expressed as the sum of a positive real-valued measure θ1 in the usual sense and a mea0. The notion of measurability sure θ0 that takes only the values 0 and ¯ from Section 1 for R-valued functions coincides with the usual one. Let f be a measurable and bounded below R-valued function, and let F ∈ AR . For a neighborhood w = ε ∈ W the step functions s ∈ vε are invertible, and for a step function h ∈ SR (X, R)( such that( h ≤ f + vε we have h ≤ f with h = h − s ∈ SR (X, R) and F h dθ ≤ F h dθ + ε. This shows ' ' (ε) f dθ ≤ sup h dθ h ∈ SR (X, P), h ≤ f + ε F

F

and consequently ' ' f dθ = sup h dθ h ∈ SR (X, P), h ≤ f , F

F

the usual deﬁnition. 6.4 Real- or Complex-Valued Functions and Measures. In the preceding example we integrated R-valued functions with respect to positive real-valued measures. Alternatively, we may consider real- or complex-valued functions, that is P = K for K = R or K = C with the usual Euclidean topology and the equality as order. The vector space dual PK∗ , of K is of ∗ cone consists of the course K itself, whereas its dual P as a locally convex real parts of these evaluations see Example I.2.1(c) . For Q we choose the of K which consists of all bounded simpliﬁed standard lattice completion K below R-valued functions on Γ, the unit circle of K, endowed with the (strictly) positive constants as neighborhoods see Example I.5.62(f) .

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II Measures and Integrals. The General Theory

We consider K-valued measures E → θK E : R → K in this case, yielding via the convention continuous linear operators θE from K to K θE a (γ) = e(γa θK E ) for E ∈ R, a ∈ K and γ ∈ Γ. According to 3.2 we calculate the modulus of such a measure for every E ∈ R as

n ! |θ| E, B (γ) = sup e(γa θK E ) |ai | ≤ 1, Ei ∈ R disjoint subsets of E i=1

= sup

n !

|θEi | Ei ∈ R disjoint subsets of E

i=1

for all γ ∈ Γ, where B ∈ V stands for the unit ball in K. This is of course the usual notation for the total variation var(θ, E) of a real- or complex-valued measure on a set E (see III.1.4 in [55] or Section 6.1 in [179]). A simple argument (see Lemmas III.1.5 and III.4.5 in [55]) shows that |θ|(E, B) ≤ 4 sup |θG | | G ∈ R, G ⊂ E < +∞ for every E ∈ R in this case. Hence any K-valued measure is R-bounded in the sense of Section 3.6 and may therefore be extended to the standard full extension PV = {a + αB | a ∈ K, α ≥ 0} of P = K, setting θE a + αB (γ) = θE a (γ) + α|θ|(E, B) = e(γa θK E ) + α|θ|(E, B) for all γ ∈ Γ. The notion of measurability from Section 1 for K-valued functions coincides with the usual one. A measurable function f is contained in FR (X, K) if on every set E ∈ R it can be uniformly approximated by a sequence of step functions. It follows from our convergence theorems that the integral of f over E is the limit of the integrals of this sequence of step functions. Integrability in the sense of 4.12 and 4.13, however reaches beyond that is this requirement. Integrals of K-valued functions are evaluated in K, as R-valued functions on Γ. However, since according to Corollary 5.9 these integrals are elements of the order closure of the embedding of K into K, hence are K-linear by I.5.60(b). We may therefore identify the integral in the usual way with a number in K, setting * ' )' f dθ = f dθ (1) F

in the real, and

R

F

6. Examples and Special Cases

211

*

)'

'

f dθ

= C

F

' f dθ (1) − i f dθ (i)

F

F

in the complex case, respectively. Moreover, given γ ∈ Γ we have * )' * )' γh dθ =γ f dθ K

F

F

K

for every step function h ∈ SR (X, K). Because h ≤ f + vw holds if and only if γh ≤ γf + vw for h ∈ SR (X, K) and f ∈ FR (X, K), we have , +' , +' (w)

(w)

γf dθ F

=γ K

f dθ F

. K

Consequently, F(F,θ) (X, K) is a vector space over K , and the mapping * )' f dθ : F(F,θ) (X, K) → K f → F

K

is linear over K. 6.5 The Case that Q Is the Standard Lattice Completion of Some Subcone Q0 . Suppose that (Q, W) is the standard lattice completion of a locally convex cone (Q0 , W0 ) (see I.5.57), and suppose that the measure θ is indeed L(P, Q0 )-valued. The closure of Q0 in Q with respect to the ∗∗ see Secorder topology was seen to be a subcone of the second dual Q 0 tions I.5.60 and I.7.3 in this case, and following Corollary 5.9, integrals of (P, V)-based integrable functions in F(X, P) are therefore elements of Q∗∗ 0. Stronger statements can be obtained for certain types of integrable functions. We shall develop these in the following remarks: Remarks 6.6. Let A be a relatively bounded subset of P, that is, A is bounded below, and bounded above relative to some element a0 ∈ P. Let E ∈ R. We observe the following: (a) The convex hull of A ∪ {0}, that is the set

n n ! ! A˜ = αi ai ai ∈ A, 0 ≤ αi ∈ R, αi ≤ 1 i=1

i=1

is also bounded below and bounded above relative to a0 . Indeed, given v ∈ V let λ, ρ ≥ 0 such that 0 ≤ a0 + λv, 0 ≤ a + λv and a ≤ ρa0 + λv for all a ∈ A. Then for any choice of ai ∈ A and 0 ≤ αi ∈ R such that ni=1 αi ≤ 1 we have n n ! ! 0≤ αi (ai + λv) ≤ αi ai + λv i=1

i=1

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II Measures and Integrals. The General Theory

and n !

αi ai ≤

i=1

n !

n ! αi (ρa0 + λv) + ρ 1 − αi (a0 + λv) ≤ ρa0 + λ(1 + ρ)v.

i=1

i=1

This yields our claim. (b) For every E ∈ R the set

n ! Z(A, E) = θEi (ai ) | ai ∈ A, Ei ∈ R disjoint subsets of E i=1

is bounded below, and bounded above relative to the element θE (a0 ), hence Z(A, E) is a relatively bounded subset of Q0 . Indeed, given w ∈ W there is v ∈ V such that |θ|(E, v) = θE (v) ≤ w. In turn, there are λ, ρ ≥ 0 such that 0 ≤ a + λv and a ≤ ρa0 + λv for all a ∈ A. We may also assume that 0 ≤ ρa0 + λv. Now let a1 , . . . , an ∈ A and let E1 , . . . , En ∈ R be disjoint subsets of E. Then 0≤

n !

θEi (ai + λv) =

i=1

n !

θEi (ai ) + λ

n !

i=1

i=1

θEi (v) ≤

n !

θEi (ai ) + λw

i=1

and n ! i=1

θEi (ai ) ≤

n !

θEi (ρa0 + λv) ≤ θE (ρa0 + λv) ≤ ρθE (a0 ) + λw.

i=1

The set Z(A, E) is therefore bounded below and bounded above relative to the element θE (a0 ), thus relatively bounded in Q0 . (c) Now recall from Section I.5.57 that the order topology of the standard lattice completion Q of Q0 coincides with the topology of pointwise convergence on the elements of Q∗0 . Thus, according to I.7.3 the limit in Q with respect to order convergence of any net in the relatively bounded set Z(A, E) ⊂ Q0 ⊂ Q from (b) is contained in the relative strong second dual (Q0 )∗∗ sr of Q0 . (d) Let E ∈ R, let ϕ1 , . . . , ϕn be non-negative measurable real-valued n functions such that i=1 ϕi ≤ χE and let a1 , . . . , an ∈ A. For each i = i 1, . . . , n let (ψk ⊗ai )k∈N be a sequence of step functions approximating ϕi ⊗ai as in 5.27 and 5.28, that is ' ' 0 ≤ ψki ≤ ϕi and lim ψki ⊗ai dθ = ϕi ⊗ai dθ. k→∞

According to 5.27, these step functions

X

ψki ⊗ai

x

are of the type

6. Examples and Special Cases

213 k !

χE (i,k) ⊗(αj ai ), j

j=1 (i,k)

with disjoint sets Ej ∈ R whose union is E, with 0 ≤ αj ≤ 1 and such k that j=1 αj χE (i,k) ≤ ϕi . For every k ∈ N let j

hk =

n !

n ! k !

ψik ⊗ai =

i=1

χE (i,k) ⊗(αj ai ). j

i=1 j=1

As n ! k !

αj χE (i,k) ≤

n !

j

i=1 j=1

ϕi ≤ χE ,

i=1

the step function hk can be expressed as hk =

p !

χFl ⊗bl ,

l=1

where F1 , . . . , Fp ∈ R are disjoint subsets of E and b1 , . . . , bp are suitable convex combinations of the elements of the relatively bounded set A˜ = A∪{0} see 6.6(a) ; more precisely bl =

n !

βi ai ,

i=1 (i,k)

where βi is the sum of all those αj , for j = 1, . . . , k, such that Fl ⊂ Ej Thus the integral ' p ! hk dθ = θFl (bl ) X

l=1

is contained in the relatively bounded subset

n ! ˜ ˜ θEi (ai ) ai ∈ A, Ei ∈ R disjoint subsets of E Z(A, E) = i=1

of Q0 . We have

' "! n X

i=1

# ϕi ⊗ai

' hk dθ,

dθ = lim

k→∞

X

.

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II Measures and Integrals. The General Theory

n hence according to (c), the integral of the function i=1 ϕi ⊗ai is contained in of Q . The same applies to integrals the relative strong second dual (Q0 )∗∗ 0 sr of this function over sets F ∈ AR , since the functions ϕi may be replaced by the functions χF ϕi in the preceding argument. (e) If for a function f ∈ F(F,θ) (X, P) there ( consisting ( is a net (fj )j∈J n ϕ ⊗ai as in (d) such that f dθ = lim of functions i j∈J i=1( F F fj dθ, then . according to (c), F f dθ is also contained in (Q0 )∗∗ sr We summarize: Proposition 6.7. Let (P, V) and (Q0 , W0 ) be locally convex cones such that (P, V) is quasi-full, and let θ be an L(P, Q0 )-valued measure. Let F ∈ AR . (a) For every (P, V)-based integrable function in f ∈ F(F,θ) (X, P) the inte( gral F f dθ is contained in Q∗∗ 0 , the second dual of Q0 . (b) Let E ∈ R and let A be a relatively bounded subset of P. If for f ∈ F(F,θ) (X, P) there is a net (fj )j∈J consisting of functions n where ϕi are non-negative measurable real-valued( functions i=1 ϕi ⊗ai , n such that i=1 ϕi ≤( χE and ai ∈ A, and such that F f dθ = ( . limj∈J F fj dθ, then F f dθ is contained in (Q0 )∗∗ sr We shall obtain a further strengthening of these observations in some special cases. 6.8 Compact and Weakly Compact Measures. Let θ be an L(P, Q0 )valued measure, where (P, V) is a quasi-full and (Q0 , W0 ) is a locally convex cone such that (Q, W) is its standard lattice completion. Such a measure θ is called compact (or weakly compact) if for every E ∈ R and every relatively bounded subset A of P the subset

n ! θEi (ai ) ai ∈ A, Ei ∈ R disjoint subsets of E Z(A, E) = i=1

in the symmetric relative topology or in the of Q0 is relatively compact weak topology σ(Q0 , Q∗0 ) of Q0 (see I.4.6). Recall from Lemma I.4.7 that the symmetric relative topology is ﬁner than σ(Q0 , Q∗0 ), and from I.5.57 that σ(Q0 , Q∗0 ) is ﬁner than the induced order topology on Q0 which is however still Hausdorﬀ. The latter two topologies coincide, if all elements of Q0 are bounded (see I.5.57). Moreover, σ(Q0 , Q∗0 ) coincides with its own relative topology (see I.4.6). We observe that every subset Z of Q0 which is relatively compact in the symmetric relative topology is also relatively weakly compact. Indeed, the closure Z of Z with w respect to the symmetric relative topology is contained in its closure Z with respect to the weak topology. Z is compact in the former, hence also in the latter topology, thus weakly closed since σ(Q0 , Q∗0 ) is Hausdorﬀ. We w infer that Z = Z , and our claim follows. Every compact measure θ is therefore also weakly compact.

6. Examples and Special Cases

215

For a set E ∈ R and a relatively bounded subset A ∈ P we denote by #

' " n n ! ! ϕi ⊗ai dθ ai ∈ A, 0 ≤ ϕi measurable, ϕi ≤ χE . I(A, E) = X

i=1

i=1

Clearly Z(A, E) ⊂ I(A, E) ⊂ Q. Conversely, we observed in Remark 6.6(d) that I(A, E) is contained in the closure of Z(A, E) with respect to the order topology of Q. If the measure θ is compact (or a weakly compact), then w the (weak) closure Z(A, E) of Z(A, E) is weakly compact and therefore also compact in the coarser induced order topology, and indeed closed in Q as the order topology is Hausdorﬀ in this case. This demonstrates that w

I(A, E) ⊂ Z(A, E) ⊂ Q0 in this case. Consequently the set I(A, E) is also (weakly) compact in Q0 . We summarize: Proposition 6.9. Let (P, V) and (Q0 , W0 ) be locally convex cones such that (P, V) is quasi-full. An L(P, Q0 )-valued measure θ is compact (or weakly compact), if and only if for every E ∈ R and for every relatively bounded subset A of P, #

' " n n ! ! ϕi ⊗ai dθ ai ∈ A, 0 ≤ ϕi measurable, ϕi ≤ χE X

i=1

i=1

is a relatively compact (or relatively weakly compact) subset of Q0 . Corollary 6.10. Let (P, V) and (Q0 , W0 ) be locally convex cones such that (P, V) is quasi-full and let θ be an L(P, Q0 )-valued relatively compact measure. Let E ∈ R, F ∈ AR and let A be a relatively bounded subset of P. n If for f ∈ F(F,θ) (X, P) there is a net (fj )j∈J consisting of functions ϕi ⊗ai , where ϕi are non-negative measurable(real-valued functions i=1 ( such that ( ni=1 ϕi ≤ χE and ai ∈ A, and such that F f dθ = limj∈J F fj dθ, then F f dθ is contained in Q0 . The following consequence of Theorem 3.15 yields that in certain special circumstances every bounded measure is weakly compact. Proposition 6.11. Suppose that (P, ) is a ﬁnite dimensional normed space and that (Q, W) is the standard lattice completion of a Banach space (Q0 , ). Then every bounded L(P, Q0 )-valued measure is weakly compact. Proof. Let (P, ) and (Q0 , ) be as stated and let θ be a bounded L(P, Q0 )-valued measure on R. We consider both P and Q0 as normed spaces over R. Given a basis {b1 , . . . , bm } of P, there is a constant ρ > 0 such that

216

II Measures and Integrals. The General Theory

-m -! max |βk | βk bk - ≥ ρ k=1,...,m k=1 for every choice of scalars β , . . . , β ∈ R see for example Lemma 2.4.1 1 m in [107] . Now let E ∈ R and let A be a bounded subset of P. According to the above then there exists λ > 0 such that

m ! βk bk βk ∈ R, |βk | ≤ λ . A ⊂ k=1

We ﬁx 1 ≤ k ≤ m. Theorem 3.15 yields that the set Zk = {θG (bk ) | G ∈ R, G ⊂ E} is relatively compact in Q0 with respect to the weak topology σ(Q0 , Q∗0 ). Now let Ei ∈ R, for i = 1, . . . , n, be disjoint subsets of E and in a ﬁrst step let 0 ≤ βk1 ≤ βk2 . . . ≤ βkn ≤ 1. Set F1 = ni=1 Ei , F2 = ni=2 Ei , and so on, and Fn = En . Then n !

βki θEi (bk ) = βk1 θF1 (bk ) +

i=1

n !

(βki − βki−1 )θFi (bk ).

i=2

n

i &k The element hull Z i=1 βk θEi (bk ) is therefore contained in the convex of the set Zk . Following a well-known theorem due to Krein see Theorem IV.11.4 in [185] this convex hull is again relatively weakly compact in Q0 . &k . Using this, we infer that indeed for every choice So, obviously is the set −Z of βki ∈ R such that |βki | ≤ 1 for all i = 1, . . . , n the element ni=1 βki θEi (bk ) &k + (−Z &k ). is contained in relatively weakly compact set Yk = Z m i Thus for every choice of elements ai = k=1 βk bk ∈ A and disjoint subsets Ei ∈ R of E we have |βki | ≤ λ for all i = 1, . . . , n and k = 1, . . . , m, hence "m # n m ! n ! ! ! i θEi (ai ) = βk θEi (bk ) ∈ λ Yk . i=1

k=1 i=1

k=1

As a ﬁnite sum of relatively weakly compact sets see I.V.2 in [185] , the set on the right-hand side is also relatively weakly compact in Q0 , and our claim follows. 6.12 The Case that P Is a Locally Convex Vector Space. Let (P, V) be a locally convex topological vector space over K = R or K = C, endowed with a basis V of balanced convex neighborhoods, that is subsets of P. Equality is the order on P, and involving the neighborhoods we have a ≤ b + v if a − b ∈ v for a, b ∈ P and v ∈ V. As a locally convex cone (P, V) is of course quasi-full (see I.6.1). The modulus of an L(P, Q)-valued measure θ is given by

6. Examples and Special Cases

|θ|(E, v) = sup

n !

217

θEi (si ) si ∈ v, Ei ∈ R disjoint subsets of E

∈ Q.

i=1

According to Lemma 2.5, the cone FR (X, P) as introduced in 2.3 consists of those P-valued functions that vanish outside some set E ∈ R and may be uniformly approximated on X by step functions; more precisely: for f ∈ FR (X, P) there is E ∈ R such that f (x) = 0 for all x ∈ X \ E and for every v ∈ V there exists a step function h ∈ SR (X, P) such that h(x) − f (x) ∈ v for all x ∈ X. Any such function f is measurable by Theorem 1.7. Consequently, the functions in FR (X, P) are uniformly bounded on all sets in R. We have αf ∈ FR (X, P) whenever f ∈ FR (X, P) and α ∈ K. Every measurable neighborhood-valued function s ∈ F(X, PV ) is however contained FR (X, PV ), since its values are positive. For a positive real-valued measurable function ϕ and a neighborhood v ∈ V, for example, the function ϕ⊗v is measurable, hence in F(X, PV ). Recall that V-valued measurable functions are integrated using the canonical extension of the measure θ to the full cone (PV , V) as elaborated in Section 3.8. According to 4.12, a P-valued function f is integrable over a set E ∈ R if for every w ∈ W and ε > 0 there are functions f(w,ε) ∈ FR (X, PV ) and s(w,ε) ∈ FR (X, V) such that ≤ f(w,ε) a.e.E ≤ γf + s(w,ε) f a.e.E ( and E s(w,ε) dθ ≤ εw for some 1 ≤ γ ≤ 1 + ε. A straightforward argument involving the uniform boundedness of the functions in FR (X, P) leads to a slight simpliﬁcation in this case, avoiding the relative topologies: A function f ∈ F(X, PV ) is integrable over a set E ∈ R if for every w ∈ W there are functions fw ∈ FR (X, PV ) and sw ∈ FR (X, V) such that ' ≤ fw a.e.E ≤ f + sw (I) f a.e.E and sw dθ ≤ w. E

The function αf is integrable over E for any α ∈ K, whenever f is. A function f ∈ F(X, P) is (P, V)-based integrable over E ∈ R (see 5.6) if there are fw ∈ FR (X, P) and sw ∈ FR (X, V) such that ' ≤ ≤ f a.e.E fw + sw , fw a.e.E f + sw and sw dθ ≤ w. E

Considering that the functions in FR (X, P) can be approximated by step functions, this is equivalent to the following condition for integrability which is only slightly stronger than (I): For every w ∈ W there is a step function hw ∈ SR (X, P) and sw ∈ FR (X, V) such that ' (BI 1) f (x) − hw (x) ∈ sw (x) a.e. on E and sw dθ ≤ w. E

218

II Measures and Integrals. The General Theory

(Set fw = hw + sw ∈ FR (X, PV ) in order to satisfy (I).) Condition (BI 1) yields indeed strong integrability in the meaning of Section 5.18, since it obviously implies that f ≤ hw + sw , hence ' ' ' ' f dθ ≤ hw dθ + sw dθ ≤ hw dθ + w G

G

G

G

for all subsets G ∈ R of E. Somewhat stronger than (BI 1) is the following suﬃcient integrability condition: For v ∈ V let v denote the corresponding seminorm on P, that is av = inf{λ ≥ 0 | a ∈ λv}. We require that for every v ∈ V and w ∈ W there is a step function h(v,w) ∈ SR (X, P) such that the positive real-valued function x → f (x) − h(v,w) (x)v is measurable and ' f − h(v,w) v ⊗ v dθ ≤ w. (BI 2) E

Condition (BI 2) obviously implies (BI 1) since, given w ∈ W we choose any v ∈ V and set sw (x) = f (x) − h(v,w) (x) v. Then obviously f (x) − h(v,w) (x) ∈ sw (x) holds for all x ∈ E, hence (BI 1). Moreover, a function f ∈ F(X, P) satisfying (BI 2) is (P, V0 )-based integrable over E with respect to θ for every one-dimensional neighborhood subsystem V0 = {ρv0 | ρ > 0}, for v0 ∈ V. This is one of the requirements in Theorem 5.36. In the special case that (P, V) is a normed space, that is V = {ρB | ρ > 0}, where B is the unit ball in P, condition (BI 2) leads to the well-known notion of Bochner (or Dunford and Schwartz) integrability (see for example III.2.17 in [55] or II.2 in [43]). This will be further elaborated in Section 6.18 below. In all of the above cases, integrability is then extended to sets F ∈ AR as in 4.13. Convergence for sequences of P-valued functions as required in Theorems 5.23 to 5.25 and 5.34 refers to pointwise convergence with respect to the vector space topology of P. If the measure θ is strongly additive and if the order continuous linear functionals on the locally convex complete lattice cone (Q, W) support the separation property (see I.5.32), then the strong convergence statements of Theorem 5.36 apply to functions satisfying (BI 2). We already observed that the functions which are integrable over a set E ∈ R with respect to any of the above criteria form also a vector space over K in this case. Now suppose in addition to the above that (Q, W) is the standard lattice completion of some subcone (Q0 , W0 ) and the measure θ is L(P, Q0 )-valued (see 6.5). Then, according to Theorem 3.11 countable additivity for θ refers to the strong operator topology of L(P, Q0 ). Moreover, following Proposition 6.7(a), integrals of (P, V)-based integrable functions in F(X, P) are elements of the second dual Q∗∗ 0 of Q0 . We shall make a few supplementary observations for the case that (Q0 , W0 ) is indeed a vector space over K = R or K = C : (i) If (Q0 , W0 ) is a locally convex topological vector space over K, then the (P, V)-based integrable functions in F(X, P) form a vector space

6. Examples and Special Cases

219

F(F,θ,BI1) (X, P) over K. The integrals of functions in F(F,θ,BI1) (X, P) for in the order closure of Q0 in Q, hence are K-linear F ∈ AR are contained see I.5.60(b) and therefore elements of the second vector space dual Q∗∗ 0K of Q0 . (ii) If the locally convex space Q0 is indeed topologically complete, and if a function f ∈ F(X, P) ( fulﬁlls the integrability criterion (BI1), then for in the symmetevery E ∈ R its integral E f dθ in Q may ( be approximated h dθ of integrals over step ric (modular) topology of Q by a net E i i∈I functions. Integrals over step functions are however contained in the com( plete subspace Q0 of Q. The Cauchy sequence E hn dθ n∈N is therefore ( convergent in Q0 and its limit, that is E f dθ is also contained in Q0 . (iii) If the locally convex space Q0 is reﬂexive, then every bounded L(P, Q0 )-valued measure θ is seen to be weakly compact. Indeed, for every E ∈ R and every bounded subset A of P the set

n ! Z(A, E) = θEi (ai ) ai ∈ A, Ei ∈ R disjoint subsets of E i=1

from 6.8 is bounded in Q0 see Remark 6.8(a) , hence relatively weakly compact, since this holds for all bounded subsets in reﬂexive spaces. (iv) If both P and Q0 are locally convex topological vector spaces over K, then we denote by LK (P, Q0 ) the space of all continuous K-linear operators from P into Q0 . If the measure θ is indeed LK (P, Q0 )-valued, then for every F ∈ AR the operator ' f dθ : F(F,θ,BI1) (X, P) → Q∗∗ f → 0K F

is also linear over K. According to I.5.60(d) we need to verify two conditions for this. The ﬁrst one is obvious, because the additivity of the operator is given. Likewise, the second condition in I.5.60(d) is evident for all α ≥ 0. Thus all left to verify is that ' ' γf dθ (μ) = f dθ (γμ). F

F

holds for all f ∈ F(F,θ,BI1) (X, P), μ ∈ Q∗0 and γ ∈ Γ, the unit circle in K. Indeed, this obviously holds true for every step function h ∈ SR (X, P). Because the neighborhoods in V and in W are supposed to be balanced, h ≤ f + vw holds for h ∈ SR (X, P) and f ∈ FR (X, P) if and only if γh ≤ γf + vw . Therefore and because the lattice operations are taken pointwise in Q, we infer that "' # "' # (w) (w) γf dθ (μ) = f dθ (γμ). F

F

Now Deﬁnition 4.13 yields our claim. We shall formulate this special case as a separate Proposition:

220

II Measures and Integrals. The General Theory

Proposition 6.13. Let (P, V) and (Q0 , W0 ) be locally convex topological vector spaces over K = R or K = C and let θ be a bounded LK (P, Q0 )-valued measure. Then the functions in F(X, P) satisfying (BI1) form a vector space F(F,θ,BI1) (X, P) over K, their integrals are contained in the second vector space dual Q∗∗ 0 K of Q0 , and the operator ' f → f dθ : F(F,θ,BI1) (X, P) → Q∗∗ 0K F

is linear over K. 6.14 Algebra Homomorphisms. Let us consider a special case of 6.13. Suppose that the locally convex vector spaces (P, V) and (Q0 , W) are indeed topological algebras over K, and that (Q, W) is the standard lattice completion of Q0 . A topological algebra P is an algebra and a locally convex topological vector space such that for a ﬁxed element a ∈ P (or b ∈ P) the linear operator c → ac (or c → cb) from P into P is continuous (see for example 8.1 in [137]). Recall that for a linear operator continuity implies weak continuity. Thus P is also a topological algebra in its weak topology. Indeed, for a ﬁxed a ∈ P and μ ∈ P ∗ , the mapping c → μ(ac) : P → R is a continuous linear functional. Thus, if the net (ci )i∈I in P converges weakly to c ∈ P, then μ(aci )i∈I converges to μ(ac) in R. The net (aci )i∈I therefore converges weakly to ac ∈ P. Now suppose that θ is an R-bounded measure such that its values θE for all E ∈ R of are continuous K-linear operators from P to Q0 satisfying the following condition: (A) θE (a) θE (b) = θE (ab) and disjoint sets E, G ∈ R.

θE (a) θG (b) = 0 for all a, b ∈ P and

Both requirements in Condition (A) may be reformulated and combined as (A’) θE (a) θG (b) = θ(E∩G) (ab)

for all

E, G ∈ R

and

a, b ∈ P.

Indeed, (A’) implies (A), and if (A) holds, then for a, b ∈ P and E, G ∈ R we have θE (a) θG (b) = θ(E\G) (a) + θ(E∩G) (a) θ(G\E) (b) + θ(E∩G) (b) = θ(E∩G) (ab), hence (A’). Endowed with the canonical, that is pointwise multiplication, the P-valued step functions form an algebra, and we obtain '

' (hl) dθ = X

' h dθ

X

l dθ

X

for all h, l ∈ SR (X, P) as an immediate consequence of (A). Indeed, nthe func tions h and l can be expressed as h = ni=1 χEi ⊗ai and l = i=1 χEi ⊗bi with disjoint sets Ei ∈ R and elements ai , bi ∈ P. Then hl = ni=1 χEi ⊗ai bi and

6. Examples and Special Cases n !

221

" θEi (ai bi ) =

i=1

#" n # n ! ! θEi (ai ) θEi (bi ) , i=1

i=1

that is our claim. Now let us denote by ER (X, P) the vector subspace of F(X, P) generated by all elementary functions. Recall that elementary functions are of the type ϕ⊗a, where ϕ is a bounded non-negative measurable real-valued function supported by a set in R, and a is an element of P. Obviously, ER (X, P) forms also an algebra, as the product of two elementary functions ϕ⊗a and ψ ⊗b is the elementary function (ϕψ)⊗(ab). We would like to establish that the integral deﬁnes a multiplicative operator on ER (X, P) as well. However, because integrals of these functions are generally contained in the strong second dual Q∗∗ 0 of Q0 rather than in Q0 itself, we shall a introduce a continuation of the multiplication to Q∗∗ 0 ⊂ Q in the following way: For elements l, m ∈ Q we denote by l • m the set of all elements q ∈ Q for which we can ﬁnd nets (li )i∈I and (mj )j∈J in Q0 ⊂ Q such that limi∈I li = l, limj∈J mj = m and lim lim li mj = lim lim li mj = lim lim li mj = lim lim li mj = q. i∈I j∈J

i∈I j∈J

j∈J i∈I

j∈J i∈I

Our introductory remark shows that for elements l, m ∈ Q0 we have l • m = {lm}, since on Q0 ⊂ Q weak and order convergence coincide (see I.5.57). In general, the set l • m may be empty or contain more than one element of Q. However, if q ∈ l • m and if μ ∈ Q∗ is a multiplicative linear functional, then q(μ) = lim lim li mj (μ) = lim lim li (μ) mj (μ) = l(μ) m(μ). i∈I j∈J

i∈I j∈J

Now let f = ϕ⊗a and g = ψ ⊗b be two elementary functions. Their product f g is the elementary function (ϕψ)⊗(ab). Let (ϕn )n∈N and (ψn )n∈N be the sequences of real-valued step functions converging to ϕ and ψ as in 5.27 and 5.28. Thus ' ' ' ' ϕn ⊗a dθ = f dθ and lim ψn ⊗b dθ = g dθ lim n→∞

X

n→∞

X

X

X

by 5.28. For every ﬁxed m ∈ N the sequence (ϕm ψn )n∈N converges pointwise to the function ϕm ψ, and we have 0 ≤ ϕm ψn ≤ ϕm ψ for all n ∈ N. This shows ' ' ' ϕm ⊗a ψn ⊗b dθ = lim (ϕm ψn )⊗(ab) dθ = (ϕm ψ)⊗(ab) dθ

' lim

n→∞

X

X

n→∞

X

X

by Corollary 5.28 and the above. Furthermore, the sequence (ϕm ψ)m∈N converges pointwise to the function ϕψ, and we have 0 ≤ ϕm ψ ≤ ϕψ for all

222

II Measures and Integrals. The General Theory

m ∈ N. Again using 5.28, this yields ' ' ' lim (ϕm ψ)⊗(ab) dθ = (ϕψ)⊗(ab) dθ = (f g) dθ, m→∞

X

X

'

hence lim lim

m→∞ n→∞

lim lim

n→∞ m→∞

Thus indeed

' ' ϕm ⊗ a ψn ⊗b dθ = (f g) dθ.

X

Similarly, one veriﬁes

'

X

X

X

' ' ϕm ⊗ a ψn ⊗b dθ = (f g) dθ.

X

X

'

'

'

(f g) dθ ∈ X

X

f dθ

g dθ .

•

X

X

0 0 fi and g = kk=1 fk with eleFinally, let f, g ∈ ER (X, P), that is f = ii=1 mentary functions fi , gk . For each of these functions there are approximating sequences (hin )n∈N and (ekn )n∈N of step functions as in the preceding step 0 i 0 k of our argument. We set hn = ii=1 fn and en = kk=1 en . The sequences ' ' hn dθ and en dθ X

X

n∈N

(

n∈N

(

in Q0 then converge to X f dθ and X g dθ, respectively. For all n, m ∈ N we have ' ! ' k0 ' io ! hm dθ en dθ = (him ekn ) dθ, X

X

and for ﬁxed i and k

'

' him ekn

lim lim

n→∞ n→∞

X

i=1 k=1

X

dθ =

(fi gk ) dθ X

by the above. This yields

' ' ! ' io ! k0 ' hm dθ en dθ = (fi gk ) dθ = (f g) dθ. lim lim m→∞ n→∞

X

X

i=1 k=1

X

X

Reversing the parts of n and m leads to the same result. Thus indeed ' ' ' • (f g) dθ ∈ f dθ g dθ X

X

X

holds for all f, g ∈ ER (X, P), provided that the measure θ satisﬁes (A).

6. Examples and Special Cases

223

If both (P, V) and (Q0 , W) are topological algebras with an involution, that is a continuous operator a → a∗ such that (a + b)∗ = a∗ + b∗ , (αa)∗ = α ¯ a∗ , (a∗ )∗ = a and (ab)∗ = b∗ a∗ for a, b in P or in Q0 , respectively, and if the L(P, Q0 )-valued measure θ satisﬁes ∗ (A*) θE (a∗ ) = θE (a) for all E ∈ R and a ∈ P in addition to (A), then a similar property can be derived for the integrals of functions in ER (X, P). Analogously to the above extension of the multiplication in Q0 , for an elements l ∈ Q we denote by l the set of all elements q ∈ Q for which we can ﬁnd a net (li )i∈I in Q0 ⊂ Q such that limi∈I li = l and lim li∗ = q. i∈I

The continuity of the involution in Q0 shows that for l ∈ Q0 we have l = {l∗ }. Otherwise, the set l may be empty or contain more than one element of Q. Canonically, for f ∈ F(X, P) we denote by f ∗ ∈ a function ∗ F(X, P) the function x → f (x) . Then an argument similar to that for the multiplication yields '

'

∗

f dθ ∈ X

f dθ X

for all f ∈ ER (X, P) and every L(P, Q0 )-valued measure θ which satisﬁes (A) and (A*). Because both χF ⊗f, χ(F ⊗g ∈ ER((X, P) whenever f, g ∈ ER (X, P) and F ∈ AR , and because F f dθ = X χF ⊗f dθ, the above properties apply also to integrals over measurable subsets F of X. We formulate this as a further proposition: Proposition 6.15. Let (P, V) and (Q0 , W0 ) be topological algebras over K = R or K = C and let θ be a bounded LK (P, Q0 )-valued measure such that θE (a) θG (b) = θ(E∩G) (ab) holds for all E, G ∈ R and a, b ∈ P. Then '

' (f g) dθ ∈ X

' f dθ

X

g dθ

•

X

holds for all f, g ∈ ER (X, P). If both (P, V) and (Q0 , W) are topological ∗ algebras with an involution a → a∗ and if θ satisﬁes θE (a∗ ) = θE (a) for all E ∈ R and a ∈ P, then ' ' ∗ f dθ ∈ f dθ X

holds for all f ∈ ER (X, P).

X

224

II Measures and Integrals. The General Theory

The case that Q0 = K. If Q0 = K, that is if the values θE of the measure θ are K-linear functionals on the algebra P, then Condition (A) means that all functionals θE are multiplicative and that for disjoint sets E, G ∈ R we have either θE = 0 or θG = 0. Thus θ takes at most one nonzero value, that is some multiplicative K-linear functional in P ∗ . In special cases (see also Section 4.7 in Chapter III below) we infer that θ is indeed some point evaluation measure. Condition (A*) for an algebra with involution means that θE (a∗ ) = θE (a) holds for all E ∈ R and a ∈ P. 6.16 Lattice Homomorphisms. In Section 5.1 of Chapter I we deﬁned a locally convex ∨-semilattice cone to be a locally convex cone (P, V) with the following properties: The order in P is antisymmetric, for any two elements a, b ∈ P their supremum a ∨ b exists in P and (∨1) (a + c) ∨ (b + c) = a ∨ b + c holds for all a, b, c ∈ P. (∨2) a ≤ c + v and b ≤ c + w for a, b, c ∈ P and v, w ∈ V implies that a ∨ b ≤ c + (v + w). In case that the locally convex cone (P, V) is quasi-full, (∨2) may be replaced by the somewhat simpler condition (∨2 ) a ≤ v for a ∈ P and v ∈ V implies that a ∨ 0 ≤ v. Indeed, suppose that (∨1) and (∨2 ) hold in a quasi-full cone (P, V), and that a ≤ c + v and b ≤ c + w for a, b, c ∈ P and v, w ∈ V. Then a ≤ c + s and b ≤ c+t for some elements s ≤ v and t ≤ w by (QF1) in I.6.1. By (∨2 ) we have s ∨ 0 ≤ v and t ∨ 0 ≤ w as well. Now a ≤ c + s ∨ 0 + t ∨ 0 and b ≤ c + s ∨ 0 + t ∨ 0 implies a ∨ b ≤ c + s ∨ 0 + t ∨ 0 ≤ c + (v + w) as required in (∨2). Recall from Proposition I.5.2 that in a locally convex ∨-semilattice cone the lattice operation, that is the mapping (a, b) → a ∨ b : P × P → P is continuous with respect to the symmetric relative topology. Topological vector lattices and locally convex complete lattice cones in the sense of I.5 are locally convex ∨-semilattice cones. Further speciﬁc examples include R and R+ Examples I.1.4(a) and (b) and cones of non-empty convex subsets of a topological vector space with the set-inclusion as order Example I.1.4(c) . The supremum of two convex sets is their convex hull in this case while inﬁma do not always exist. In the following let us suppose that (P, V) is a quasi-full locally convex ∨-semilattice cone and that θ is an R-bounded L(P, Q)-valued measure whose values θE for all E ∈ R of are continuous linear operators from P to Q satisfying the following condition: and θE (a) ∨ θG (b) = θE (a) + θG (b) (L) θE (a) ∨ θE (b) = θE (a ∨ b) for all a, b ≥ 0 in P and disjoint sets E, G ∈ R. We shall verify below that (L) implies that its ﬁrst requirement, that is to say

6. Examples and Special Cases

225

θE (a) ∨ θE (b) = θE (a ∨ b), holds indeed for all, not only the positive elements of P. First we observe that Condition (L) implies θE (a) ∧ θG (b) ≤ O (θE (a) ∨ θG (b))

(i)

for disjoint sets E, G ∈ R and 0 ≤ a, b ∈ P, as well as (ii)

sup θEi (ai ) =

i=1,...,n

n !

θEi (ai )

i=1

for disjoint sets Ei ∈ R and ai ≥ 0 in P. For (i), let E, G ∈ R be disjoint and 0 ≤ a, b ∈ P. Then θE (a) ∨ θG (b) = θE (a) + θG (b) = θE (a) ∨ θG (b) + θE (a) ∧ θG (b) by (L) and Proposition I.5.3. This yields our claim via the cancellation rule in I.5.10(a). We shall prove (ii) by induction: For n = 1 there is nothing to prove. Suppose our claim holds for n ∈ N, and let E1 , . . . , En+1 ∈ R be disjoint sets, and 0 ≤ a1 , . . . , an+1 ∈ P. The inequality sup θEi (ai ) ≤

i=1,...,n+1

n+1 !

θEi (ai )

i=1

is obvious. For the converse, using Proposition I.5.3 we infer n+1 !

θEi (ai ) = sup θEi (ai ) + θEn+1 (an+1 ) i=1,...,n

i=1

=

sup θEi (ai ) + sup θEi (ai ) ∧ θEn+1 (an+1 ).

i=1,...,n+1

i=1,...,n

We have sup θEi (ai ) ∧ θEn+1 (an+1 )

i=1,...,n

≤ sup θEi (ai ) ∧ θEn+1 (an+1 ) + O i=1,...,n

by Proposition I.5.15(b), and for each i = 1, . . . , n θEi (ai ) ∧ θEn+1 (an+1 ) ≤ O (θEi (ai ) ∨ θEn+1 (an+1 )) ≤ O by (i). Thus Propositions I.5.10(c) and I.5.11 yield n+1 ! i=1

θEi (ai ) ≤

sup θEi (ai )

i=1,...,n+1

sup θEi (ai )

i=1,...,n+1

sup θEi (ai )

i=1,...,n+1

226

II Measures and Integrals. The General Theory

as claimed. Next we shall verify that " n # " n # n ! ! ! (iii) θEi (ai ) ∨ θEi (bi ) = θEi (ai ∨ bi ) i=1

i=1

i=1

holds for disjoint sets Ei ∈ R and ai , bi ∈ P. Indeed, let E = ni=1 Ei . Given w ∈ W there is v ∈ V such that θE (v) ≤ w and λ ≥ 0 such that 0 ≤ ai + λv and 0 ≤ bi + λv for all i = 1, . . . , n. Because the locally convex cone P is supposed to be quasi-full, there are si , ti ∈ P such that si , ti ≤ λv and 0 ≤ ai + si and 0 ≤ bi + ti . Then si ∨ 0, ti ∨ 0 ≤ λv by our assumptions for a semi lattice cone. We set s = ni=1 (si ∨ 0) + (ti ∨ 0) and conclude that 0 ≤ s ≤ nλ v as well as 0 ≤ ai + s and 0 ≤ bi + s for all i = 1, . . . , n. Using this and (ii) from above, we conclude that # " n # " n ! ! θEi (ai ) ∨ θEi (bi ) + θE (s) i=1

i=1

" =

n !

# θEi (ai + s)

" ∨

i=1

n !

# θEi (bi + s)

i=1

= sup θEi (ai + s) ∨ sup θEi (bi + s) i=1,...,n

i=1,...,n

= sup θEi (ai + s) ∨ θEi (bi + s) i=1,...,n

= sup θEi (ai + s) ∨ (bi + s) =

i=1,...,n n !

θEi (ai + s) ∨ (bi + s)

i=1

=

n !

θEi (ai ∨ bi ) + θE (s).

i=1

Considering that O θE (s) ≤ w and that w ∈ W was arbitrarily chosen, now the cancellation law from Proposition I.5.10(a) yields (iii). Note that (iii) implies a strengthening of the ﬁrst requirement in (L): θE (a)∨θE (b)=θE (a∨b) holds for all E ∈ R and all (not necessarily positive) elements a, b ∈ P. The supremum f ∨ g ∈ F(X, P) of two functions f, g ∈ F(X, P) is canonically deﬁned as the mapping x → f (x) ∨ g(x). If we take into account the continuity of the lattice operation in P, then Theorem 1.4 yields immediately that the supremum of two measurable functions is again measurable, and consequently, a brief review of 2.3 conﬁrms that the subcone FR (X, P) of F(X, P) is closed for suprema. As an immediate consequence of (iii) we ' ' ' infer that (h ∨ l) dθ = h dθ ∨ l dθ X

X

X

σ holds for all step functions h, l ∈ SR (X, P). Now let us denote by SR (X, P) the subcone of all functions f ∈ FR (X, P) for which there exists a sequence

6. Examples and Special Cases

227

(hn )n∈N of step functions that is bounded below and bounded above relative to f( and such (that hn −→ f. According to Corollary 5.26, this implies lim X hn dθ = X f dθ. Lemma 5.27 and Corollary 5.28 yield in particular n→∞

that ER (X, P), the subcone generated by all elementary functions, is conσ (X, P). We proceed to establish that the integral with respect to tained in SR a measure satisfying (L) deﬁnes a ∨-semilattice homomorphism (see I.5.30) σ (X, P) into Q : from SR σ (X, P), and let (hn )n∈B and (ln )n∈B be the corresponding Let f, g ∈ SR sequences of step functions approaching f and g as required above. Because of the continuity (with respect to the symmetric relative topology) of the lattice operation in P, this implies hn ∨ ln −→ f ∨ g, that is the sequence (hn ∨ ln )n∈N of step functions converges pointwise to the function f ∨ g ∈ F(X, P). We shall proceed to verify that this sequence is bounded below and bounded above relative to f ∨ g, hence the function f ∨ g is also contained σ (X, P). Indeed, let v be an inductive limit neighborhood for F(X, P). in SR There are λ, ρ, σ ≥ 0 such that all of the following hold true: 0 ≤ f + λv, 0 ≤ g + λv see Lemma 2.4(a) , as well as 0 ≤ hn + λv, 0 ≤ ln + λv, hn ≤ ρf + λv and ln ≤ σg + λv for all n ∈ N. We may indeed assume that σ = ρ, since otherwise, for example if σ < ρ, we can suitably adjust ln ≤ (σg + λv) + (ρ − σ)(g + λv) = ρg + λ v. Using this, we argue as follows: Firstly, the preceding conditions imply that 0 ≤ hn ∨ ln + λv holds for all n ∈ N. Secondly, there are V-valued functions sn , tn ∈ v such that hn ≤ ρf + λsn and ln ≤ ρg + λtn . Let x ∈ X. Because P is quasi-full, there are 0 ≤ un , vn ∈ P such that un ≤ sn (x), vn ≤ tn (x) and hn (x)≤ ρf (x) + λun and ln (x) ≤ ρg(x) + λvn . Thus both hn (x), ln (x) ≤ ρ f ∨ g (x) + λ(un + vn ) and therefore hn ∨ ln (x) ≤ ρ f ∨ g (x) + λ(un + vn ) ≤ ρ f ∨ g (x) + λ sn + tn (x). This shows hn ∨ ln ≤ ρ(f ∨ g) + 2λv for all n ∈ N and veriﬁes our claim. We therefore have ' ' ' ' hn dθ = f dθ, lim ln dθ = g dθ lim n→∞

X

n→∞

X

hn ∨ ln dθ =

lim

n→∞

X

'

'

and

X

X

f ∨ g dθ X

by Corollary 5.26. As ' ' ' hn ∨ ln dθ = lim hn dθ ∨ ln dθ lim n→∞ X n→∞ X ' X ' = lim hn dθ ∨ lim ln dθ n→∞

X

n→∞

X

228

II Measures and Integrals. The General Theory

by the above and by Proposition I.5.25(a), we conclude that ' ' ' (f ∨ g) dθ = f dθ ∨ g dθ X

X

X

σ holds for all functions f, g ∈ SR (X, P), provided that the measure θ satisﬁes (L). In other words, the integral with respect to θ deﬁnes a ∨σ (X, P) to Q in the sense of I.5.30. Besemilattice homomorphism from SR σ of SR (X, cause both functions χF ⊗f and χF ⊗g are elements ( ( P) whenever f, g ∈ ER (X, P) and F ∈ AR , and because F f dθ = X χF ⊗f dθ, this applies also to integrals over measurable subsets F of X. We summarize:

Proposition 6.17. Let (P, V) be a quasi-full locally convex ∨-semilattice cone and let θ be a bounded LK (P, Q0 )-valued measure such that θE (a) ∨ θE (b) = θE (a ∨ b) and θE (a) ∨ θG (b) = θE (a) + θG (b) for all a, b ≥ 0 in P and disjoint sets E, G ∈ R Then ' ' ' (f ∨ g) dθ = f dθ ∨ g dθ X

X

X

σ holds for all functions f, g ∈ SR (X, P).

The case that Q = R. If Q = R, that is if the values θE of the measure θ are elements of P ∗ , then Condition (L) means that (i) all functionals θE are lattice homomorphisms and (ii) for disjoint sets E, G ∈ R we have either θE = 0 or θG = 0. Similar concepts and results could obviously developed for locally convex ∧-semilattice cones as deﬁned in Section I.5.1 and ∧-semilattice homomorphisms (see I.5.30). 6.18 Cone-Valued Functions and Positive Real-Valued Measures. If P is a subcone of Q, and if the topology induced onto P by the neighborhood system W of Q is equivalent to the topology induced by its given neighborhood system V, then for every ρ ∈ R+ the mapping a → ρa : P → Q, deﬁnes a continuous linear operator. Thus every R+ -valued measure θ on R, that is E → θE : R → R+ is an operator-valued measure in the sense of Section 3. In particular, σ-additivity in our sense follows from σ-additivity for the R+ -valued measure θ in the usual sense using Proposition n I.5.22. Indeed, let Ei ∈ R be disjoint sets, E = ∞ E and set F = n i=1 i i=1 Ei . Then θE = lim θFn ∈ R+ . n→∞

For σ-additivity of θ as an L(P, Q)-valued measure, we shall ﬁrst consider the case that θE = 0. Then θFn = 0 for all n ∈ N, as this sequence is

6. Examples and Special Cases

229

increasing. For any a ∈ P this means ∞ !

θEi (a) = lim θFn (a) = θE (a) = 0. n→∞

i=1

Otherwise, Proposition I.5.22 yields ∞ !

θEi (a) = lim θFn (a) = lim θFn a = lim θFn a = θE a = θE (a) n→∞

i=1

n→∞

n→∞

as well. For E ∈ R and w ∈ W there is v ∈ V such that a ≤ b + v for a, b ∈ P implies a ≤ b + w. Then the modulus of the measure θ is given by

n ! θEi si si ≤ v, Ei ∈ R disjoint subsets of E |θ|(E, v) = sup i=1

≤ sup

n !

θEi

Ei ∈ R disjoint subsets of E

w ≤ θE w.

i=1

The L(P, Q)-valued measure θ is therefore bounded in the sense of Section 3.6 and can be extended to the full cone (PV , V) (see Section 3.8). In case that (Q, W) is indeed the standard lattice completion of (P, V) as introduced in I.5.57, then Corollary 5.9 (see also 6.5) yields that the integrals of integrable functions in F(X, P) are indeed elements of the second dual P ∗∗ of P. 6.19 Vector-Valued Functions and Real- or Complex-Valued Measures. Let (P, V) be a locally convex topological vector space over K = R or K = C, endowed with a basis V of balanced convex neighborhoods, and let (Q, W) be the standard lattice completion of (P, V), as deﬁned in Section 5.57 of Chapter I. Then the topology induced by W onto the embedding of P into Q is equivalent to the topology induced by its given neighborhood system V (see I.5.57). For each ρ ∈ K the mapping a → ρa : P → Q, is therefore a continuous linear operator. Thus every K-valued measure θ on R, that is E → θE : R → K is an operator-valued measure in the sense of Section 3. For σ-additivity, ∞ n let Ei ∈ R be disjoint sets, E = i=1 E Fn = i=1 Ei . Then i and set θE = lim θFn ∈ K, and lim θFn a = lim θFn a holds for all a ∈ P, n→∞

n→∞

n→∞

since (P, V) is a topological vector space, hence the scalar multiplication is continuous. The L(P, Q)-valued measure θ is indeed strongly additive in the sense of 5.32 since for every decreasing sequence (En )n∈N of sets in R

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II Measures and Integrals. The General Theory

such that n∈N En = ∅ and ε > 0 there is n0 ∈ N such that |θEn | ≤ ε for all n ≥ n0 . Because for a ∈ P and w ∈ W there is v ∈ V and λ ≥ 0 such that a ≤ λv ≤ λw, hence θEn (a) = θEn a ≤ λw, holds for all n ≥ n0 . The latter follows since the neighborhoods in V are balanced and convex for P. Recall from 6.4 that the total variation var(θ, E) of a real- or complex-valued measure θ on is always ﬁnite. For E ∈ R and w ∈ W there is v ∈ V such that a ≤ b + v, that is a − b ∈ v for a, b ∈ P implies a ≤ b + w. We have γs ∈ |γ|v for all γ ∈ K whenever s ∈ v for s ∈ P and v ∈ V. According to 6.12 the modulus of the L(P, Q)-valued measure θ is therefore given by

n ! θEi si si ∈ v, Ei ∈ R disjoint subsets of E |θ|(E, v) = sup i=1

≤ sup

n !

|θEi | Ei ∈ R disjoint subsets of E

w

i=1

= var(θ, E) w. The L(P, Q)-valued measure θ is therefore bounded in the sense of Section 3.6. Integrability for P-valued functions had been characterized in 6.12. Integrals of functions that satisfy Condition (BI 1) are elements of the second vector space dual P ∗∗K of P see 6.12(i) . According to 6.12(iv), the operator ' f dθ : F(F,θ,BI1) (X, P) → P ∗∗K f → F

is linear over K. Integrals of functions that satisfy Condition (BI 2) from 6.12 are indeed elements of the closure with respect to the symmetric topology of P in P ∗∗K . In case of a topologically complete locally convex vector space P, this closure coincides with P. Neighborhood-valued measurable functions are integrated using the canonical extension of the measure θ to the full cone (PV , V) as elaborated in Section 3.8. For a positive real-valued measurable function ϕ and a neighborhood v ∈ V, for example, the function ϕ⊗v is measurable, hence in to the F ∈ R its integral may be F(X, PV ). According ( above for every ( estimated as F ϕ⊗v dθ ≤ F ϕ d var(θ) w, where var(θ) is the positive real-valued measure E → var(θ, E) : R → R and w ∈ W is a neighborhood such that a ≤ b + v, that is a − b ∈ v for a, b ∈ P implies a ≤ b + w. Because the locally convex complete lattice cone (Q, W) allows suﬃciently many order continuous linear functionals, that is the order continuous lattice homomorphisms on Q support the separation property (see I.5.32 and I.5.57), the strong convergence statements of Theorem 5.36 apply to functions satisfying Condition (BI 2) from 6.12.

6. Examples and Special Cases

231

Let us consider the special case that (P, V) is a normed space, that is V = {ρB | ρ > 0}, where B is the unit ball in P. A vector-valued function f ∈ F(X, P) is called Bochner (or Dunford and Schwartz) integrable over a set E ∈ R with respect to a scalar-valued measure θ (see for example III.2.17 in [55] or II.2 in [43]) if for every ε > 0 there is a step function hε ∈ SR (X, P) such that the mapping x → f (x) − hε (x) is measurable and ' f − hε d var(θ) ≤ ε. E

Indeed, if the P-valued function f is Bochner integrable, then given w ∈ W there is ε > 0 such that a ≤ b + εB, that is a − b ≤ εB for a, b ∈ P implies a ≤ b + w. We set h(B,w) = hε ∈ SR (X, P) and compute ' ' f − hε ⊗ B d θ ≤ f − hε d var(θ) ≤ w E

E

by our preceding considerations, hence (BI 2) from 6.12 holds for f. 6.20 Operator-Valued Functions and Operator-Valued Measures. Let N and H be cones, and let Z and Y be families of subsets of N and of H, directed upward by set inclusion. Furthermore, let (M, U) and (L, R) be two locally convex cones, and for the respective cones L(N , M) and L(H, L) of linear operators consider the neighborhoods V(Z,u) for Z ∈ Z and u ∈ U, and W(Y,r) for Y ∈ Y and r ∈ R (see Section I.7); that is S ≤ U + V(Z,u) or R ≤ T + W(Y,r) for operators S, U ∈ L(N , M) or R, T ∈ L(H, L), respectively, if S(z) ≤ U (z) + u

for all z ∈ Z,

or

R(y) ≤ T (y) + r

for all y ∈ Y.

Let H(N , M) be a subcone of L(N , M) such that all its elements are bounded below with respect to the V(Z,u) and such that neighborhoods the resulting locally convex cone H(N , M), V is quasi-full. Similarly, let H(H, L) be a subcone of L(H, L) whose elements are bounded below with respect to the neighborhoods W(Y,r) and denote the resulting locally convex $ be a locally convex complete latcone by H(H, L), W . Let H(H, L), W tice cone containing the latter, for example its (simpliﬁed) standard lattice completion (see Sections I.5.57 and I.7). Now in the context of our general theory we may consider integrals for H(N , M)-valued functions with respect to bounded L H(N , M), H(H, L) -valued measures. This is indeed a rather unwieldy setting. It does however facilitate a considerably wider choice of applications for our theory, as we shall see in Sections 6.22 to 6.23 below. Moreover, note that this point of view generalizes our original one since the given cones (P, V) and (Q, W) may be consid ered as cones of linear operators from R+ to P or to Q, respectively see Example I.7.1(c) .

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II Measures and Integrals. The General Theory

We shall study two useful special cases in further detail: (i) The case N = H and Z = Y. In this case every linear operator T ∈ L(M, L) may be reinterpreted as a linear operator T from H(N , M) into L(N , L) mapping the operator U ∈ H(N , M) into the operator T ◦U ∈ L(N , L); that is T ◦ U (z) = T U (z) ∈ L for all z ∈ N. In order to guarantee that the operator T ◦ U is bounded below with respect to the neighborhoods W(Y,r) ∈ W, and that the operator T : H(N , M) → L(N , L) is continuous with regard to the respective neighborhood systems for these cones, we shall require that T itself is continuous from (M, U) into (L, R), that is T ∈ L(M, L). Indeed, for Z ∈ Z and r ∈ R there is u ∈ U such that T (a) ≤ T (b) + r whenever a ≤ b + u for a, b ∈ M. Then for operators S, U ∈ H(N , M) such that S ≤ U + V(Z,u) we have S(z) ≤ U (z) + r, hence T ◦ S (z) ≤ T ◦ U (z) + r for all z ∈ Z. This shows T (S) ≤ T (U ) + W(Z,r) . Moreover, as for every S ∈ H(N , M) we have 0 ≤ S + λV(Z,u) for some λ ≥ 0, the above implies that 0 ≤ T (S) + λW(Z,r) . In this way, an L(M, L)-valued measure θ on R may be reinterpreted , L) -valued measure, where (H(N , L), W) is a loas an L H(N , M), H(N cally convex complete lattice cone containing all the operators θE ◦ U for E ∈ R and U ∈ H(N , M). We are using the above identiﬁcation of a continuous linear operator T ∈ L(M, L) with a continuous linear operator , L) . We proceed to calculate the modulus of such a T ∈ L H(N , M), H(N measure: For E ∈ R and V(Z,u) ∈ V we have |θ| E, V(Z,u)

n ! = sup θEi ◦ Si Si ≤ V(Z,u) , Ei ∈ R disjoint subsets of E . i=1

The supremum on the right-hand side is taken in the locally convex com , L). For R-boundedness of this measure we replete lattice cone H(N quire that for every r ∈ R and Z ∈ Z there is u ∈ U such that ≤ W |θ| E, V (Z,u) (Z,r) . Note that for N = R+ and Z = {1} , that is for H(N , M), V and H(N , L), W being isomorphic to the given cones (M, U) and (L, R) we have |θ| E, V({1},u) = |θ|(E, u)|. Countable additiv , L) -valued measure θ requires that for disjoint ity for the L H(N , M), H(N sets Ei ∈ R for every U ∈ H(N , M) the series θ( i∈N

Ei )

◦U =

∞ ! θEi ◦ U i=1

6. Examples and Special Cases

233

, L). In case that H(N , L) is the simpliﬁed stanis order convergent in H(N dard lattice completion of H(N , L) as constructed in I.7.1, this means that for disjoint sets Ei ∈ R μ θ( i∈N

Ei )

∞ ! U (a) = μ θEi U (a) i=1

holds for all U ∈ H(N , M), a ∈ Z∈Z Z and μ ∈ L∗ . Also in this case, Corollary 5.9 yields together with Remark I.7.1 that integrals of H(N , M), V -based integrable functions are indeed linear operators from N into L∗∗ , the second dual of L. (ii) The case M = L and U = R. In this case every linear operator T ∈L(H, N ) may be reinterpreted as a linear operator T% from H(N , M) into L(H, M), mapping the operator U ∈ H(N , M) into the operator U ◦ T ∈ L(H, M); that is U ◦ T (z) = U T (z) ∈ M for all z ∈ H. In order to guarantee that the operator U ◦ T is bounded below with respect to the neighborhoods W(Y,r) ∈ W, and that the operator T% : H(N , M) → L(H, M) is continuous with regard to the respective neighborhood systems, we shall require that for every Y ∈ Y there is some Z ∈ Z such that f (Y ) ⊂ Z. Indeed, for Y ∈ Y and u ∈ U let Z ∈ Z such that f (Y ) ⊂ Z. Then for operators S, U ∈ H(N , M) such that S ≤ U + V(Z,u) we have S(z) ≤ U (z)+u for all z ∈ Z, hence S ◦T (y) ≤ U ◦T (z)+u for all y ∈ Y. This shows T%(S) ≤ T%(U ) + W(Y,u) . Moreover, as for every S ∈ H(N , M) we have 0 ≤ S + λV(Z,u) for some λ ≥ 0, the above implies that 0 ≤ T (S) + λW(Y,u) . In this way, an L(H, N )-valued measure θ satisfying the above require ment may be reinterpreted as an L H(N , M), H(H, M) -valued measure, $ is a locally convex complete lattice cone containing all where (H(H, M), W) the operators U ◦ θE for E ∈ R and U ∈ H(N , M), and using the above identiﬁcation. The modulus of such a measure is calculated for E ∈ R and V(Z,u) ∈ V as

n ! Si ◦ θEi Si ≤ V(Z,u) , Ei ∈ R disjoint subsets of E . |θ| E, V(Z,u) = sup i=1

The supremum on the right-hand side is taken in the locally convex complete lattice cone H(H, M). For R-boundedness of this measure we require that for every u ∈ U and Y ∈ Y there is Z ∈ Z such that |θ| E, V(Z,u) ≤ W(Y,u) . Countable additivity for the measure θ requires that for disjoint sets Ei ∈ R for every U ∈ H(N , M) the series

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II Measures and Integrals. The General Theory

U ◦ θ( i∈N

Ei )

=

∞ ! U ◦ θEi i=1

is order convergent in H(H, M). In case that H(H, M) is the simpliﬁed standard lattice completion of H(H, M)as constructed in I.7.1, Corollary 5.9 and Remark I.7.1 yield that integrals of H(N , M), V -based integrable functions are linear operators from H into M∗∗ , the second dual of M. If both H and M are vector spaces over K = R or K = C, then these integrals are indeed K-linear operators from M into the second vector space dual M∗∗ K of M (see I.7.1). 6.21 Positive, Real or Complex-Valued Functions and OperatorValued Measures. This is a special case for the preceding section. Let (P, V) and (Q, W) be locally convex cones, and let K = R+ , or K = R or K = C if P and Q are indeed locally convex topological vector spaces over R or C, respectively, endowed with their symmetric topologies. We choose N = M = P and H(N , M) = K in the setting of Section 6.20. Depending on the suitable choice for the family Z of bounded below subsets of P, the following upper neighborhoods for an element α ∈ K will render K into a quasi-full locally convex see Example I.7.2(c) : For K = R+ the family of all Buε (α) = [0, α + ε] for ε > 0, or the single neighborhood and Bu0 (α) = [0, α], both yielding the natural order; for K = R or K = C the Euclidean neighborhoods Bε (α) = {β ∈ K | |β − α| ≤ ε} with equality as the order on K. In order to deal with these cases simultaneously, let us denote by B either Bu1 (0), Bu0 (0) or B1 (α), that is the respective unit neighborhoods of 0 ∈ K, and let Γ = {0}, Γ = {0, 1} or Γ = {γ ∈ K | |γ| = 1}} be the corresponding units spheres. We set L = Q and use the special case (i) in Section 6.20 in order to integrate K-valued functions with respect to an L(P, Q)-valued measure. For H(P, Q) we choose the simpliﬁed standard lattice completion of L(P, Q). For E ∈ R and the above neighborhoods we calculate the modulus of an L(P, Q)-valued measure θ as follows:

n ! γi θEi γi ∈ Γ, Ei ∈ R disjoint subsets of E . |θ|(E, B) = sup i=1

The supremum on the right-hand side of these expressions is taken in the locally convex complete lattice cone H(P, Q), that is a cone of R-valued functions with the pointwise algebraic and lattice operations. For K = R+ and B = Bu0 we have of course |θ|(E, B) = 0. For the remaining cases boundedness of the L K, L(P, Q) -valued measure θ requires that for every E ∈ R, the modulus |θ|(E, B) is bounded in H(P, Q) with respect to all neighborhoods W(Z,w) for Z ∈ Z and w ∈ W. Let us recall the construction in I.7 of the standard lattice completion H(P, Q) of L(P, Q) to understand this further: The elements of H(P, Q) are R-valued functions on the set

6. Examples and Special Cases

235

Υ = Z × Q∗ . An element ϕ ∈ H(P, Q), that is an R-valued function Z∈Z on Υ is bounded relative to a neighborhood W(Z,w) if there is λ ≥ 0 such that ϕ(a, μ) ≤ λ holds for all a ∈ Z and μ ∈ w◦ . Thus for boundedness of the measure θ we require that for every choice of disjoint subsets Ei ∈ R of E and γi ⊂ Γ we have n n ! ! μ γi Ei (a) = e(γi )μ Ei (a) ≤ λ i=1

i=1

for all a ∈ Z and μ ∈ w◦ ; or equivalently, that for every Z ∈ Z the subset

n ! θEi (a) Ei ∈ R disjoint subsets of E, a ∈ Z i=1

is bounded above in Q. Indeed, in case that K = R or K = C, both (P, V) and (Q, W) are locally convex vector spaces, and we have γμ ∈ w◦ for all γ ∈ Γ whenever μ ∈ w◦ for w ∈ W. Recall that all sets Z ∈ Z are required to be bounded below in P. The choice of all these sets for Z results in the uniform operator topology for L(P, Q) see I,7.1(i) . If the sets in Z are also bounded above (as is indeed implied in the case that P is a locally convex vector space in its symmetric topology), then boundedness of θ as an L K, L(P, Q) -valued measure is already implied by its boundedness as an L(P, Q)-valued measure. Indeed, given E ∈ R and w ∈ W there is v ∈ V such that |θ|(E, v) ≤ w (see Sections 3.2 to 3.6). Then for every Z ∈ Z there is λ ≥ 0 such that z ≤ λv for all z ∈ Z. This implies the above condition for the boundedness of θ. If Z consists of all ﬁnite subsets of P, that is if we consider the strong operator topology for L(P, Q) see I,7.1(ii) , then boundedness is a much weaker condition for θ : For every a ∈ P the subset

n ! θEi (a) Ei ∈ R disjoint subsets of E i=1

is required to be bounded above in Q. Countable additivity for the L K, L(P, Q) -valued measure θ demands that for disjoint sets Ei ∈ R μ θ( i∈N

∞ ! (a) = μ θEi (a) Ei ) i=1

holds for all a ∈ Z∈Z Z and μ ∈ Q∗ . Our notion of measurability for K-valued functions coincides with the usual one (see also Examples 6.3 and 6.4). For K = R+ all measurable K-valued functions are in F(X, K), hence integrable. For K = R or K = C with the Euclidean topology, a measurable K-valued function is in F(X, K) if on

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II Measures and Integrals. The General Theory

every set E ∈ R it can be uniformly approximated by step functions. This implies of course strong integrability in the sense of 5.18. Because H(P, Q) was supposed to be the simpliﬁed standard lattice completion of L(P, Q), the integral to a function ϕ ∈ F(X, K) with respect to an L(P, Q)-valued measure over a set E ∈ R is a linear operator from P into Q∗∗ , contained in the closure of L(P, Q) in H(P, Q) with respect to the symmetric relative topology. Thus, if the cone L(P, Q) is topologically complete with respect to this topology, then this integral is indeed an element of L(P, Q). Let us proceed to discuss the convergence theorems from Section 5: For the sake of simplicity we shall restrict ourselves to the case of a single measure, that is θn = θ for all n ∈ N in Theorems 5.23 to 5.25: Let (ϕn )n∈N be a sequence of integrable K-valued functions that converges pointwise θ-almost everywhere on a set F ∈ AR to a function ϕ in the symmetric relative topology of K. This is of course the usual (Euclidean) notion of convergence, Bu0 which except for the case of K = R+ endowed with the neighborhood renders 0 ∈ R+ into an isolated point see Example I.4.37(b) . The boundedness conditions from Theorem 5.25 for the sequence (ϕn )n∈N read somewhat diﬀerently for the diﬀerent choices for K : We set ϕ∗∗ = ϕ∗ = 0 in all cases. ≤ ϕ∗ holds for all n ∈ N with some inteFor K = R+ we require that ϕna.e.F ∗ grable function ϕ . For K = R or K = C with the Euclidean topology and the order we use an integrable positive-valued function ϕ∗ and the function f ∗ = ϕ∗⊗ B whose values are in the full cone KV = {α+ρB | α ∈ K, ρ ≥ 0} to ≤ ϕ∗ which Theorem 5.25 applies in this case. We therefore require that |ϕn |a.e.F holds for all n ∈ N in this of Theorem are ( case. The assumptions ( ( 5.25 ∗ ∗ = ϕ dθ, T = ϕ dθ and T = ϕ dθ, or now satisﬁed. Let T n F n F F ( T ∗ = F (ϕ∗⊗ B) dθ in case K = R or K = C. These integrals are in general elements of H(P, Q). The conclusion of Theorem 5.25 now states that T ≤ lim Tn

and

n→∞

in H(P, Q), that is for all a ∈

lim Tn ≤ T + O (T ∗ )

n→∞

T (a, μ) ≤ lim Tn (a, μ) n→∞

∗

Z∈Z

Z and μ ∈ P , and indeed T (a, μ) = lim Tn (a, μ) n→∞

whenever T ∗ (a, μ) < +∞. Note that for linear operators T ∈ L(P, Q) as elements of H(P, Q) we have T (a, μ) = μ T (a) . Now let us investigate the additional assumptions of Theorem 5.36 which will lead to convergence of (Tn )n∈N towards T in the symmetric topology of H(P, Q) : We require that F = E is in R. Strong additivity of the L K, L(P, Q) -valued measure θ in the sense of 5.30 means

that for every decreasing sequence (En )n∈N of sets in R such that n∈N En = ∅, for

6. Examples and Special Cases

237

Z ∈ Z and w ∈ W there is n0 ∈ N such that θEn (a) ≤ O θE1 (a) + w holds for all a ∈ Z and n ≥ n0 . Recall that in case K = R or K = C we assume that both P and Q are locally convex vector spaces in their respective symmetric topologies, thus O θE1 (a) = 0, and θEn (a) ≤ w implies that θEn (γa) ≤ w for all γ ∈ Γ. The above therefore means that the sequence of (θ En )n∈N of linear operators converges to 0 in the symmetric topology L(P, Q), W . We also need to require that the functions ϕn , ϕ and ϕ∗ or ϕ∗⊗ B are (K, V)-based integrable in the sense of 5.6. Measurability in the classical sense and boundedness below almost everywhere on the set E is suﬃcient for this. This condition also yields strong integrability for the functions (ϕ∗ or ϕ∗⊗ B. Finally, ( according to 5.36 we require that the element Q). Under these T ∗ = E ϕ∗ dθ or T ∗ = E (ϕ∗⊗ K) dθ is bounded in H(P, additional assumptions then Theorem 5.36 yields T = lim Tn n→∞

in the symmetric topology of H(P, Q). If as in most cases of interest the integrals Tn and T are actually elements of L(P, Q), then we infer convergence in the symmetric operator topology of (L(P, Q), W). Operator algebras. If H = P = Q is a locally convex topological vector space, then the space of continuous linear operator L(P, P) forms a topological algebra, endowed with the composition of operators as its multiplication (see 6.4). We integrate K-valued functions with respect to an L(P, P)-valued measure θ in this case. The values of the integrals are con tained in the simpliﬁed standard completion H(P, P) of L(P, P). For the integral to determine a multiplicative linear operator from ER (X, K) = P) in the sense of Example 6.14 we need to require FR (X, K) into H(P, that the measure θ satisﬁes Condition (A), that is θE (a) θE (b) = θE (ab) and θE (a) θG (b) = 0 holds for all a, b ∈ K and disjoint sets E, G ∈ R. As θE (a) = aθE in this case, Condition (A) reads as follows: (A) (θE )2 = θE and θE θG = 0 for disjoint sets E, G ∈ R, that is the operators θE ∈ L(P, P) are required to be idempotent and pairwise orthogonal for disjoint sets E, G ∈ R. Spectral Measures. For a concrete example, let H = P = Q be a complex Hilbert space with unit ball U and the neighborhood system V={ρU | ρ>0}. Let R be a weak σ-ring, and as in spectral theory, let θ be a projection valued measure on R. We consider θ as an L C, L(H, H) -valued measure in the above sense. Such a measure is seen to be R-bounded, even if we choose the uniform operator topology for L(H, H) see I.7.2(i) , that is the family of all bounded subsets of H for Z. Indeed, let a ∈ H such that

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II Measures and Integrals. The General Theory

a ≤ 1 and let Ei ∈ R, for i = 1, . . . , n be disjoint sets. For a spectral subspaces of P. measure the θEi are projections onto mutually orthogonal Thus the elements ai = θEi (a) are orthogonal and ni=1 ai 2 ≤ a2 = 1 by the Bessel inequality see Theorem 1 in I.5 of [82] . Thus n n n -2 - ! -! -2 ! θEi (a)- = ai - = ai 2 ≤ 1. i=1

The set

n !

i=1

i=1

θEi (a) Ei ∈ R disjoint subsets of E, a ∈ H, a ≤ 1

i=1

is therefore indeed bounded above in H. Countable additivity for a spectral measure is however required only with respect to the strong operator topology for L(H, H), which arises if we choose the family of all ﬁnite subsets of H for Z. (Because projection operators in L(H, H) are of norm 1, countable additivity with respect to the uniform operator topology can of course only apply to ﬁnite sums of such operators.) Theorem 5.36 therefore yields convergence in the strong but not in the uniform operator topology of L(H, H) for spectral measures. Spectral measures satisfy Condition (A) from above and also Condi∗ tion (A*) from 6.14, that is θE (a∗ ) = θE (a) for all E ∈ R and a ∈ P. ∗ ∗ ∗ ¯θE and θE (a) = aθE = a ¯ θE in this case, this is As θE (a∗ ) = a equivalent to ∗ (A*) θE = θE for all E ∈ R. This condition holds because the projection operators θE ∈ L(H, H) are self( adjoint. The linear operator f → X f dθ is therefore multiplicative on the measurable K-valued functions and preserves space FR (X, C) of bounded ( ∗ ( the involution, that is X f ∗ dθ = X f dθ . 6.22 Operator-Valued Functions and Cone-Valued Measures. This is again a special case of 6.20. Let P be a cone, (Q, W) a locally convex complete lattice cone. We choose N = P, M = L = Q and H = R+ in the setting of 6.20 and use the special case (ii). For Z we choose a family of subsets of P, directed upward by set inclusion such that Z∈Z Z = P, and suppose that the locally convex cone (H(P, Q), V) of linear operators from P into Q is quasi-full. Let Y consist of the singleton subset {1} of . Then the locally convex cone L(R+ , Q), W) is isomorphic to (Q, W) R + see Example I.7.2(d) , hence a locally convex complete lattice cone. Similarly, because the cone P can be identiﬁed with the cone L(R+ , P), we may consider the elements of P to be linear operators from some quasi-full cone Z and H(P, Q) into L(R+ , Q), that is into Q. Our choice for the families Y guarantees that these operators are continuous see 6.20 (ii) . Using these

6. Examples and Special Cases

239

settings, case (ii) from 6.20 therefore permits us to consider H(P, Q)-valued functions together with P-valued measures. Countable additivity requires for a P-valued measure θ that for disjoint sets Ei ∈ R and for every linear operator T ∈ H(P, Q) the series T θ( i∈N

Ei )

=

∞ ! T θEi i=1

is order convergent in Q. The modulus of θ is calculated for E ∈ R and V(Z,w) ∈ V as |θ| E, V(Z,w)

n ! = sup Ti θEi Ti ≤ V(Z,w) , Ei ∈ R disjoint subsets of E . i=1

R-boundedness in the sense of Section 3.6 requires that for every E ∈ R and u ∈ W there is V(Z,w) ∈ V such that |θ| E, V(Z,w) ≤ u. A bounded P-valued measure then integrates H(P, Q)-valued functions, and the values of these integrals are elements of L(R+ , Q), that is Q itself. If (Q, W) is indeed the standard lattice completion of some locally convex cone Q0 and if the concerned function is H(P, Q0 ) -based integrable, then its integral is an element of the subcone Q∗∗ 0 of Q. Let us further consider the special case that (P, V) is a locally convex vector space, that (Q, W) is the standard lattice completion of a locally convex vector space (Q0 , W0 ) and that H(P, Q) ⊂ L(P, Q0 ). Then countable additivity of an P-valued n measure θ is guaranteed by weak convergence of the Indeed, weak convergence in P concerned series i=1 θEi in P in this case. implies weak convergence in Q for the series ni=1 θEi (T ) = ni=1 T (θEi ) for every operator T ∈ L(P, Q0 ). (see IV.2.1 in [185]). Weak convergence in Q0 , however, coincides with order convergence in Q in this case (see I.5.57) as required for countable additivity. Moreover, Theorem 3.11 (or Corollary 3.13), that is our version of Pettis’ theorem yields that for a vector-valued measure, countable additivity with respect to weak convergence implies countable additivity with respect to strong convergence, that is convergence in the symmetric topology of P. Every such measure is therefore strongly additive in the sense of 5.30. 6.23 Positive, Real or Complex-Valued Functions and Cone- or Vector-Valued Measures. This is a special case for the preceding section. Let (P, V) be a locally convex cone and let and (Q, W) be its standard lattice completion, Let K = R+ , or K = R or K = C if P is indeed a locally convex vector space over R or C, respectively, endowed with its symmetric topology. We choose H(P, Q) = K endowed with one of the suitable topologies arising from the choice for the family Z of bounded below subsets of P (see Example I.7.2(c) and 6.21 above), that is topologies generated by the

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neighborhoods B as discussed for the respective cases in 6.21. We shall also use the notation for the unit sphere Γ from 6.21. Using this, the modulus of a P-valued measure θ is given by

n ! γi θEi γi ∈ Γ, Ei ∈ R disjoint subsets of E . |θ|(E, B) = sup i=1

The supremum on the right-hand side of this expression is taken in the locally convex complete lattice cone Q, that is a cone of R-valued functions with the pointwise algebraic and lattice operations. Boundedness is of course guaranteed in the case of B = Bu0 . In the remaining cases it requires (see the corresponding detailed argument in 6.21) that the set

n ! θEi Ei ∈ R disjoint subsets of E i=1

is bounded in P. We will be able to verify that every P-valued measure θ is R-bounded in this instance. For this call to mind that the elements θE ∈ P, for all E ∈ R, are considered to be continuous linear operators from K into N , thus are required to be bounded elements of P. Furthermore, recall from I.5.57 that the neighborhood system V for P is a generating subset of the neighborhood system W for the standard lattice completion Q of P. For E ∈ R let us consider the subset A = {θE | E ∈ R, E ⊂ E} of P. We shall use Proposition I.4.25 (which is derived from the Uniform Boundedness Theorem 3.4 in [172]) in order to verify that A is bounded above in P. For this, let μ ∈ P ∗ . Because the elements of P ∗ are also order continuous linear functionals on the standard lattice completion Q of P, we know from 3.9 that μ ◦ θ is an L(K, R)-valued, that is an R-valued measure on R. This measure is indeed real-valued, since the elements θE , for all E ∈ R, were seen to be bounded elements of P. A countably additive real-valued measure is however known to be bounded, that is μ(a) | a ∈ A = (μ ◦ θ)E | E ∈ R, E ⊂ E is a bounded subset of R. Because this holds true for all linear functionals μ ∈ P ∗ , Proposition I.4.25 yields that the set A is bounded above relative to 0 ∈ P, that is bounded above, as claimed. Thus, given v ∈ V, there is indeed λ ≥ 0 such that θE ≤ λv holds for all subsets E ∈ R of E. We claim that this implies |θ|(E, V ) ≤ 4λv. For this let us recall the construction of the standard lattice completion (Q, W) of (P, V). Its elements are R-valued functions ϕ on the dual P ∗ of P, and we have ϕ ≤ v if ϕ(μ) ≤ 1 for all μ ∈ v ◦ . For any such μ ∈ v ◦ , μ ◦ θ was seen to be a real-valued countably

6. Examples and Special Cases

241

additive measure on R. As (μ ◦ θ)E ≤ λ for all subsets E ∈ R of E, we know that is total variation on E, that is var μ ◦ θ, E is bounded by the constant 4λ (see 6.4). Thus θ E, V (μ) = var μ ◦ θ, E ≤ 4λ

for all μ ∈ v ◦ . This demonstrates |θ|(E, V ) ≤ 4λv, as claimed. Integrals of K-valued functions with respect to a P-valued measure θ were seen to be elements of P ∗∗ . If (P, V) is indeed a locally convex vector space that is complete in its symmetric topology and if as required in some integrability conditions in the literature (see for example IV.10.7 in [55]) the K-valued function ϕ can be approximated by a sequence of step functions converging pointwise almost everywhere towards ϕ and such that the sequence of integrals over these step functions is convergent in P, then this additional requirement guarantees that the value of the integral of ϕ is also contained in P rather than in P ∗∗ . Let us discuss the convergence theorems from Section 5: Let (ϕn )n∈N be a sequence of integrable K-valued functions that converges pointwise θ-almost everywhere on a set F ∈ AR to a function ϕ in the symmetric relative topology of K. This is the usual notion of convergence, except for the case of K = R+ endowed with the neighborhood Bu0 which renders 0 ∈ R+ as an isolated point. The boundedness conditions from Theorem 5.25 are as follows: ≤ ϕ∗ We set ϕ∗∗ = ϕ∗ = 0 in all cases. For K = R+ we require that ϕna.e.F ∗ holds for all n ∈ N with some integrable function ϕ . For K = R or K = C with the Euclidean topology and the order we use an integrable positivevalued function ϕ∗ and the function f ∗ = ϕ∗⊗ B whose values are in the full cone KV = {α + ρB | α ∈ K, ρ ≥ 0} to which Theorem 5.25 applies. We ≤ ϕ∗ holds for all n ∈ N in this case. The assumptions require that |ϕn |a.e.F ( ( of Theorem 5.25 are now (satisﬁed. Let an = F ϕn dθ, a = F ϕ dθ and ( a∗ = F ϕ∗ dθ, or a∗ = F (ϕ∗⊗ K) dθ in case K = R or K = C. These integrals are in general elements of P ∗∗ . The conclusion of Theorem 5.25 now states that a ≤ lim an n→∞

in P ∗∗ , that is

and

lim an ≤ a + O (a∗ )

n→∞

a(μ) ≤ lim an (μ) n→∞

∗

for all μ ∈ P , and indeed a(μ) = lim an (μ) n→∞

whenever a∗ (μ) < +∞. For elements a ∈ P ⊂ P ∗∗ we have a(μ) = μ(a). If (P, V) is indeed a locally convex topological vector space and if F = E ∈ R, then the assumptions of Theorem 5.36 apply: The measure θ is

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II Measures and Integrals. The General Theory

strongly additive by Theorem 3.11. Measurability in the classical sense and boundedness below almost everywhere on the set E is suﬃcient for the functions ϕn , ϕ and ϕ∗⊗ B to be (K, V)-based integrable in the sense of 5.6. The latter is indeed strongly integrable in the sense of 5.18. All integrals involved are elements of P and Theorem 5.36 yields a = lim an n→∞

in the symmetric, that is the given topology of P. Algebra-valued measures. If P is a topological algebra, that is a locally convex topological vector space over K = R or K = C with a compatible multiplication, then Conditions (A) and (A*) from 6.14 read as follows: (A) (θE )2 = θE and θE θG = 0 for disjoint sets E, G ∈ R. (A*) (θE )∗ = θE for all E ∈ R. According to 6.14, Condition (A) guarantees the multiplicativity of the integral as an( operator from ER (X, K) = FR (X, K) into P ∗∗ , that is ( ( • , Condition (A*) the compatibility with X (f g) dθ ∈ X f dθ ( X g dθ ( ∗ ∗ an involution, that is X f dθ = X f dθ for K-valued functions f, g ∈ FR (X, K). Lattice-valued measures. Now suppose that P is a lattice cone over K = R or K = R+ in the sense of 6.16, that is a quasi-full locally convex cone containing suprema for any two of its elements and satisfying the properties speciﬁed in 6.16. For the integral to determine a ∨-semilattice homomorphism from ER (X, K) = FR (X, K) into P ∗∗ in the sense of 6.16 we need to require that the measure θ satisﬁes Condition (L), that is θE (a) ∨ θE (b) = θE (a ∨ b) and θE (a) ∨ θG (b) = θE (a) + θG (b) holds for all 0 ≤ a, b ∈ K and disjoint sets E, G ∈ R. As θE (a) = aθE in this case, Condition (L) reads as follows: (L) θE ≥ 0 and θE ∨ θG = θE + θG for disjoint sets E, G ∈ R, that is the elements θE ∈ P are positive and mutually disjoint for disjoint sets E, G ∈ R. According to 6.16, Condition (L) guarantees that the integral σ P ∗∗ is a ∨-semilattice as an operator from SR ( K) into ( ((X, K) = FR (X, homomorphism, that is X (f ∨ g) dθ = X f dθ ∨ X g dθ for functions f, g ∈ FR (X, K). 6.24 Positive Linear Operators on Cones of R-Valued Functions. Let P = R, let X and R be as before, and let W be a neighborhood system for F(X, R), consisting of non-negative functions w ∈ F(X, R). Let Q = FW (X, R) be the subcone of functions in F(X, R) that are bounded below with respect to W. Then FW (X, R), W is a full locally convex complete lattice cone, provided that for every x ∈ X there is w ∈ W such that w(x) < +∞ (see Example I.5.7(c)). There are two distinct types of continuous linear operators from R into FW (X, R). Firstly, for a non-negative real-valued function ϕ such that both ϕ, −ϕ ∈ FW (X, R), let Tϕ (a) = aϕ

6. Examples and Special Cases

243

for a ∈ R. (In particular, this means Tϕ (+∞)(x) = +∞ for all x ∈ X such that ϕ(x) > 0 and Tϕ (+∞)(x) = 0 else.) Secondly, for a function ψ ∈ FW (X, R) that takes only the values 0 and +∞, set Tψ0 (a) = 0 for a ∈ R and Tψ0 (+∞) = ψ. Then every linear operator T ∈ L R, FW (X, R) can be expressed as T = Tϕ +Tψ0 with some ϕ, ψ ∈ FW (X, R) as above. Con sequently, an L R, FW (X, R) -valued measure θ on R can be expressed as a sum of two FW (X, R)-valued measures θ1 and θ0 , both yielding functions 1 is positive in FW (X, R), and such that for each E ∈ R the function θE 1 1 0 and both θE , −θE ∈ FW (X, R), and the function θE takes only the values 0 and +∞. For a step function h=

n !

χEi ai ∈ SR (X, R),

i=1

where a1 , . . . , an ∈ R, we have in particular ' n ! ! 1 0 h dθ = ai θE + θE ∈ FW (X, R). i i X

i=1

i=1,...,n s.th.ai =+∞

On FR (X, R), the mapping ' f → f dθ : FR (X, R) → FW (X, R) X

deﬁnes a linear operator, continuous with respect to the locally convex cone topologies induced by the neighborhood system W, that is ' ' f dθ ≤ g dθ + w whenever f ≤ g + vw , X

X

funcfor f, g ∈ FR (X, R). Recall from Section 4 that v(w consists of all step 1 ≤ w. tions s = ni=1 χEi ⊗ai for 0 < ai ∈ R such that X s dθ = ni=1 ai θE i According to 6.17, the linear operator determined by the integral is indeed a ∨-semilattice homomorphism, if Condition (L) holds, that is if θE (a) ∨ θE (b) = θE (a ∨ b) and θE (a) ∨ θG (b) = θE (a) + θG (b) for all a, b ≥ 0 in R and disjoint sets E, G ∈ R. The ﬁrst part of this condition holds always true for an L R, FW (X, R) -valued measure θ as introduced above, since the operators involved, Tϕ and Tψ0 , are deﬁned using non-negative functions ϕ and ψ. Let us investigate the second part of the condition in (L): For disjoint sets E, G ∈ R let θE = TϕE + Tψ0 E and θG = TϕG + Tψ0 G . Then (L) requires that the functions ϕE and ϕG are orthogonal, that is ϕE (x)ϕG (x) = 0 for all x ∈ X. (There are no additional conditions for the functions ψE and ψG .) If this condition is satisﬁed, then we have ' ' ' (f ∨ g) dθ = f dθ ∨ g dθ X

for all f, g ∈ FR (X, R).

X

X

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II Measures and Integrals. The General Theory

6.25 Bounded Linear Operators on Spaces of Real- or ComplexValued Functions. Now let P = K for K = R or K = C, endowed with the equality as order and the usual topology, that is V = {ρB | ρ > 0}, and a ≤ b+ρB if |a−b| ≤ ρ for a, b ∈ K. Let X and R be as before. Let W be a system of nonnegative R-valued functions on X, closed for addition and multiplication by (strictly) positive scalars and directed downward. Suppose that for every x ∈ X there is v ∈ V such that v(x) < +∞. Let Q0 = FW (X, K) be the vector space over K of all functions f ∈ F(X, K) that are bounded with respect to the functions in W, that is for every w ∈ W there is λ ≥ 0 such that |f (x)| ≤ λw(x) for all x ∈ X. The above condition on W guarantees that for every x ∈ X the point evaluation εx is contained in the vector space dual Q∗0 K of Q0 . Let Q be the standard lattice completion of Q0 . We shall consider an L(K, Q0 )-valued measure θ such that for all E ∈ R the operators θE ∈ L(K, Q0 ) are linear over K. According to 6.12(iii) then the operator ' f dθ : FR (X, K) → Q f → X

is linear over K in the sense that ' ' af dθ (μ) = f dθ (aμ) X

X

for every f ∈ F(F,θ) (X, P), μ ∈ Q∗0 and a ∈ K. If we set ' ' ' f dθ (x) ≡ f dθ (εx ) − i f dθ (iεx ) F

F

F

for x ∈ X, then these integrals may be reinterpreted as K-valued functions on X and the integral is a K-linear operator from FR (X, K) into FW (X, K). Moreover, ' f dθ ≤ w holds whenever f ≤ vw F

for f ∈ FR (X, K) and w ∈ W. (Recall that vw consists of ( all step n α χ for 0 < α ∈ R such that ⊗B functions s = i E i i i=1 X s dθ = n α |θ|(E , B) ≤ w.) i i i=1 Obviously, K-linear operators in L K, FW (X, K) correspond to functions ϕ ∈ FW (X, K). They operate as Tϕ (a) = aϕ

for a ∈ K.

An L K, FW (X, K) -valued measure θ on R may therefore be considered as an FW (X, K)-valued set function on R. Boundedness means that for every E ∈ R and w ∈ W there is ρ ≥ 0 such that

7. Extended Integrability

|θ|(E, B) = sup

245

n !

|θEi | Ei ∈ R disjoint subsets of E

≤ ρw.

i=1

Measurability for a function in F(X, K) in the sense of Section 1 coincides with measurability in the usual sense. Both P = K and Q0 = FW (X, K) are indeed topological algebras. Thus according to 6.14, the integral is an algebra homomorphism if Condition (A) holds, that is if θE (a)θE (b) = θE (ab) and θE (a)θG (b) = 0 holds for all a, b ∈ K and disjoint sets E, G ∈ R. The ﬁrst part of this condition means that the function θE ∈ FW (X, K) takes only the values 0 or 1, i.e. is the characteristic function of some subset Φ(E) of X. The second part of (A) requires that for disjoint sets E, G ∈ R the functions θE and θG are orthogonal, that is θE (x)θG (x) = 0 for all x ∈ X, that is the sets Φ(E) and Φ(G) are disjoint. If this condition is satisﬁed, then we have ' ' ' (f g) dθ = f dθ g dθ X

X

X

for all f, g ∈ FR (X, K). The extension of the multiplication from Q0 to Q that was introduced in 6.14 implies pointwise multiplication for the corresponding K-valued functions. Thus, under Condition (A) the integral deﬁnes a K-linear bounded and multiplicative operator from FR (X, K) into FW (X, K). It also preserves the involution since Condition (A*) is obviously implied by (A).

7. Extended Integrability We can further extend integrability to a wider class of functions f ∈ F(X, P). Obviously, if there is g ∈ F(F,θ) (X, P) such that ( both f + g and g are contained in F(F,θ) (X, P) and if the element F g dθ is invertible in Q, then we may set ' ' ' f dθ = (f + g) dθ − g dθ. F

F

F

The class of these functions f ∈ F(X, P) will be denoted by F(F,θ) (X, P). The following is straightforward to verify: Theorem 7.1. F(F,θ) is a subcone of F(X, P) containing F(F,θ) (X, P). If f, g ∈ F(F,θ) and 0 ≤ α ∈ R, then ( ( (a) (F (αf ) dθ = α (F f dθ ( (b) (F (f + g) dθ ( = F f dθ + F g dθ (c) F f dθ ≤ F g dθ + w whenever f ≤ g + vw for w ∈ W.

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II Measures and Integrals. The General Theory

8. Notes and Remarks The beginnings of modern measure theory date back to the late 19th century, some of the foundations being laid by Riemann, Harnack, Peano, Jordan, Borel, Baire, Lebesgue, Carath´eodory and Radon, to name just a few of the mathematicians involved. Excellent expositions about the early history of measure theory can be found in the works of Lebesgue [114], [115] and [116], Carath´eodory [30], Hahn and Rosenthal [80], Halmos [83] and Saks [182]. Vector-valued measure theory originated in the ﬁrst half of the twentieth century in treatises by Clarkson, Bochner, Dunford, Morse, Pettis and Gelfand among others. Since its appearance in 1977 the book by Diestel and Uhl [43] about vector measures has become a standard reference on the subject and is also often cited in this text. It contains various sections with detailed surveys of the history of the ﬁeld. There is also an extensive literature on ﬁnitely additive measures. The books by Dunford and Schwartz [55], [56], [57] and Diestel and Uhl [43] contain some sections about these. However, ﬁnitely additive measures appear to be less suitable for analytic purposes, and we therefore do not address them in this text. The (total) variation of a Banach space-valued measure θ on a σ-ﬁeld R is usually deﬁned as the positive R-valued set-function |θ| by

n ! θ(Ei ) Ei ıR disjoint subsets of E |θ|(E) = sup i=1

for E ∈ R (See III.1.4 in [55] or I.1.4 in [43]). The semivariation of a vectorvalued measure was introduced by Gowurin [74] and is given by

- n - - -! γi θ(Ei )θ(E) = sup - |γi | ≤ 1, Ei ∈ R disjoint subsets of E , i=1

Clearly θ(E) ≤ |θ|(E), and every countably additive vector measure is known to be bounded, that is |θ|(X) < +∞ (see IV.10.2 in [55]). The setfunction |θ| is seen to be σ-additive, whereas θ is generally only subadditive. On the other hand, the deﬁnition of the modulus of θ from Section 3.2, if applied to this situation, reads as

n ! γi θEi |γi | ≤ 1, Ei ∈ R disjoint subsets of E , |θ|(E, B) = sup i=1

where B denotes the unit ball of P = R or P = C. Recall that |θ|(E, B) is an element of the standard lattice completion of the given Banach space, that is an R-valued function on its dual unit ball B∗ . Since this function is non-negative, it cannot be considered as an element of the second dual of

8. Notes and Remarks

247

this Banach space. However, its supremum norm - |θ|(E, B)- as a function on B∗ is the semivariation of the measure. Indeed, -|θ|(E, B)# "

n ! = sup sup γi θEi |γi | ≤ 1, Ei ∈ R disjoint subsets of E (μ) μ∈B∗

i=1

= sup sup μ∈B∗

= sup

" n !

sup

i=1 " n !

μ∈B∗

# γi θEi # γi θEi

(μ) |γi | ≤ 1, Ei ∈ R disjoint subsets of E (μ) |γi | ≤ 1, Ei ∈ R disjoint subsets of E

i=1

- n - -! - = sup γi θEi - |γi | ≤ 1, Ei ∈ R disjoint subsets of E i=1

= θ(E). This observation establishes the relationship between the modulus of a vector-valued measure according to Section 3.6 and its classical semivariation. However, while the modulus is a countably additive set-function, the semivariation, as its norm is only subadditive. Boundedness in the sense of Section 3.6 means that supμ∈B∗ |θ|(X, B)(μ) < +∞, hence that the semivariation θ(X) is ﬁnite. Boundedness guarantees that the linear operators from P into Q which are the values of the measure can be extended to linear operators from the standard full extension PV into Q. In the literature there is no shortage of diﬀerent concepts of integrability for scalar-valued functions with respect to vector-valued functions or measures and variations in the resulting deﬁnitions of the integral. The best known are perhaps those by Bochner [19], Pettis [144], Bartle [8], [9] and Dunford and Schwartz [55]. There are also some corresponding diﬀerences in the deﬁnition of measurability. Again, a comprehensive treatment of the relevant deﬁnitions and their implications can be found in Chapter II of Diestel and Uhl [43]. Due to its well-understood properties, the Bochner integral is probably most used in applications. Not surprisingly, our very general approach in this chapter covers many of the above-mentioned notions. This is because we are using locally convex cones in our settings and order convergence for most of our deﬁnitions and results, and order convergence is generally weaker than the originally given topological convergence. Stronger results are pointed out when possible. Since Q, the range of the integrals, is required to be a locally convex complete lattice cone, if applied to the case of a vector space, the results of this chapter often refer to the second dual of this vector space (Section 6.5). This situation is well-understood for vector-valued measures. It would probably be worthwhile, though demanding, to explore the Radon-Nikod´ ym property in the settings of this chapter. This refers to a

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II Measures and Integrals. The General Theory

special case of the Application 6.20 from above. Let P and Q be locally convex cones satisfying our standard assumptions, and let μ be a scalarvalued (positive, real or complex-valued) measure on R. These scalars can be interpreted as linear operators from L(P, Q) into itself. One can therefore integrate certain L(P, Q)-valued functions with respect to μ, and the integral is evaluated in the standard completion L(P, Q) of L(P, Q). Given a suitable function ϕ of this type, this can be used to deﬁne an L(P, Q)valued set function by ' ϕ dμ for E ∈ R. θE = E

The convergence theorems then will guarantee that θ is countably additive and indeed an L(P, Q)-valued measure. Now investigations would have to be carried out, under which conditions a given L(P, Q)-valued measure θ can be expressed in this way using a given scalar-valued measure μ and some L(P, Q)-valued density function ϕ.

Chapter III

Measures on Locally Compact Spaces

This ﬁnal chapter of the book is concerned with topological measure theory and integral representation. The domain X of the operator-valued measure θ will be a locally compact topological space and the σ-ring R will consist of all relatively compact Borel subsets of X. As usual, measures on topological spaces are supposed to fulﬁll certain regularity requirements. Sections 1 and 2 of this chapter will probe diﬀerent notions of continuity for locally convex cone-valued functions on a topological space. Inductive limit-type topologies lead to the identiﬁcation of certain cones of continuous functions. This is motivated by the concept of weighted spaces of continuous real-valued functions which is due to Nachbin [136] and Prolla [155]. Continuous linear operators on such cones of cone-valued functions will be investigated in Section 4. The main result is a generalized Riesz-type integral representation theorem for this type of operators in Section 5. Section 6 contains a long list of special cases and examples, including a generalization of the classical Spectral representation theorem for normal linear operators on a complex Hilbert space. Generally, the notations introduced in Chapters I and II will be used. As usual, for a subset Y of a topological space X, the sets Y , Y ◦ and ∂Y = Y \ Y ◦ denote its topological closure, interior and boundary in X, respectively. The core support of a cone-valued function f on X is the set {x ∈ X | f (x) = 0} and denoted by supp*(f ). Its closure, supp(f ) is the usual support of f. Throughout the following, let (P, V) be a locally convex cone.

1. Relatively Continuous Cone-Valued Functions Due to the possible presence of unbounded elements in a locally convex cone (P, V), continuity for P-valued functions on a topological space X with respect to any of the given topologies of P is a rather restrictive requirement. For example, even if ϕ is a continuous real-valued positive function on X, W. Roth, Operator-Valued Measures and Integrals for Cone-Valued Functions, Lecture Notes in Mathematics 1964, c Springer-Verlag Berlin Heidelberg 2009

249

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III Measures on Locally Compact Spaces

the mapping ϕ⊗a ∈ F(X, P) need not be continuous if the element a is not bounded see Proposition I.4.2(iii) and Remark I.4.38(b) . We shall provide an example for this below. As with our deﬁnition of measurability in Section II.1 we shall therefore use the slightly coarser relative topologies on P which will better suit our purposes. Continuity with respect to the symmetric relative topology, r-continuity for short, turns out to be a suﬃciently generous concept. We shall denote the set of all r-continuous functions in F(X, P) by C r (X, P). Clearly, every P-valued function f ∈ F(X, P) which is continuous with respect to the given symmetric topology of P is also r-continuous, but the reverse does not hold true if the values of f are not bounded in P. However, if f (x) is bounded for x ∈ X, then continuity at x coincides for both topologies, as their neighborhood systems for f (x) ∈ P are equivalent (see Proposition I.4.2(iv)). We observe that a function f ∈ F(X, P) is r-continuous at a point x ∈ X if and only if for every v ∈ V and ε > 0 there is a neighborhood U of x such that f (y) ∈ vε f (z) for all y, z ∈ U. Similarly, we shall say that a function f ∈ F(X, P) is r-lower respectively r-upper continuous at a point x ∈ X if it is continuous with respect to the lower or upper relative topologies. Obviously, r-continuity is the combination of both r-lower and r-upper continuity. Lemma 1.1. The r-lower, the r-upper and the r-continuous functions form subcones of F(X, P). Proof. Clearly, αf is r-lower (r-upper) continuous whenever f is and α ≥ 0. It is however less obvious that these properties are preserved by addition. For this, let f, g ∈ F(X, P) and x ∈ X. For v ∈ V and ε > 0 there is λ ≥ 0 such that both 0 ≤ f (x) + λv and 0 ≤ g(x) + λv. First suppose that both functions f, g ∈ F(X, P) are r-upper continuous at x. We set δ = 1/(2 + 2λ) and choose a neighborhood U of x such that both f (y) ∈ vδ f (x) and g(y) ∈ vδ g(x) for all y ∈ U. Following Lemma I.4.1(d), we conclude that f (y) + g(y) ∈ vε f (x) + g(x) for all y ∈ U, hence the function f + g is indeed r-upper continuous at x. Now for the second case, suppose that both f, g ∈ F(X, P) are r-lower continuous at x. We set δ = min{1, a neighborhood 1/(4 + 2λ)} and choose f (y) , and g(y) ∈ U of x such that both f (y) ∈ f (x) v , that is f (x) ∈ v δ δ g(x) vδ , that is g(x) ∈ vδ g(y) for all y ∈ U. Following Lemma I.4.1(c), this implies 0 ≤ f (y) + (λ + δ)v and 0 ≤ g(y) + (λ + δ)v. Now using Lemma I.4.1(d), as 2δ(1 + λ + δ) ≤ ε, we conclude that f (x) + g(x) ∈ vε f (y) + g(y) for all y ∈ U, hence the function f + g is r-lower continuous at x.

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251

Lemma 1.2. An invertible function f ∈ F(X, P) is r-lower (or r-upper) continuous if and only if its inverse −f ∈ F(X, P) is r-upper (or r-lower) continuous. Proof. Let f ∈ F(X, P) be invertible, and let x ∈ X. As the element f (x) ∈ P is invertible, hence bounded, the notions of continuity and r-continuity coincide for f (see Proposition I.4.2(iv)). We have f (y) ∈ v f (x) , that is f (y) ≤ f (x)+v if and only if −f (x) ≤ −f (y)+v that is −f (x) ∈ v −f (y) . This demonstrates that the function f is upper continuous at x if and only if −f is lower continuous at x, and vice versa. If (P, V) is a locally convex topological vector space over K = R or K = C, then Lemma 1.2 applies to all functions in F(X, P). Moreover, then f (y) ∈ v f (x) , that is f (y) − f (x) ∈ v(0) for a function f ∈ F(X, P) and x, y ∈ X implies that γ (f (y) − f (x) ∈ v(0), that is γf (y) ∈ v γf (x) for all γ ∈ Γ, the unit circle in K. Thus f is upper continuous at x if and only if γf is upper continuous of x for all γ ∈ Γ see I.1.4(d) . This yields: Lemma 1.3. If (P, V) is a locally convex topological vector space over K = R or K = C, then the notions of r-lower, r-upper and r-continuity coincide, and the r-continuous functions form a K-linear subspace of F(X, P). We return to the general case: Lemma 1.4. For an r-lower, r-upper or r-continuous function f ∈ F(X, P) and for v ∈ V, there is an upper semicontinuous, lower semicontinuous or continuous positive real-valued function ϕ on X, respectively, such that f +ϕ⊗v ≥ 0. In particular, if f is r-lower continuous, then for every compact subset K of X and v ∈ V there is λ ≥ 0 such that f (x) + λv ≥ 0 for all x ∈ K. Proof. For a f ∈ F(X, P) and v ∈ V set ψ(x) = inf{λ ≥ 0 | f (x) + λv ≥ 0}. For ﬁxed x, y ∈ X and ε > 0 such that f (x) ∈ vε f (y) we have γf (y) + εv ≥ f (x), hence γf (y) + (2ε + ψ(x))v ≥ 0 for some 1 ≤ γ ≤ 1 + ε. Thus ψ(y) ≤ (1/γ)(ψ(x) + 2ε) ≤ ψ(x) + 2ε. If f is r-lower at x then there is aneighborhood U of x such continuous that f (y) ∈ f (x) vε , that is f (x) ∈ vε f (y) for all y ∈ U. The above shows that the function ψ is upper semicontinuous in this case. A similar argument shows that ψ is lower semicontinuous if f is r-upper continuous. The combination of both yield continuity. Our claim holds with the function ϕ = ψ + 1. Given a neighborhood v ∈ V, in Section I.4 we deﬁned a disjoint partition of P into its v-boundedness components, subsets Bvs (a) that are both open and closed with respect to the symmetric relative topology of P and

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whose elements are v-bounded relative to each other (Proposition I.4.19). The inverse images of these boundedness components under an r-continuous mapping f therefore yield a corresponding disjoint partition of X into both open and closed segments. Proposition 1.5. Let f ∈ C r (X, P). For every v ∈ V there is a disjoint partition of X into segments which are both open and closed and such that the values of f at points in the same segment are v-bounded relative to each other. If the function f ∈ C r (X, P) has a compact support, then a simple compactness argument shows that there can be only ﬁnitely many segments in the above partition of X. Moreover, for every such segment Xi ⊂ X the function fi = χXi ⊗f is also r-continuous, and f may be expressed as the sum of these functions. The values of each fi are v-bounded relative to each other at points in Xi . Similarly, the global boundedness components B s (a) provide a disjoint partition of P into closed (but not necessarily also open) subsets. However, if the range of an r-continuous function f is covered by only ﬁnitely many of these boundedness components, then their inverse images furnish a corresponding ﬁnite disjoint partition of X into closed subsets. Finiteness implies that these sets are also open, hence the statement of Proposition 1.5 also holds for the global boundedness components in this case. The same holds true, if the locally convex cone (P, V) is locally connected in the symmetric relative topology (see Section I.4.20), as its boundedness components are open and closed in this case. Proposition 1.6. Let f ∈ C r (X, P). There is a disjoint partition of X into closed segments such that the values of f at points in the same segment are bounded relative to each other. If either P is locally connected in the symmetric relative topology or if the range of f is covered by ﬁnitely many global boundedness components of P, then these segments are also open. Let us recall that the quasi-component of a point x in a topological space X is the intersection of all closed and open subsets of X which contain the point x. The quasi-components contain the components, that is the maximal connected subsets of X, and constitute a decomposition of X into pairwise disjoint and closed subsets (see VIII.26 in [198] or VI.1 in [59]). In compact spaces, the quasi-components and components coincide (see Theorem 6.1.22 in [59]). In locally connected spaces they coincide and are both open and closed (see Corollary 27.10 in [198]). Hence in locally connected compact spaces they form a ﬁnite partition into disjoint open, closed and connected subsets. Corollary 1.7. Two points x, y ∈ X are contained in the same quasi-component of X if and only if f (x) and f (y) are bounded relative to each other for every r-continuous function f from X into a locally convex cone (P, V).

1. Relatively Continuous Cone-Valued Functions

253

Proof. If the points x, y ∈ X are not contained in the same quasi-component of X, then there is an open and closed subset U of X such that x ∈ U but y ∈ / U. For P = R the function f deﬁned as f (x) = 0 for x ∈ U and f (x) = +∞ else, is in C r (X, R), and the values of f in x and in y are obviously not bounded relative to each other. For the converse, suppose that for x, y ∈ X and f ∈ C r (X, P) for some convex cone (P, V) the function values f (x) and f (y) are not bounded relative to each other, that is f (x) and f (y) are not v-bounded relative to each other for some neighborhood v ∈ V. Following Proposition 1.5 there is an open and closed subset of X containing x but not y. Then x and y are not contained in the same quasi-component of X. Proposition 1.8. Let f = ϕ⊗g ∈ F(X, P) for a positive real-valued continuous function ϕ and g ∈ F(X, P). (a) The function f is r-lower continuous at all points x ∈ X where g is r-lower continuous. (b) The function f is r-upper continuous at all points x ∈ X where g is r-upper continuous and where either ϕ(x) > 0 or g(x) is bounded in P. Proof. For Part (a), suppose that the function g is r-lower continuous at x ∈ X. Let v ∈ V and ε > 0. There is λ ≥ 0 such that 0 ≤ g(x) + λv. We choose any δ > 0. Because g is r-lower continuous and ϕ is r-continuous at x, there is a neighborhood ≤ (1 + δ)ϕ(y), ϕ(y) ≤ U of x such that ϕ(x) ϕ(x) + δ and g(y) ∈ g(x) vδ , that is g(x) ∈ vδ g(y) for all y ∈ U. Then for all z ∈ U this yields with some 1 ≤ γ ≤ 1 + δ ϕ(x) g(x) + λv ≤ (1 + δ)ϕ(y) γg(y) + (λ + δ)v ≤ γ(1 + δ)ϕ(y)g(y) + ϕ(x) + δ (1 + δ)(λ + δ)v. We may choose δ >0 suﬃciently small such that both γ(1 + δ) ≤ (1 + δ)2 ≤ 1 + ε and ϕ(x) + δ (1 + δ)(λ + δ) < λϕ(x) + ε. Then, using the cancellation property for the positive element v (see Lemma I.4.2 in [100]), we conclude that ϕ(x)g(x) ≤ γ(1 + δ)ϕ(z)g(y) + εv, that is f (x) ∈ vε f (y) , that is f (y) ∈ f (x) vε holds for all y ∈ U. Hence f is indeed r-lower continuous at x. Now for Part (b), suppose that g is r-upper continuous at x ∈ X, and for the ﬁrst part of our claim, that ϕ(x) > 0. Let v ∈ V and ε > 0. There is λ ≥ 0 such that 0 ≤ g(x) + λv. We choose any δ > 0. Because g is r-upper continuous and ϕ is continuous at x, and as ϕ(x) > 0, there is a neighborhood U of x such that ϕ(y) ≤ (1 + δ)ϕ(x), ϕ(x) ≤ ϕ(y) + δ and g(y) ∈ vδ g(x) for all y ∈ U. Following Lemma I.4.1(c), the latter implies 0 ≤ g(y) + (λ + δ)v for all y ∈ U. Then for all y ∈ U this yields with some 1≤γ ≤1+δ

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III Measures on Locally Compact Spaces

ϕ(y)g(y) + ϕ(x)(λ + δ)v

≤ ϕ(y)g(y) + ϕ(y) + δ (λ + δ)v ≤ ϕ(y) g(y) + (λ + δ)v + δ(λ + δ)v ≤ (1 + δ)ϕ(x) γg(x) + (λ + 2δ)v + δ(λ + δ)v ≤ γ(1+δ)ϕ(x)g(x) + (1 + δ)(λ + 2δ)ϕ(x) + δ(λ + δ) v.

Again, we may choose δ suﬃciently small such that both γ(1+δ) ≤ (1+δ)2 ≤ 1 + ε and (1 + δ)(λ + 2δ)ϕ(x) + δ(λ + δ) < (λ + δ)ϕ(x) + ε. Then, using the cancellation property for the positive element v we conclude that ϕ(y)g(y) ≤ γ(1 + δ)ϕ(z)g(z) + εv, that is f (y) ∈ vε f (x) holds for all y ∈ U. Hence f is indeed r-upper continuous at x. For the second part of our claim in (b), suppose that g is r-upper continuous at x ∈ X, that ϕ(x) = 0, hence f (x) = 0, and that g(x) is bounded in P, and let v ∈ V. There is λ ≥ 0 such that g(x) ≤ λv. We choose a neighborhood U of x such that 0 ≤ ϕ(y) ≤ 1/(2λ+1) and g(y) ∈ v1 g(x) , for all y ∈ U. With some 1 ≤ γ ≤ 2, the latter means g(y) ≤ γg(x) + v ≤ (γλ + 1)v ≤ (2λ + 1)v. Then ϕ(y)g(y) ≤ ϕ(y)(2λ + 1)v ≤ v, hence f (y) ∈ v f (x) ⊂ vε f (x) for all ε > 0 and y ∈ U. Thus f = ϕ⊗g is indeed r-upper continuous at x. n Proposition 1.9. Let f = i=1 ϕi ⊗gi for positive real-valued continuous functions ϕi and gi ∈ C r (X, P). The function f is r-continuous if and only if for every v ∈ V there is a disjoint partition of X into segments which are both open and closed and such that the values of f at points in the same segment are v-bounded relative to each other. Proof. Let f = ni=1 ϕi ⊗gi be as stated, and for each i ∈ I = {1, . . . , n} let Oi = {x ∈ X | ϕ(x) > 0} be the (open) core support of the positive realvalued function ϕi . If the function f is r-continuous, then Proposition 1.5 yields the existence of a partition of X with the claimed properties for each v ∈ V. For the converse, Proposition 1.8(a) shows that each of the functions ϕi ⊗gi is r-lower continuous, hence following Lemma 1.1, the function f is also r-lower continuous on X. Now let us assume that for every v ∈ V there exists a disjoint partition of X into open and closed segments such that the values of f at points in the same segment are v-bounded relative to each other. We shall proceed to verify that f is r-upper continuous in this case. / ∂Oi }, For this, let x ∈ X, let I1 = {i ∈ I | x ∈ ∂Oi } and I2 = {i ∈ I | x ∈ where the open sets Oi ⊂ X, i = 1, . . . , n, are deﬁned as above and ∂Oi is

1. Relatively Continuous Cone-Valued Functions

the topological boundary of Oi . Set ! ϕi ⊗gi and h1 = i∈I1

255

h2 =

!

ϕi ⊗gi .

i∈I2

Then f = h1 + h2 and h1 (x) = 0. For every i ∈ I2 the function ϕi either vanishes on a neighborhood of x, or ϕi (x) > 0. In both cases, Proposition 1.8(b) shows that the function ϕi ⊗gi is r-upper continuous at x. In turn, Lemma 1.1 yields that the function h2 is also r-upper continuous at x. For the function h1 we continue our argument as follows: Let v ∈ V and ε > 0. By our assumption on f there is a neighborhood O of x such that all values of f on O are v-bounded relative to f (x). Moreover, because the functions gi are r-continuous, following Proposition 1.5 we may assume in addition that the values of gi on O are v-bounded relative to gi (x) for all i ∈ I1 . For every i ∈ I1 , on the other hand, there is an element yi ∈ O such that ϕi (yi ) > 0. Thus f (yi ) =

n !

ϕi (yi )gi (yi ) ∈ Bv f (x)

i=1

components were seen implies that gi (yi ) ∈ Bv f (x) , as the boundedness to be faces in P see Proposition I.4.11(b) . There is λ > 0 such that gi (x) ∈ vλ f (x) for all i ∈ I1 . We may also assume that 0 ≤ f (x) + λv. 1 ε , (1+λ)(3+λ) } and a neighborhood U of x such Now we choose δ = min{ 1+λ that ϕi (y) ≤ δ/n, gi (y) ∈ vδ gi (x) for all i ∈ I1 , and h2 (y) ∈ vδ h2 (x) holds for all y ∈ U. Recall that f (x) = h2 (x) as h1 (x) = 0. Thus the latter implies with Lemma I.4.1(b) h2 (y) ≤ (1 + δ)f (x) + δ(1 + λ)v. Using I.4.1(a) and our assumption that δ(λ + 1) ≤ 1, the former yields gi (y) ∈ v(δ+λ+δλ) f (x) ⊂ v(1+λ) f (x) for all y ∈ U and i ∈ I1 . Hence, again using I.4.1(b) gi (y) ≤ (2 + λ)f (x) + (1 + λ)2 v ≤ (2 + λ) f (x) + (1 + λ) v and therefore, as f (x) + (1 + λ) v ≥ 0 and ϕi (y) ≤ δ/n ! ϕi (y)gi (y) h1 (y) = i∈I1

≤

! i∈I1

ϕi (y)(2 + λ) f (x) + (1 + λ) v

≤ δ(2 + λ) f (x) + (1 + λ) v

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for all y ∈ U. Thus, combining f (y) = h1 (y) + h2 (y) ≤ δ(2 + λ) + (1 + δ) f (x) + δ(2 + λ)(1 + λ) + δ(1 + λ) v ≤ 1 + δ(3 + λ) f (x) + δ(1 + λ)(3 + λ)v. Because 1 ≤ 1 + δ(3 + λ) ≤ 1 + ε and δ(1 + λ)(3 + λ) ≤ ε, this demonstrates f (y) ∈ vε f (x) for all y ∈ U. The function f is therefore r-upper continuous at x, as claimed. 1.10 Elementary Functions. We are particularly interested in elementary functions of the type ϕ⊗a for a positive real-valued continuous function ϕ on X and an element a ∈ P, and in sums of functions of this type. These functions are r-lower continuous by Proposition 1.8(a). If the element a is unbounded, then the inverse image under ϕ⊗a of the boundedness component B s (a) is the set supp*(ϕ) = {x ∈ X | ϕ(x) > 0}, that is the core support of the function ϕ. This set is open since ϕ is continuous and indeed also closed whenever ϕ⊗a is r-continuous, since B s (a) is closed in the symmetric relative topology of X. Because the range of a function f = ni=1 ϕi ⊗ai is covered by the ﬁnitely many global boundedness components B s (aI , ) where aI = i∈I ai and I ⊂ {1, . . . , n}, the second part of Proposition 1.6 applies. In combination with Proposition 1.9, we obtain the following characterization of r-continuous elementary functions: Proposition 1.11. Let ϕ, ϕi be positive real-valued continuous functions on X and let a, ai ∈ P. (a) The elementary function ϕ⊗a is relatively continuous if and only if either the element a ∈ P is bounded or the set {x ∈ X | ϕ(x) > 0} is both open and closed in X. (b) The function f = ni=1 ϕi ⊗ai is r-continuous if and only if there is a disjoint partition of X into segments which are both open and closed and such that the values of f at points in the same segment are bounded relative to each other. Examples 1.12. Let P be the cone of all real-valued continuous functions on R which are uniformly bounded below, endowed with the pointwise algebraic operations and order. With the neighborhood system V consisting of all strictly positive constant functions in P, then (P, V) becomes a full locally convex cone. The function f ∈ P such that f (t) = t2 for t ∈ R is obviously not bounded in P. If we choose X = [0, 1) and the real-valued function

2. Cone-Valued Functions on Locally Compact Spaces

257

ϕ(x) = x on X, then the P-valued function ϕ⊗f is r-continuous on (0, 1) but not at x = 0. Continuity with respect to any of the given locally convex topologies on P fails at all points of X.

2. Cone-Valued Functions on Locally Compact Spaces We shall further specialize the assumptions of the previous section. Throughout the following we shall assume that X is a locally compact Hausdorﬀ space. Let K be the family of all compact subsets of X, and by K0 the subfamily of K consisting of those subsets of X that are both open and compact. We shall denote the cone of all continuous positive real-valued functions on X with compact support by K(X) , and by K0 (X) the subcone of those functions ϕ ∈ K(X) such that supp*(ϕ) ∈ K0 . The latter notion is motivated by the observation in Proposition 1.11. Note that every 0 = ϕ ∈ K0 (X) attains a minimal non-zero value. The functions in K0 (X) may alternatively be characterized as continuous R+ -valued functions with compact support, where continuity is required with respect to the topology of R+ as introduced in Example I.1.4(b). Throughout the following, let R be the weak σ-ring of all relatively compact Borel subsets of X. Then the corresponding σ-algebra AR , consisting of all sets A ⊂ X such that A ∩ E ∈ R for all E ∈ R, is just the σ-algebra of all Borel subsets of X (see Lemma 13.9 in [178]). As before, (P, V) is a locally convex cone without any further requirements. 2.1 Inductive Limit Topologies. Recall from Section II.2.2 that an Rcompatible inductive limit neighborhood is a convex subset v of measurable functions in F X, V such that for every E ∈ R there is vE ∈ V such that χE ⊗vE ≤ v, that is χK ⊗v ≤ s for some s ∈ v. An inductive limit topology on F(X, P) is generated by a system V of R-compatible inductive limit neighborhoods, closed for addition and multiplication by strictly positive scalars and directed downward with respect to the order relation ≤ as deﬁned in II.2.2. In this case R-compatibility means that for every compact set K we have χK ⊗v ≤ v for some v ∈ V. An R-compatible inductive limit neighborhood v is called r-lower, r-upper or r-continuous if all its elements s ∈ v are r-lower, r-upper or r-continuous V-valued functions, respectively. (Recall that (V, V) itself is also a locally convex cone.) Correspondingly, an inductive limit topology V on F(X, P) is called r-lower, r-upper or r-continuous if it contains a basis of r-lower, r-upper or r-continuous neighborhoods, respectively. For our further investigations, r-lower continuous topologies will be of particular interest. Considering this, we observe that for a non-negative R-valued lower semicontinuous function ϕ and some v ∈ V, the neighborhood-valued function ϕ⊗v is r-lower continuous

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III Measures on Locally Compact Spaces

(see 1.8(a)). In this context, 0 · v = 0 and +∞ · v = ∞. (Recall from II.2.1 that V contains 0 ∈ P as well as the maximal element ∞). We record a wide variety of r-lower continuous inductive limit topologies, the ﬁnest of which is the standard inductive limit topology for functions on a locally compact space, consisting of all R-compatible inductive limit neighborhoods. This topology is r-lower continuous, as for every inductive limit neighborhood v there is an r-lower continuous neighborhood u such that u ≤ v. Indeed, for every relatively compact open set O ⊂ X choose vO ∈ V such that χO ⊗vO ≤ s for some s ∈ v, and let u consist of all convex combinations of the functions χO ⊗vO . If, for another example, the elements of V are just the singleton sets containing the constant mappings x → v for v ∈ V, then V generates the topology of uniform convergence. If, on the other hand, these singleton sets consist of mappings x → v for x ∈ K and x → ∞ else, for some K ∈ K, then we obtain the topology of compact convergence. If we use ﬁnite sets instead of compact ones in the last example, the topology of pointwise convergence emerges. All these topologies are obviously r-lower continuous. (See also Examples 2.11 below.) 2.2 Functions that Vanish at Inﬁnity. Given an inductive limit topology, we shall say that a function f ∈ F(X, P) vanishes at inﬁnity (relative to a system V of inductive limit neighborhoods) if for every v ∈ V there is K ∈ K such that χ(X\K) ⊗f ≤ v

and

0 ≤ χ(X\K) ⊗f + v.

Lemma 1.4 shows that every r-continuous function f that vanishes at inﬁnity is bounded below relative to the neighborhoods in V. Indeed, for v ∈ V, choose the compact set K ⊂ X from above. There is v ∈ V such that χK ⊗v ≤ v. Following 1.4 there is λ ≥ 0 such that 0 ≤ f (x) + λv for all x ∈ K, that is 0 ≤ χK ⊗f + λv. This demonstrates 0 ≤ χ(X\K) ⊗f + v + χK ⊗f + λv = f + (λ + 1)v. 2.3 The Cones E(X, P) and E0 (X, P). We shall use E(X, P) to denote the subcone of F(X, P) consisting of all ﬁnite sums of elementary functions ϕ⊗a, deﬁned by an element a ∈ P and a continuous positive real-valued function ϕ ∈ K(X). Recall that for a function ϕ⊗a ∈ E(X, P) to be r-continuous, it is necessary and suﬃcient that either the element a ∈ P is bounded or that ϕ ∈ K0 (X). We shall denote the subcone of E(X, P) generated by all rcontinuous elementary functions by E0 (X, P). Note that E0 (X, P) does not necessarily contain all r-continuous functions in E(X, P), which had been characterized in Proposition 1.11(b). 2.4 The Cones FV (X, P) and FV0 (X, P). We denote the closure of E(X, P) in F(X, P) with respect to the symmetric relative topology generated by the inductive limit neighborhoods in V, by FV (X, P). It consists of those functions f ∈ F(X, P) such that for every v ∈ V and ε > 0 there

2. Cone-Valued Functions on Locally Compact Spaces

259

is some g ∈ E(X, P) such that f ≤ γg + εv

and

g ≤ γ f + εv

for some 1 ≤ γ, γ ≤ 1 + ε; that is f ∈ vsε (g). Recall from Section 2 that the relative topologies of a locally convex cone are in general no longer locally convex cone topologies. These topologies are however still compatible with the addition and multiplication by positive scalars as shown in Lemma I.4.1(d). The closure FV (X, P) of the subcone E(X, P) is therefore again a subcone of F(X, P). Moreover, since all functions in FV (X, P) obviously vanish at inﬁnity, hence are bounded below relative to the neighborhoods in V, we infer that FV (X, P), V forms a locally convex cone. Similarly, we shall denote the closure (with respect to the same topology) of E0 (X, P) in F(X, P) by FV0 (X, P). r r 2.5 The Cones CV (X, P) and CV (X, P). We are particularly inter0 ested in the respective subcones of r-continuous functions in FV (X, P) and r r (X, P) and CV (X, P), respecFV0 (X, P) which we shall denote by CV 0 tively. A complete characterization of their elements appears to be diﬃcult to achieve, due to the generality of our approach and the wide variety of choices for inductive limit topologies. The following Theorem 2.6, however, provides a rather general and powerful criterion for an r-continuous function to belong to either of these cones. Recall that B denotes the subcone of all bounded elements of P.

Theorem 2.6. Let f ∈ C r (X, P) and let S = {x ∈ X | f (x) ∈ B}. (a) If for every v ∈ V there is K ∈ K such that ∂K ⊂ S ◦ and such that r (X, P). χ(X\K) ⊗f ≤ v and 0 ≤ χ(X\K) ⊗f + v, then f ∈ CV (b) If for every v ∈ V there are K ∈ K and a subset K0 ∈ K0 of K such that K \ K0 ⊂ S ◦ , and such that χ(X\K) ⊗f ≤ v and 0 ≤ χ(X\K) ⊗f + v, r (X, P). then f ∈ CV 0 Proof. We shall proceed in several steps. First we observe that the condition of Part (b) implies the condition of Part (a). Indeed, K0 ⊂ K and K0 ∈ K0 implies that K0 ⊂ K ◦ , hence ∂K = K \ K ◦ ⊂ K \ K0 ⊂ S ◦ , as required in (a). Next we consider a few special cases for functions in C r (X, P). (i) In a ﬁrst step we assume that the function f ∈ C r (X, P) has a compact support K ∈ K. Let v ∈ V and ε > 0. Let U be a relatively compact open set containing K and let v ∈ V such that χU ⊗v ≤ v. Following Lemma 1.4 there is λ ≥ 0 such that 0 ≤ f (x) + λv for all x ∈ X. We proε ceed to employ a partition of the unit for the compact set K. Let ε = 1+λ and γ = 1+ε . As f is r-continuous, there are open subsets O1 , . . . , On ∈ R

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of U whose union covers K, and such that f (y) ∈ (vε ) f (x) , that is (using Lemma I.4.1(b)) f (y) ≤ γf (x) + εv whenever x, y ∈ Oi for some i = 1, . . . , n. There is a corresponding set of functions ϕ1 , . . . , ϕn ∈ K(X) such that supp(ϕi ) ⊂ Oi , and for ϕ = ni=1 ϕi we have 0 ≤ ϕ(x) ≤ 1 for all x ∈ X, and ϕ(x) = 1 for all x ∈ K. We choose ai = f (xi ) for some xi ∈ Oi . All the functions ϕi ⊗ai are in E(X, P), and if all the values of f are bounded in P, then these functions are even contained in E0 (X, P). We observe that ϕi (x)ai ≤ ϕi (x)γf (x) + ϕi (x)εv

and

ϕi (x)f (x) ≤ ϕi (x)γai + ϕi (x)εv

holds for all x ∈ X and set g = ni=1 ϕi ⊗ai ∈ E(X, P). If in addition all the values of f are bounded, then g is contained in E0 (X, P). The above yields g(x) ≤ ϕ(x)γf (x) + ϕ(x)εv

and

ϕ(x)f (x) ≤ γg(x) + ϕ(x)εv

and

f (x) ≤ γg(x) + εv

for all x ∈ X, hence g(x) ≤ γf (x) + εv

for all x ∈ K. For x ∈ U \ K we have f (x) = 0 and 0 ≤ ϕ(x) ≤ 1, hence by the above g(x) ≤ εv = γf (x) + εv

and

f (x) = 0 ≤ γg(x) + εv.

As f (x) = g(x) = 0 for all x ∈ / U, this yields g ≤ γf + ε χU ⊗v ≤ γf + εv

and

f ≤ γg + ε χU ⊗v ≤ γg + εv,

thus f ∈ vε (g) and g ∈ vε (f ). As g ∈ E(X, P) and as v ∈ V and ε > 0 r (X, P). were arbitrarily chosen, this yields f ∈ CV r (ii) If the f ∈ C (X, P) has a compact support and if in addition all of its values are bounded, then the function g from step (i) was indeed seen to r (X, P). be an element of E0 (X, P). This yields f ∈ CV 0 (iii) In the next case, let K ∈ K0 and v ∈ V, and let f ∈ C r (X, P) be a function whose support is contained in K and whose values at points in K are v-bounded relative to each other. Let 0 < ε ≤ 1. A simple compactness argument shows that we can ﬁnd λ ≥ 0 such that f (x) ∈ vλ f (y) for all x, y ∈ K. Following Lemma 1.4 we may also assume that 0 ≤ f (x) + λv holds for all x ∈ K. By Lemma I.4.1(b) this implies f (x) ≤ (1 + λ) f (y) + λv ε for all x, y ∈ K. We set ε = 2(λ+1) , γ = 1 + ε and as in (i) choose an open cover by subsets O1 , . . . , On of K such that f (y) ∈ vε f (x) , that is

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261

following Lemma I.4.1(b) ε f (y) ≤ (1 + ε )f (x) + ε (1 + λ)v = γf (x) + v. 2 whenever x, y ∈ Oi for some i = 1, . . . , n. Let ϕ1 , . . . , ϕn ∈ K(X) be the corresponding partition of the unit, that is supp(ϕi ) ⊂ Oi , and for ϕ = ni=1 ϕi we have 0 ≤ ϕ(x) ≤ 1 for all x ∈ X, and ϕ(x) = 1 for all x ∈ K. We choose ai = f (xi ) for some xi ∈ Oi . But as the elements ai are not necessarily bounded in P, the functions ϕi ⊗ai need not be r-continuous at points x where ϕi (x) = 0. We therefore use the functions ψi = ϕi + ε . Because supp*(ψi ) = K ∈ K0 , we have ε χK instead, where ε = 2n(1+λ) ψi ∈ K0 (X), hence the functions ψi ⊗ai are r-continuous and contained in E0 (X, P). For each i = 1, . . . , n we observe that ε ai ≤ γf (x) + v 2

and

ε f (x) ≤ γai + v 2

for all x ∈ Oi and ai ≤ (1 + λ)(f (x) + λv)

and

0 ≤ ai + λv

for all x ∈ K. Therefore, as supp(ϕi ) ⊂ Oi , we have ε ε f (x) + λv ψi (x)ai = ϕi (x)ai + ε ai ≤ ϕi (x) γf (x) + v + 2 2n and

ε ε ϕi (x)f (x) ≤ ϕi (x) γai + v +γε ai +λv = γψi (x)a+ ϕi (x) + γλε v 2 2 n for all nx ∈ K. Next we consider the function g = i=1 ψi ⊗ai ∈ E0 (X, P). As i=1 ϕi (x) = 1 for all x ∈ K, we obtain from the above ε ε ε λ + + nγλε v g(x) ≤ (γ +ε )f (x)+ v and f (x) ≤ γg(x)+ 2 2 2 for all x ∈ K, and even for all x ∈ X, as f (x) = g(x) = 0 for all x ∈ X \ K. Next we observe that both ε λ ε ≤ 2 2

and

nγλε =

γε γλε ε ≤ ≤ ε ≤ . 2(1 + λ) 2 2

With 1 ≤ γ = γ + ε = 1 + 2ε ≤ 1 + ε, this shows g ≤ γ f + ε χK ⊗v

and

f ≤ γg + ε χK ⊗v.

(iv) Now let K ∈ K0 and let f ∈ C r (X, P) be a function whose support is contained in K. Let v ∈ V and let 0 < ε ≤ 1. There is v ∈ V such

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that χK ⊗v ≤ v, and following the remark after Proposition 1.5, f can be expressed as the sum of ﬁnitely many functions fi ∈ C r (X, P), for i = 1, . . . , n, whose support is contained in subsets Ki ∈ K0 of K respectively, and such that the values of fi at points in Ki are v-bounded relative to each other. Let λ ≥ 0 such that 0 ≤ fi + λv for all i = 1, . . . , n. Now we apply step (iii) to each of the functions fi with the neighborhood v and ε instead of ε : We ﬁnd gi ∈ E0 (X, P) such that ε = n(1+λ) gi ≤ γi fi + ε χKi ⊗v

and

fi ≤ γi gi + ε χKi ⊗v

with 1 ≤ γi , γi ≤ 1 + ε , that is gi ∈ vε (fi ) and fi ∈ vε (gi ) for all i = 1, . . . , n. We set g = g1 + . . . + gn ∈ E0 (X, P). Then Lemma I.4.1(d) yields indeed g ∈ vε (f ) and f ∈ vε (g). Because this may be obtained for r (X, P). any choice of v ∈ V and ε ≥ 0, we conclude that f ∈ CV 0 r (v) Now we consider the case of a function f ∈ C (X, P) that fulﬁlls the assumptions of Part (a). Let v ∈ V and ε > 0. There is λ ≥ 0 such ε and corresponding to the neighborhood that 0 ≤ f + λv. We set ε = 2(1+λ) ε v choose the set K ∈ K with the assumed properties. There is a relatively compact open set O ⊂ S containing ∂K. Then K is contained in the relatively compact open set U = O ∪ K ◦ . Thus we ﬁnd a continuous realvalued function ϕ on X such that 0 ≤ ϕ(x) ≤ 1 for all x ∈ X, such that the support of ϕ is contained in U and ϕ(x) = 1 for all x ∈ K. We set f1 = ϕ⊗f and f2 = (1 − ϕ)⊗f and use Proposition 1.8 to verify that both functions are r-continuous. Indeed, f1 is r-continuous at all points x ∈ X where ϕ(x) > 0 or f (x) is bounded. If ϕ(x) = 0 and f (x) is unbounded, then we have neither x ∈ K nor x ∈ O. Thus x ∈ / U, hence x ∈ / supp(ϕ). Thus ϕ(y) = 0 throughout a neighborhood of x, and f1 is continuous at x as well. Similarly, the function f2 is r-continuous at all points x ∈ X where ϕ(x) < 1 or f (x) is bounded. But ϕ(x) = 1 implies that x ∈ U, and if f (x) is unbounded, even x ∈ K ◦ . Therefore ϕ(y) = 1 throughout a neighborhood of x, and f2 is continuous at x as well. Using this, we observe that the function f1 is r-continuous and has a compact support. r (X, P), as shown in step (i). Thus there is It is therefore contained in CV g ∈ E(X, P) such that f1 ∈ vsε (g). The function f2 , on the other hand, vanishes on K, hence f2 ≤ ε v and 0 ≤ f2 + ε v, that is f2 ∈ vsε (0) by our assumption. Finally, as f = f1 + f2 , Lemma I.4.1(d) yields f ∈ vs2ε (1+λ) (g) = vsε (g). r Because v ∈ V and ε > 0 were arbitrary, this shows f ∈ CV (X, P). r (vi) Finally, we consider the case of a function f ∈ C (X, P) that fulﬁlls the assumptions of Part (b). We shall adapt some of the arguments from step (v). Let v ∈ V and ε > 0. There is λ ≥ 0 such that 0 ≤ f +λv. We set ε and corresponding to the neighborhood ε v choose the set K ∈ K ε = 3(1+λ) and the subset K0 ∈ K0 of K with the assumed properties. Because K0 is

2. Cone-Valued Functions on Locally Compact Spaces

263

both open and closed, both functions f1 = χ(X\K0 ) ⊗f and f2 = χK0 ⊗f are rr (X, P) continuous. The support of f2 is contained in K0 ∈ K0 , thus f2 ∈ CV 0 as was shown in step (iv). For the function f1 we shall argue as follows: There is a relatively compact open set U in S containing the compact set K = K \K0 . Thus we ﬁnd ϕ ∈ K(X) such that 0 ≤ ϕ(x) ≤ 1 for all x ∈ X, the support of ϕ is contained in U and ϕ(x) = 1 for all x ∈ K . We set f1 = ϕ⊗f1 and f1 = (1 − ϕ)⊗f1 and as in step (v) use Proposition 1.8 to verify that both functions are r-continuous. The function f1 is r-continuous, has bounded values and compact support and following step (ii) is therefore r (X, P). The function f1 vanishes on K and on K0 , hence contained in CV 0 on K, and f1 ≤ ε v and 0 ≤ f1 + ε v, that is f2 ∈ vsε (0) holds by our r (X, P), there are g1 , g2 ∈ E0 (X, P) such that assumption. As f1 , f2 ∈ CV 0 f1 ∈ vsε (g1 )

and

f2 ∈ vsε (g2 ).

Finally, as f = f1 + f1 + f2 , Lemma I.4.1(d) yields with g = g1 + g2 ∈ E0 (X, P) f ∈ vsε 3(1+λ) (g) = vsε (g). r Because v ∈ V and ε > 0 were arbitrary, this shows f ∈ CV (X, P). 0

Corollary 2.7. Suppose that the function f inﬁnity.

∈ C r (X, P) vanishes at

(a) If there is K ∈ K such that f takes only bounded values on X \ K, r then f ∈ CV (X, P). (b) If there is K0 ∈ K0 such that f takes only bounded values on X \ K0 , r (X, P). then f ∈ CV 0 Proof. (a) Let f ∈ C r (X, P) and K ∈ K be as stated, that is X \ K ⊂ S = {x ∈ X | f (x) ∈ B}. For the criterion in Theorem 2.6(a), given v ∈ V, as the function f vanishes at inﬁnity, there is K1 ∈ K such that χ(X\K1 ) ⊗f ≤ v and 0 ≤ χ(X\K1 ) ⊗f +v, Indeed, we may choose K1 such that K is contained in its interior K1◦ . As X \ K ⊂ S ◦ in this case, we have ∂K1 = K1 \ K1◦ ⊂ X \ K ⊂ S ◦ as required for the set K1 ∈ K in 2.6(a). (b) For the criterion of Theorem 2.6(b) we have X \ K0 ⊂ S ◦ and may choose the set K ∈ K from the criterion such that K0 ⊂ K. Then K \ K0 ⊂ S ◦ as required. In some special cases the criteria of Theorem 2.6 and Corollary 2.7 can be simpliﬁed or are even necessary for a function f ∈ C r (X, P) to belong to r r (X, P) or CV (X, P), respectively. CV 0 Corollary 2.8. If either (P, V) is locally connected in the symmetric relative topology or if the range of the function f ∈ C r (X, P) is covered by ﬁnitely many global boundedness components of P, then the conditions on f in 2.7(a) and 2.7(b) are equivalent.

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III Measures on Locally Compact Spaces

Proof. In this case, Proposition 1.6 provides a partition of X into open and closed components such that the values of f at points in the same component are globally bounded relative to each other. In particular, the set S = {x ∈ X | f (x) ∈ B} is both open and closed in X. Condition 2.7(a) states that its complement K0 = X \ S is relatively compact, hence in K0 , since it is both open and closed. Thus the criterion in 2.7(b) holds as well. We shall say that an inductive limit neighborhood v yields boundedness at inﬁnity if f ≤ g + v for functions f, g ∈ F(X, P) implies that there is K ∈ K such that f (x) ∈ B g(x) holds for all x ∈ X \ K. Corollary 2.9. If V contains a neighborhood which yields boundedness at inﬁnity, then the condition in 2.7(b) is also necessary for a function to belong r (X, P). If in addition, (P, V) is locally connected in the symmetric to CV r r (X, P) and CV (X, P) coincide. relative topology, then CV 0 Proof. Suppose that V contains a neighborhood v0 which yields boundedr ness at inﬁnity. We shall verify that every f ∈ CV (X, P) satisﬁes 2.7(b). Obviously, f is continuous and vanishes at inﬁnity. For the neighborhood v0 ∈ V and ε = 1 there is a function g ∈ E(X, P) such that f ∈ (v0 )sε (g), that is and g ≤ γ f + v0 f ≤ γg + v0 for some 1 ≤ γ, γ ≤ 2. Thus there is K ∈ K such that f (x) ∈ B g(x) and g(x) ∈ B f (x) for all x ∈ X \ K. The function g ∈ E(X, P) has a compact support Kg ∈ K. From this we infer that f takes only bounded values on X \ (K ∪ Kg ) and therefore satisﬁes 2.7(a). If in addition, (P, V) is locally connected in the symmetric relative topology, then Conditions 2.7(a) and 2.7(b) are equivalent by 2.8 and imply that a function f ∈ C r (X, P) that vanishes at inﬁnity and fulﬁlls these criteria r r r (X, P) ⊂ CV (X, P). Every f ∈ CV (X, P), on the other is contained in CV 0 hand satisﬁes 2.7(a) by our preceding argument. We shall consider some further special cases in the following remarks and examples. Remarks 2.10. (a) If all elements of the locally convex cone (P, V) are r (X, P) and bounded, then we infer from Corollary 2.7 that the cones CV 0 r (X, P) coincide and consist of all r-continuous (hence continuous) funcCV tions that vanish at inﬁnity. (b) Proposition 1.9 and Corollary 2.7(b) imply in particular that all rr (X, P). Indeed, for an continuous functions in E(X, P) are contained in CV 0 / B} is r-continuous function f ∈ E(X, P), the set K0 = {x ∈ X | f (x) ∈ both open and closed by 1.9. As f has a compact support, that is K0 ∈ K0 , r (X, P). Corollary 2.7(b) yields f ∈ CV 0 r r (X, P) = CV (X, P) = (c) If the space X is indeed compact, then CV 0 r C (X, P). Indeed, in Theorem 2.6(b) we may choose K = K0 = X, and our criterion holds trivially for every f ∈ C r (X, P).

2. Cone-Valued Functions on Locally Compact Spaces

265

(d) If the locally compact space X is not compact but connected, then the criterion in Theorem 2.6(b) requires that K0 = ∅, that is K ⊂ S. Following Corollary 1.7 a function f ∈ C r (X, P) therefore fulﬁlls this criterion if it vanishes at inﬁnity and all of its values are bounded elements of P. Depending on the choice for the inductive limit topology V, this requirement r (X, P). may however not be a necessary for f to belong to CV (e) If X is a discrete space (for example X = N), then we may choose r r (X, P) and CV (X, P) thereK0 = K in the criterion of Theorem 2.6(b). CV 0 r fore coincide and consist of all functions in C (X, P) that vanish at inﬁnity. Examples 2.11. (a) We obtain the ﬁnest inductive limit topology if V consists of all R-compatible inductive limit neighborhoods. This topology was seen to be r-lower continuous (see 2.1). Theorem 2.6 provides suﬃcient criter r (X, P) or CV (X, P). If the locally compact ria for a function to belong to CV 0 space X is σ-compact, that is the union of a sequence of compact sets, then r (X, P) in this we can obtain a (nearly) complete description of the cone CV case: It follows from 2.7(a) that every r-continuous function with compact r (X, P). On the other hand we shall verify that for support is contained in CV r every f ∈ CV (X, P) there is K ∈ K such that f (x) ≈ 0, that is f (x) 0 and 0 f (x), holds for all x ∈ X \K. Indeed, a simple argument (see Proposition 1, Section 8.5 in [178]) shows that there is an increasing sequence of relatively compact open subsets Yn of X such that X = ∞ i=1 Yn . Let us assume to the contrary of our claim that there is a sequence xn ∈ Yn+1 \ Yn such that f (xn ) ≈ 0. Then there is a corresponding sequence (vn )n∈N in V such that either f (xn ) vn or 0 f (xn ) + vn . Now we consider the inductive limit neighborhood v = {s} containing the single function s such that s(x) = vn for x ∈ Yn+1 \ Yn . Because every compact subset K of X is contained in the union of ﬁnitely many of the open sets Yi , hence χ(X\K) ⊗f ≤ v and 0 ≤ χ(X\K) ⊗f + v can not hold for any K ∈ K. Thus f r (X, P). does not vanish at inﬁnity and is therefore not contained in CV (b) The topology of uniform convergence is generated by the neighborhood system V whose elements are just the singleton sets v(v) , corresponding to a neighborhood v ∈ V, and containing the constant mapping x → v for v ∈ V. (c) The topology of compact convergence is generated by the neighborhood system V whose elements are all singleton sets v(v,K) , corresponding to a neighborhood v ∈ V and a compact subset K ⊂ X, and containing the mapping x → v for x ∈ K and x → ∞ else. This topology was also seen to be r-lower continuous. (d) The topology of pointwise convergence is generated by a neighborhood system V whose elements are all singleton sets v(v,Y ) , corresponding to a neighborhood v ∈ V and a ﬁnite subset Y ⊂ X, and containing the mapping x → v for x ∈ Y and x → ∞ else. It is r-lower continuous. In this case r (X, P) = C r (X, P), as for a given neighborhood v ∈ V corresponding to CV a ﬁnite subset Y ⊂ X we may choose K = K0 = Y ∈ K0 in the criterion of Theorem 2.6(b).

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(e) A family Ω of non-negative real-valued upper semicontinuous functions on X is called a family of weights (see [136] and [155]) if for all ω1 , ω2 ∈ Ω there are ω3 ∈ Ω and ρ > 0 such that ω1 ≤ ρ ω3 and ω2 ≤ ρ ω3 . We obtain an r-lower continuous inductive limit topology in the sense of 2.1 in the followingway: For v ∈ V and ω ∈ Ω we set sω,v (x) = ∞ if ω(x) = 0 and sω,v (x) = 1/ω(x) v else. Thus VΩ = vω,v = {sω,v } | v ∈ V, ω ∈ Ω forms a basis for an inductive limit topology. For functions f, g ∈ F(X, P) we have f ≤ g + vω,v

if

ω(x)f (x) ≤ ω(x)g(x) + v

for all x ∈ X.

The family of weight functions may be chosen in a way which yields the preceding Examples 2.11(b), (c) and (d). Further examples can be found in [155]. (f) Let X = N with the discrete topology and P = R. For 1 ≤ p ≤ ∞, let the inductive limit neighborhood vp consist of all positive real-valued sequences (αn )n∈N in the unit ball of the sequence space lp . Let V = {εvp | r r (N, R) = CV (N, R) consists of all sequences (xn )n∈N in ε > 0}. Then CV 0 R such that for every ε > 0 there is n0 ∈ N such that |xn | ≤ αn for all n ≥ n0 with some sequence (αn )n∈N ∈ εvp . The latter means that ∞ p (1/p) ≤ ε for p < ∞ and supn≥n0 |xn | ≤ ε for p = ∞. Apart n0 |xn | from taking the value +∞ ﬁnitely many times, (xn )n∈N itself is therefore a sequence in lp .

3. Continuous Linear Operators on Cones of Functions As before, let X be a locally compact Hausdorﬀ space, (P, V) a locally convex cone, and let V be a system of inductive limit neighborhoods for F(X, P) as introduced in the preceding section. Let (Q, W) be a locally convex complete lattice cone. We shall proceed to investigate continuous linear operators deﬁned on (FV (X, P), V) or any of the subcones introduced in Section 2, into (Q, W). We are particularly interested in extensions of these operators. If in addition (Q, W) is a full cone, if all elements of Q other than +∞ are invertible, and W = consists of the (strictly) positive multiples of a single neighborhood w ∈ W, then a Hahn-Banach type argument (see Theorem I.5.55) yields that every continuous linear operator on a subcone of FV (X, P) can be extended to a continuous linear operator on FV (X, P). For more general ranges (Q, W), however, this need no longer be the case. We proceed to investigate a few further cases were such extensions are possible. First, as an immediate consequence of Theorem I.5.56 we obtain: Theorem 3.1. Let V be a system of inductive limit neighborhoods for F(X, P).

3. Continuous Linear Operators on Cones of Functions

267

(a) Every continuous linear operator T : E0 (X, P) → Q can be uniquely extended to FV0 (X, P). (b) Every continuous linear operator T : E(X, P) → Q can be uniquely extended to FV (X, P). Our following extension results are however less obvious. We shall have to impose some additional requirements on the locally convex cone (P, V) and the inductive limit neighborhood system V. For our ﬁrst result we need to require that the locally convex cone (P, V) is both quasi-full and has uniform boundedness components (see Section I.4.23). Both requirements can be combined into the following conditions: (QU1) a ≤ b + v for a, b ∈ P and v ∈ V if and only if a ≤ b + s for some bounded element s ∈ P such that s ≤ v. (QU2) a ≤ u + v for a ∈ P and u, v ∈ V if and only if a ≤ s + t for some bounded elements s, t ∈ P such that s ≤ u and t ≤ v. Obviously, every (ordered) locally convex topological vector space satisﬁes (QU1) and (QU2); likewise, every full locally convex cone whose neighborhood system consists only of the positive multiples of a single neighborhood. Theorem 3.2. Suppose that (P, V) is quasi-full and has uniform boundedness components. Let V be a system of inductive limit neighborhoods such that for all v ∈ V and every s ∈ v the set Ks = {x ∈ X | s(x) = ∞} is contained in K0 . Then every continuous linear operator T : E0 (X, P) → Q can be extended to E(X, P). Proof. Following Theorem 3.1, the operator T can be extended to a continr uous linear operator on CV (X, P), which by Corollaries 2.7 to 2.9 together 0 r (X, P) in this case, and consists with Proposition I.4.22 coincides with CV of all r-continuous functions in F(X, P) that vanish at inﬁnity and take unbounded values only on a compact subset of X. For any neighborhood w ∈ W there is some uw ∈ V such that T (f ) ≤ T (g) + w holds whenever r (X, P). In a ﬁrst step, for a function f ∈ E(X, P) f ≤ g + uw for f, g ∈ CV 0 we set r (i) T (f ) = inf T (f + g) | 0 ≤ g ∈ E(X, P) such that f + g ∈ CV (X, P) . 0 r For f ∈ E(X, P) ∩ CV (X, P), this deﬁnition is of course consistent with 0 the given value of T. As usual, the inﬁmum over the empty set, is set to be ∞ ∈ Q. On the other hand, if for a function f ∈ E(X, P) there is r (X, P), then the above inﬁmum exists 0 ≤ g ∈ E(X, P) such that f +g ∈ CV 0 as the concerned subset of Q is bounded below. Indeed, for f ∈ E(X, P) and every w ∈ W we ﬁnd λ ≥ 0 such that 0 ≤ f +λuw , hence 0 ≤ (f +g)+λuw r for every 0 ≤ g ∈ E(X, P) such that f + g ∈ CV (X, P). This implies 0 0 ≤ T (f + g) + w. All left to show is that this extension of the operator T to the cone E(X, P) is again linear and continuous. For this, T (αf ) = αT (f ) and T (f + g) ≤ T (f ) + T (g) for f, g ∈ E(X, P) and α ≥ 0 are evident from

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our deﬁnition. The reverse of the latter inequality is however less obvious. We shall proceed as follows: (ii) Proposition 1.9 states that a function g = ni=1 ϕi ⊗ai ∈ E(X, P) is r (X, P) if and only if the inverse images under g r-continuous, hence in CV 0 of all boundedness components of P are both open and closed subsets of X. Because these inverse images are determined by the core supports of the functions ϕi , it is evident that the above criterion remains valid if we replace the functions ϕi ∈ K(X) by strictly positive multiples; precisely: If the ai ∈ E(X, P) is r-continuous, and if λ1 , . . . , λn > 0, function g = ni=1 ϕi ⊗ then the function g = ni=1 λi ϕi ⊗ai ∈ E(X, P) is also r-continuous. We shall make use of this observation in the following. (iii) Now let f = ni=1 ϕi ⊗ai ∈ E(X, P). Given w ∈ W and a compact set K which contains the support of f, there is v ∈ V such that χK ⊗v ≤ uw 1 and λ ≥ 0 such that 0 ≤ ai + λ 2 v for all i = 1, . . . , n. We may also 1 assume that 0 ≤ ϕi (x) ≤ λ 2 /n holds for all x ∈ X and i = 1, . . . , n. By our assumption on P then we can ﬁnd elements si ∈ P such that 1 si ≤ λ 2 v and ai +si ≥ 0 for all i = 1, . . . , n. As P has uniform boundedness components, the elements si are even (globally) bounded, hence the function f = ni=1 ϕi ⊗si is in E0 (X, P). Furthermore, f + f ≥ 0, and as 1

ϕi (x)si ≤ ϕi (x)λ 2 v ≤

λ v n

holds for all x ∈ X and i = 1, . . . , n, we have f ≤ λv, hence T (f ) ≤ λw. (iv) Now let f, g ∈ E(X, P) and 0 ≤ h ∈ E(X, P) such that (f +g)+h ∈ r (X, P). Given w ∈ W choose functions f , g ∈ E0 (X, P) as in (iii), that CV 0 is (f + f ), (g + g ) ≥ 0 and T (f ), T (g ) ≤ λv for some λ ≥ 0. Then r (X, P) as well, and likewise f + ε(g + g + h) ∈ f + (g + g + h) ∈ CV 0 r (X, P) for every 0 < ε ≤ 1/2, as we had observed in step (ii). Because CV 0 0 ≤ ε(g + g + h), this yields T (f ) ≤ T f + ε(g + g + h) by (i). Similarly, we obtain T (g) ≤ T g + ε(f + f + h) . r As the operator T is linear on the cone CV (X, P), and as all the functions 0 f + ε(g + g + h), g + ε(f + f + h), (1 + ε)(f + g) + 2εh and f + g are r (X, P), adding the preceding inequalities yields in CV 0

T (f ) + T (g) ≤ T f + ε(g + g + h) + T g + ε(f + f + h) = T (1 + ε)(f + g) + ε(f + g ) + 2εh ≤ T (1 + ε)(f + g) + 2εh + εT (f + g ) ≤ (1 + ε)T (f + g) + h + ελw.

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269

Because both w ∈ W and 0 < ε ≤ 1 were arbitrarily chosen, we infer that T (f ) + T (g) ≤ T (f + g) + h . Now the deﬁnition of T (f + g) in (i) yields T (f ) + T (g) ≤ T f + g), hence together with the above, the linearity of the operator T on E(X, P). (v) All left to show is the continuity of T. For this, let w ∈ W and f, g ∈ E(X, P) such that f ≤ g + uw , that is f ≤ g + s for some s ∈ v. By our assumption on the neighborhood system V, the set Ks = {x ∈ X | s(x) = ∞} is contained in K0 . In a ﬁrst case, suppose that f (x) = g(x) = 0 for all x ∈ Ks . Let λ ≥ 0 such that 0 ≤ T (g) + λv and let 0 ≤ h ∈ E(X, P) such that g + h ∈ r (X, P) and let ε > 0. Let g = (1+ε)g + h and f = f + εg + h. Then g CV 0 is again r-continuous, and f ≤ g +s. As s(x) = ∞ for all x ∈ X \Ks and as P has uniform boundedness components, we conclude that f (x) ∈ B g1 (x) for all x ∈ X. On the other hand g (x) = (1 + ε)g(x) + h(x) is evidently bounded relative to g(x)+h(x), hence relative to f (x) = f (x)+εg(x)+h(x) as well. This demonstrates that both f and g create the same inverses of boundedness components in X, and following Proposition 8.8 therefore f is r-continuous as g is. Then by (i) T (f ) ≤ T (f ) + εT (g) + ελw ≤ T (f ) + ελw ≤ T (g ) + (1 + ελ)w ≤ (1 + ε)T (g + h) + (1 + ελ)w. This holds for all ε > 0, and we conclude that T (f ) ≤ T (g + h) + w. Taking the inﬁmum over all such functions 0 ≤ h ∈ E(X, P) yields T (f ) ≤ T (g) + w. To prepare our second case, suppose that a function h ∈ E(X, P) is supported by Ks . Because the range of h is spanned by ﬁnitely many elements a1 , . . . , an ∈ P, we have h(x) ∈ B(a), where a = a1 + . . . + an for all x ∈ Ks . Note that χKs ∈ K0 (X). Let λ ≥ 0 such that 0 ≤ T (χKs ⊗a) + λw and choose ε > 0. We set h = h + χKs ⊗a ∈ E(X, P). Then all the values r (X, P) by of h on Ks are bounded relative to each other, hence h ∈ CV 0 Proposition 1.9. As h ≤ εs, we have T (h ) ≤ εw, hence T (h) ≤ T (h) + εT (χKs ⊗a) + ελw = T (h ) + ελw ≤ ε(1 + λ)w for all ε > 0. Likewise 0 ≤ T (h) + T (χKs ⊗a) ≤ T (h) + εw as χKs ⊗a ≤ εs for all ε > 0. Now in our second case, suppose that both functions f, g ∈ E(X, P) are supported by Ks . Then our preparing remarks yield T (f ) ≤ T (g) + εw for all ε > 0.

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III Measures on Locally Compact Spaces

Now, ﬁnally let us consider the general case. Both functions χ1 = χ(X\Ks ) and χ2 = χKs are r-continuous. The functions f1 = χ1 ⊗f, f2 = χ2 ⊗f, g1 = χ1 ⊗g and g2 = χ2 ⊗g are therefore all contained in E(X, P). Our ﬁrst step shows that T (f1 ) ≤ T (g1 ) + w, the second step T (f2 ) ≤ T (g2 ) + εw for all ε > 0. Combining, this yields T (f ) = T (f1 ) + T (f2 ) ≤ T (g1 ) + T (g2 ) + (1 + ε)w = T (g) + (1 + ε)w, for all ε > 0, hence T (f ) ≤ T (g) + w, as claimed.

The following extension result applies in particular to locally convex topological vector spaces (P, V). Recall the construction of the canonical embedding of a quasi-full locally convex (P, V) cone into the full cone (PV , V), that is its standard full extension, as elaborated in Section 6.2 of Chapter I. Theorem 3.3. Suppose that (P, V) is quasi-full and that all elements of P are bounded. Let V be a system of r-lower continuous inductive limit neighborhoods for F(X, P). Then every continuous linear operator T : E(X, P) → Q can be extended to a continuous linear operator from E(X, PV ) to Q. Proof. Note that under the conditions of the theorem the cones E0 (X, P) and E(X, P) coincide and that all functions in E(X, P) are r-continuous see Proposition 1.11(a) and indeed continuous with respect to the given symmetric topology of P. The V-valued functions comprising the neighborhood system V are r-lower continuous. The functions in E(X, V), on the other hand, satisfy the following: For a neighborhood-valued function ϕ⊗v ∈ E(X, V), a point x ∈ X and γ > 1 there is a neighborhood Ux of x such that (1/γ)ϕ(x) ≤ ϕ(y), hence (1/γ)ϕ(x)v ≤ ϕ(y)v for all y ∈ Ux . Consequently, for a function s ∈ E(X, V), x ∈ X and γ > 1 one ﬁnds a neighborhood Ux of x such that (1/γ)s(x) ≤ s(y) for all y ∈ Ux . Now let T : E(X, P) → Q be a continuous linear operator. For a PV valued function f ∈ E(X, PV ) we set T (f ) = sup{T (h) | h ∈ E(X, P), h ≤ f }. For V ⊂ P, that is f ∈ E(X, P), this deﬁnition is clearly consistent, and immediately yields T (αf ) = αT (f )

and

T (f + g) ≥ T (f ) + T (g)

for f, g ∈ E(X, PV ) and α ≥ 0. Subadditivity, on the other hand, is not as obvious. To prepare our argument for this, let us consider the following construction: Let f, g ∈ E(X, PV ), let u be an r-lower continuous V-valued function, and let h ∈ E(X, P) such that h ≤ f +g +u. Then f = f +s and g = g +t for functions f , g ∈ E(X, P) and s , g ∈ E(X, V). We choose a compact

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271

set K ∈ K which supports all the functions involved, f, g, h, s and t. Let v ∈ V and ε > 0. There is λ ≥ 1 such that both 0 ≤ f + λχK ⊗v and 0 ≤ g + λχk ⊗v. We set γ = 1 + ε/λ ≤ 1 + ε. For every x ∈ K, following or assumption that P is quasi-full, there are elements ax , bx and cx in P such that ax ≤ s(x), bx ≤ t(x), cx ≤ u(x) and h(x) ≤ f (x) + g (x) + ax + bx + cx . The diﬀerent types of continuity of the concerned functions guarantees that there is an open neighborhood Ux of x in X such that (1/γ)ax ≤ s(y), as well as

(1/γ)bx ≤ t(y)

cx ≤ γu(y) + εv,

and

h(y) ≤ f (y) + g (y) + ax + bx + cx + εv

holds for all y ∈ Ux . In turn there is a ﬁnite cover Ux1 , . . . , Uxn of these open sets for the compact set K and a corresponding partition of the unit, consisting of continuous real-valued functions ϕ1 , . . . , ϕn such that supp(ϕi ) ⊂ Uxi for all i = 1, . . . , n and ϕ1 (x) + . . . + ϕn (x) = 1 for all x ∈ K. We observe that (1/γ)ϕi (x)bxi ≤ ϕi (x)t(x) (1/γ)ϕi (x)axi ≤ ϕi (x)s(x), and

ϕi cxi ≤ ϕi (x) γu(x) + εv ,

as well as ϕi (x)h(x) ≤ ϕi (x) f (x) + g (x) + ax + bx + cx + εv holds for all x ∈ X and i = 1, . . . , n. We set f = f +

n 1! ϕi ⊗axi γ i=1

g = g +

and

n 1! ϕi ⊗bxi . γ i=1

Then f , g ∈ E(X, P) and f ≤ f + s = f

g ≤ g + t = g

and

as well as h ≤ f + g +

ϕi ⊗(axi + bxi + cxi ) + εχK ⊗v

i=1

" ≤

n !

f +

n ! i=1

ϕi ⊗axi

#

" +

g +

n ! i=1

# ϕi ⊗bxi

+ γu + 2εχK ⊗v.

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III Measures on Locally Compact Spaces

We observe that f ≤ f + (ε/λ)(f + λχK ⊗v) = γf + εχK ⊗v and, likewise

g ≤ γg + εχK ⊗v.

Combining with the above, this yields h ≤ γ(f + g + u) + 3εχK ⊗v. We shall in the sequel use this construction of the functions f , g ∈ E(X, P) in order to prove both subadditivity and continuity of the extended operator T : For subadditivity, we choose f, g ∈ E(X, PV ) and set u = 0 in the above. Let h ∈ E(X, P) such that h ≤ f + g. Given w ∈ W for the set K ∈ K from above there is an inductive limit neighborhood v ∈ V such that T (j) ≤ T (l) + w holds whenever j ≤ l + v for functions j, l ∈ E(X, P), and there is v ∈ V such that χK ⊗v ≤ v. With this insertion for v ∈ V and any ε > 0 we construct the functions f , g ∈ E(X, P) as before. Then T (f ) ≤ T (f )

and

T (g ) ≤ T (g),

since f ≤ f and g ≤ g, and T (h) ≤ γ T (f ) + T (g ) + 3εw ≤ γ T (f ) + T (g) + 3εw, since h ≤ γf + γg + 3εv, and consequently T (h) ≤ T (f ) + T (g), because w ∈ W and ε > 0 were arbitrarily chosen. Finally, taking the supremum over all h ∈ E(X, P) such that h ≤ f + g on the left hand side of the last inequality yields indeed T (f + g) ≤ T (f ) + T (g) by our deﬁnition of the extension of the operator T. For continuity of this extension, let w ∈ W and v ∈ V such that T (j) ≤ T (l) + w holds whenever j ≤ l + v for functions j, l ∈ E(X, P). We shall verify that this property is preserved by the extension of T to the cone E(X, PV ). Indeed, let f, g ∈ E(X, PV ) such that f ≤ g + v, that is f ≤ g + u for some V-valued r-lower continuous function u ≤ v. Let h ∈ E(X, P) such that h ≤ f. Then h ≤ g + u by the deﬁnition of the order in the extended cone PV as deﬁned in Section I.6.2. We shall now use the above construction with the function f = 0, hence f = 0: Given v ∈ V such that χK ⊗v ≤ v and ε > 0, there is g ∈ E(X, P), satisfying g ≤ g and

4. Measures on Locally Compact Spaces

273

h ≤ γ(g + u) + 3εχK ⊗v ≤ γg + (γ + 3ε)v. This yields T (h) ≤ γT (g ) + (γ + 3ε)w ≤ γT (g) + (γ + 3ε)w, and consequently T (h) ≤ T (g) + w, because ε > 0 was arbitrarily chosen. Finally, taking the supremum over all h ∈ E(X, P) such that h ≤ f on the left hand side of the last inequality yields indeed T (f ) ≤ T (g) + w, thus completing our argument.

Proposition 3.4. Suppose that (P, V) is quasi-full and that X carries the discrete topology. Let V be a system of r-lower continuous inductive limit neighborhoods for F(X, P). Then every continuous linear operator T : E(X, P) → Q can be extended to a continuous linear operator from E(X, PV ) to Q. Proof. This is a straightforward consequence of Theorem I.6.3. R consists of all ﬁnite subsets of X in this case. For every x ∈ X the operator Tx : P → Q, that is a → T (χ{x} ⊗a) is linear and continuous. According to I.6.3 there is a continuous linear extension to PV , that is Tx : PV → Q. Since all compact subsets of X are ﬁnite, the functions f ∈ E(X, PV ) are of the type f = ni=1 χ{xi } ⊗ai for xi ∈ X and ai ∈ PV . Then the formula # " n n ! ! χ{xi } ⊗ai = Txi (ai ) T i=1

i=1

provides the stated extension of T to E(X, PV ).

4. Measures on Locally Compact Spaces We shall now return to the concepts of integration theory from Chapter II. As in the preceding sections, let X be a locally compact Hausdorﬀ space. Let R be the weak σ-ring of all relatively compact Borel subsets of X and correspondingly, AR the σ-algebra of all Borel subsets of X. As before, (P, V) is a locally convex cone. Proposition 4.1. Let f ∈ F(X, P). (a) If f is r-continuous, then f is measurable.

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III Measures on Locally Compact Spaces

(b) If f is r-lower continuous and measurable, then χE ⊗f ∈ FR (X, P) for every E ∈ R. Proof. (a) The symmetric relative topology of P is the common reﬁnement of all symmetric relative v-topologies, and therefore an r-continuous function f ∈ F(X, P) is continuous with respect to any of these topologies. It is therefore clear that f −1 (O) ∈ AR for any set O in P which is open in any of the relative v-topologies. This is Condition (M1) from II.1.2. Furthermore, since all sets E ∈ R are relatively compact in X, the sets f (E) are relatively compact in every symmetric relative v-topology. But these topologies are generated by pseudo-metrics (see Section I.4), and relatively compact sets are therefore separable, hence (M2). For Part (b), let f ∈ F(X, P) be r-lower continuous and measurable, and let E ∈ R. First we recall from Theorem II.1.8(a) that the function χE ⊗f is also measurable. The closure ¯ of E is compact and also contained in R. We shall have to verify that E f can be reached from below by step functions in the sense of II.2.3: Given an inductive limit neighborhood v there is v ∈ V such that χE¯ ⊗v ≤ v. By ¯ Given Lemma 1.4 then there is λ ≥ 0 such that 0 ≤ f (x)+λv for all x ∈ E. 1 0 ≤ ε ≤ 1+λ , as the function f is r-lower continuous, a compactness argu¯ and corresponding neighborhoods ment yields that there are x1 , . . . , xn ∈ E ¯ and such that f (x) ∈ f (xi ) vε , points, covering E E1 . . . , En ∈ R of these that is f (xi ) ∈ vε f (x) whenever x ∈ Ei . We set ai = f (xi ), and for any x ∈ Ei we have ai ≤ γf (x) + εv for some 1 ≤ γ ≤ 1 + ε, hence ai ≤ (1 + ε)f (x) + ε(1 + λ)v by Lemma I.4.1(b), and (1 + ε)−1 ai ≤ f (x) + v. i−1 Fk for i = 2, . . . , n. Then We set F1 = E ∩ E1 and Fi = (E ∩ Ei ) \ ∪k=1 h = (1 + ε)−1

n !

χFi ⊗ai ∈ SR (X, P),

i=1

and h ≤ χE ⊗f + χE ⊗v ≤ χE ⊗f + v. Our claim follows.

Part (b) of Proposition 4.1 applies in particular to the functions in E(X, P) since every function of the type ϕ⊗a for a continuous positive real-valued function ϕ and a ∈ P is measurable by Theorem II.1.8(a) and r-lower continuous by Proposition 1.8(b). In addition to the general assumptions of the preceding we shall for the remainder of this section require that (P, V) is a quasi-full locally convex cone, and that (Q, W) is a locally convex complete lattice cone. 4.2 Regularity of Measures. Following the usual terminology, we shall say that an operator-valued measure θ : R → L(P, Q) is inner regular or outer regular on a set E ∈ R if, with respect to the order convergence in Q, θE (a) = lim θK (a) K⊂E

or

θE (a) = lim θO (a) O⊃E

4. Measures on Locally Compact Spaces

275

holds for all a ∈ P, respectively. The limits are taken over the upward directed family of all compact sets of K ⊂ E in the ﬁrst case, and the downward directed family of all relatively compact open sets O ⊃ E in the second case. A measure is outer or inner regular if it is outer or inner regular for all E ∈ R, respectively. A measure which is both outer regular for all E ∈ R and inner regular for all open sets E ∈ R is called quasi regular. Lemma 4.3. Let θ be a bounded measure on R. Let E, F ∈ R be disjoint sets and let a ∈ P. (a) If θ is outer regular for F and inner regular for E ∪ F, then lim θK (a) ≤ θE (a) ≤ lim θK (a) + O θF (a) . K⊂E

K⊂E

(b) If θ is inner regular for F and outer regular for E ∪ F, then θE (a) ≤ lim θO (a) ≤ lim θO (a) ≤ θE (a) + O θF (a) . O⊃E

O⊃E

These limits are taken over the upward directed family of all compact sets K ⊂ E or over the downward directed family of all relatively compact open sets O ⊃ E, respectively. Proof. Because the R-bounded measure θ can be extended to PV , we may assume that the locally convex cone (P, V) is full. For Part (a), let E, F ∈ R be disjoint sets and suppose that θ is outer regular for F and inner regular for E ∪ F. In a ﬁrst step let us consider an element 0 ≤ a ∈ P. Let S be any compact subset of E ∪ F and let O ∈ R be any open set containing F. Let K = S \ O. Then K is compact, K ⊂ E, and as S = K ∪ S ∩ O) we have θS (a) = θK (a) + θ(S∩O) (a) ≤ θK (a) + θO (a), hence θS (a) ≤ sup{θK (a) | K ⊂ E, K ∈ K} + θO (a) = lim θK (a) + θO (a). K⊂E

First taking the supremum over all compact subsets S of E ∪ F and using the inner regularity of θ for the set E ∪ F on the left-hand side of this inequality, and then taking the inﬁmum over all open sets O ∈ R containing F and using the outer regularity of θ for the set F on the right-hand side leads to θ(E∪F ) (a) ≤ lim θK (a) + θF (a). K⊂E

As θ(E∪F ) (a) = θE (a)+θF (a), the cancellation law from Proposition I.5.10(a) yields θE (a) ≤ lim θK (a) + O θF (a) . K⊂E

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III Measures on Locally Compact Spaces

For the general case, let a ∈ P, and for a neighborhood w ∈ W we choose v ∈ V such that θ(E∪F ) (v) ≤ w and λ ≥ 0 such that 0 ≤ a + λv. Then using the above we infer that θE (a) + θE (λv) ≤ lim θK (a + λv) + O θF (a + λv) K⊂E ≤ lim θK (a) + lim θK (λv) + O θF (a + λv) K⊂E K⊂E ≤ lim θK (a) + θE (λv) + O θF (a) + O θF (λv) . K⊂E

For thelatter we used Proposition I.5.11. Now because both O θE (λv) ≤ w and O θF (λv) ≤ w, the cancellation law in Proposition I.5.10(a) yields θE (a) ≤ lim θK (a) + O θF (a) + 2w. K⊂E

This holds for all w ∈ W and therefore demonstrates θE (a) ≤ lim θK (a) + O θF (a) , K⊂E

hence the right-hand part of the inequality in Lemma 4.3(a). For the left-hand part, using Lemma I.5.19 and the above we observe that lim θK (a) + θE (λv) ≤ lim θK (a) + lim θK (λv) + O θF (λv) K⊂E

K⊂E

K⊂E

≤ lim θK (a + λv) + w K⊂E

≤ θE (a + λv) + w = θE (a) + θE (λv) + w. Thus lim θK (a) ≤ θE (a) + 2w

K⊂E

using the cancellation law in I.5.10(a). This holds for all w ∈ W, yielding lim θK (a) ≤ θE (a),

K⊂E

our claim in Part (a). The argument for Part (b) of Lemma 4.3 follows along similar lines: Let E, F ∈ R be disjoint and suppose that θ is inner regular for F and outer regular for E ∪ F, and let 0 ≤ a ∈ P. Let G ∈ R be an open set containing E ∪ F and let K be a compact subset of F. The set O = G \ K ∈ R is open and contains E. Hence θK (a) + lim θO (a) ≤ θK (a) + θ(G\K) (a) = θG (a), O⊃E

4. Measures on Locally Compact Spaces

277

and therefore by the inner regularity of θ for F and the outer regularity for E ∪ F, θF (a) + lim θO (a) ≤ θ(E∪F ) (a) = θF (a) + θE (a). O⊃E

The cancellation law I.5.10(a) yields lim θO (a) ≤ θE (a) + O θF (a) .

O⊃E

For the general case, let a ∈ P, and for w ∈ W choose v ∈ V such that θG (v) ≤ w and λ ≥ 0 such that 0 ≤ a + λv. Using the above we infer that lim θO (a) + θE (λv) ≤ lim θO (a) + lim θO (λv)

O⊃E

O⊃E

O⊃E

≤ lim θO (a + λv) O⊃E

≤ θE (a + λv) + O θF (a + λv) = θE (a) + θE (λv) + O θF (a) + O θF (λv) . Now the cancellation law I.5.10(a) yields lim θO (a) ≤ θE (a) + O θF (a) + 2w

O⊃E

for all w ∈ W, hence lim θO (a) ≤ θE (a) + O θF (a) .

O⊃E

For the left-hand part of the inequality in 4.3(b), using I.5.19 and the above we observe that θE (a) + θE (λv) ≤ lim θO (a + λv) O⊃E

≤ lim θO (a) + lim θO (λv) O⊃E

O⊃E

≤ lim θO (a) + θE (λv) + O θF (λv). O⊃E

Thus θE (a) ≤ lim θO (a) + w O⊃E

for all w ∈ W, yielding

θE (a) ≤ lim θO (a), O⊃E

our claim in Part (b).

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III Measures on Locally Compact Spaces

Proposition 4.4. Let θ be a bounded measure on R. Let E ∈ R and a ∈ P. (a) If θ is outer regular, then lim θK (a) ≤ θE (a) ≤ lim θK (a) + O θ(G\E) (a)

K⊂E

K⊂E

for any set G ∈ R containing E such that θ is inner regular for G. (b) If θ is inner regular, then θE (a) ≤ lim θO (a) ≤ lim θO (a) ≤ θE (a) + O θ(G\E) (a) O⊃E

O⊃E

for any set G ∈ R containing E such that θ is outer regular for G. (c) If θ is quasi regular, then lim θK (a) ≤ θE (a) ≤ lim θK (a) + O θ (E\G) (a)

K⊂E

K⊂E

for any subset G ∈ R of E such that θ is inner regular for G. Proof. Let θ be a bounded measure, let E ∈ R and a ∈ P. We may again assume that the cone (P, V) is full. Parts (a) and (b) of the Proposition follow directly from Parts (a) and (b) of Lemma 4.3, respectively, if we set F = G \ E. For Part (c), suppose that θ is quasi regular and let H ∈ R be an open set containing E. With F = H \ E ∈ R the assumptions of Lemma 4.3(a) are satisﬁed, and we have lim θK (a) ≤ θE (a) ≤ lim θK (a) + O θ(H\E) (a) K⊂E K⊂E ≤ lim θK (a) + O θH (a) . K⊂E

The latter follows from Proposition II.4.15(c) and holds for all open sets H ∈ R containing E. We have lim O θH (a) ≤ O lim θH (a) = O θE (a) H⊃E

H⊃E

by Proposition I.5.24 and by the outer regularity of θ. This shows θE (a) ≤ lim θK (a) + O θE (a) . K⊂E

Next, for w ∈ W let v ∈ V such that θE (v) ≤ w and λ ≥ 0 such that 0 ≤ a + λv. Let G ∈ R be a subset of E such that θ is inner regular for G. Then θG (a) ≤ θG (a + λv) = lim θK (a + λv) ≤ lim θK (a + λv) ≤ lim θK (a) + λw. K⊂G

K⊂E

K⊂E

4. Measures on Locally Compact Spaces

279

This shows θG (a) ∈ B lim θK (a) and implies that K⊂E

lim θK (a) + O θG (a) = lim θK (a)

K⊂E

K⊂E

using Proposition I.5.14. Because O θE (a) = O θG (a) + O θ(E\G) (a) by I.5.11(a), we have indeed demonstrated that θE (a) ≤ lim θK (a) + O θ(E\G) (a) K⊂E

holds as claimed.

Note that the set G ∈ R in Proposition 4.4 may, for example, be chosen as E in Part (a), as any open set containing E in Part (b) and as E ◦ or as any closed subset of E in Part (c). For a set E ∈ R and a function ϕ ∈ K(X) we abbreviate E ≺ ϕ if χE ≤ ϕ, and ϕ ≺ E if ϕ ≤ χE and supp(ϕ) ⊂ E. Note that the families {ϕ ∈ K(X) | ϕ ≺ E} and {ϕ ∈ K(X) | ϕ ≺ E} are directed downward and upward, respectively. Both are therefore suitable index sets for nets. The order topology for a locally convex complete lattice cone was deﬁned in Section I.5.43. Proposition 4.5. Let θ be a bounded measure on R. (a) If θ is inner regular for an open set 0 ∈ R, then ' θO (a) = lim

ϕ⊗a dθ

ϕ≺O

for all

a ∈ P.

X

(b) If θ is outer regular for a compact set K ∈ R, then ' θK (a) = lim

ϕ⊗a dθ

ϕ#K

for all

a ∈ P.

X

(c) If θ is quasi regular, then for every E ∈ R and every open set O ∈ R containing E there exists a net (ϕi )i∈I in K(X) such that ϕi ≺ O for all i ∈ I, and ' θE (a) = lim i∈I

ϕi ⊗a dθ

for all

a∈P

X

in the order topology of Q. Proof. We may assume that the cone (P, V) is full. Let θ be a bounded measure on R. Propositions 1.8(a) and 4.1(b) imply that all functions in E(X, P) are integrable over any set F ∈ AR with respect to θ. For Part (a)

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III Measures on Locally Compact Spaces

of our proposition, let O ∈ R be an open set such that θ is inner regular for O. Let us ﬁrst consider the case of a positive element 0 ≤ a ∈ P. Let K be a compact subset of O. Following Urysohn’s lemma (see 2.12 in [179]) there is ϕ ∈ K(X) such that K ≺ ϕ ≺ O. Thus ' θK (a) ≤

ϕ⊗a dθ ≤ θE (a). X

This shows θK (a) ≤ lim

(

ϕ⊗a dθ, hence

ϕ≺O X

' θO (a) = lim θK (a) ≤ lim K⊂O

and therefore θO (a) = lim

ϕ⊗a dθ ≤ θO (a),

ϕ≺O

(

ϕ≺O X

X

ϕ⊗a dθ. Now for the general case, let a ∈ P.

For w ∈ W there is v ∈ V such that θO (v) ≤ w, and there is λ ≥ 0 such that a + λv ≥ 0. As argued before, we have ' θO (a + λv) = lim

ϕ≺O

' ϕ⊗(a + λv) dθ

and

θO (v) = lim

ϕ≺O

X

ϕ⊗v dθ. X

Thus using Lemma I.5.19 ' lim

ϕ≺O

' ϕ⊗a dθ + λθO (v) = lim

ϕ≺O

X

' ϕ⊗a dθ + lim ϕ≺O

X

ϕ⊗v dθ X

' ≤ lim

ϕ≺O

ϕ⊗(a + λv) dθ X

= θO (a) + λθO (v). The cancellation law in Proposition I.5.10(a) now yields ' ϕ⊗a dθ ≤ θO (a) + εw

lim

ϕ≺O

X

for all ε > 0, since θO (v) ≤ w. The latter holds true for all w ∈ W and therefore demonstrates ' lim ϕ⊗a dθ ≤ θO (a). ϕ≺O

Similarly, one argues that

X

4. Measures on Locally Compact Spaces

281

' θO (a) + λθO (v) = lim

ϕ≺O

ϕ⊗(a + λv) dθ '

≤ lim ϕ≺O

ϕ⊗a dθ + λ lim

ϕ≺O

X

'

= lim ϕ≺O

'

X

ϕ⊗v dθ X

ϕ⊗a dθ + λθO (v) X

implies

' θO (a) ≤ lim ϕ≺O

ϕ⊗a dθ. X

This completes the proof of Part (a). For Part (b), suppose that θ is( outer regular for the compact set K ∈ R. First, let 0 ≤ a ∈ P. The net ϕ ⊗a dθ is decreasing and bounded X ϕ#K below, hence convergent in Q. For every open set O ∈( R containing K there is ϕ ∈ K(X) such that K ≺ ϕ ≺ O. This shows X ϕ⊗a dθ ≤ θO (a), ( hence limϕ#K X ϕ⊗a dθ ≤ θO (a) and therefore ' ϕ⊗a dθ ≤ θK (a) lim ϕ#K

X

by the outer regularity of θ for the set K. For the converse inequality, let ϕ # K and γ > 1. (There is an open set O ⊃ K such that O ≺ γϕ, hence θK (a) ≤ θO (a) ≤ γ X ϕ⊗a dθ. As this holds for all ϕ # K we infer that ' ϕ⊗a dθ. θK (a) ≤ γ lim ϕ#K

X

Finally, as γ > 1 was arbitrarily chosen, and as Q is a locally convex complete lattice cone, this yields together with the above θK (a) = ( limϕ#K X ϕ⊗a dθ. Now for the general case, let a ∈ P. Given any w ∈ W there is v ∈ V such that θK (v) ≤ w and λ ≥ 0 such that 0 ≤ a + λv. As we observed before, the latter yields ' ' θK (a + λv) = lim ϕ⊗(a + λv) dθ and θK (v) = lim ϕ⊗v dθ. ϕ#K

Thus

ϕ#K

X

' lim

ϕ#K

' ϕ⊗a dθ + λθK (v) = lim

X

ϕ#K

ϕ#K

' ϕ⊗a dθ + lim ϕ#K

X

'

≤ lim

X

ϕ⊗(a + λv) dθ X

= θK (a) + λθK (v).

ϕ⊗v dθ X

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III Measures on Locally Compact Spaces

As θK (v) ≤ λw, the cancellation law in Proposition I.5.10 yields ' ' lim ϕ⊗a dθ ≤ θK (a) + εw and θK (a) ≤ lim ϕ⊗a dθ + εw ϕ#K

ϕ#K

X

X

for all ε > 0. As w ∈ W was arbitrary, this yields θK (a) = limϕ#K T (ϕ⊗a). For Part (c), suppose that θ is quasi regular and let E ∈ R. Let O0 ∈ R be an open set containing E. Then for every open set O ∈ R such that Part (a), the element θO (a) ∈ Q is the limit of the E ⊂ O( ⊂ O0 , following net X ϕ⊗a dθ ϕ≺O . The outer regularity of θ yields θE (a) =

lim

O0 ⊃O⊃E

θO (a).

Now a well-known diagonal principle for convergent nets (see for example I.6.A in [59] or 11D in [198]) yields that there is a diagonal net ( ϕ ⊗ a dθ converging to θE (a) in the order topology of Q. More preX i i∈I cisely: Let O be the family of all open sets O such that O0 ⊃ O ⊃ E. Then the index set I of the diagonal net consists of all ordered pairs (O, f ), where O ∈ O and f : O → K(X) is a mapping such that f (O) ≺ O for all O ∈ O. This index set is ordered by (O1 , f1 ) ≤ (O2 , f2 ) if O1 ⊃ O2 and f1 (O) ≤ f2 (O) for all O ∈ O. If we set ϕi = f (O) ∈ K(X) ( for i = (O, f ) ∈ I, then it can be easily veriﬁed that θE (a) = lim X ϕi ⊗a dθ. i∈I

Note that the limit in 4.5(c) refers to the order topology of Q, not necessarily to order convergence as deﬁned in I.5.18. Corollary 4.6. ( Let θ and( ϑ be bounded quasi regular measures on R and let a ∈ P. If X ϕ⊗a dθ = X ϕ⊗a dϑ for all ϕ ∈ K(X), then θE (a) = ϑE (a) for all E ∈ R. In particular, the measures θ and ϑ coincide, provided that their integrals for all functions in E(X, P) coincide. Proof. We have θO (a) = ϑO (a) for all open sets O ∈ R by Proposition 4.5(a), hence θE (a) = ϑE (a) for all sets E ∈ R due to outer regularity. 4.7 Measures as Continuous Linear Operators. We shall say that a bounded operator-valued measure θ : R → L(P, Q) is continuous relative to a system V of inductive limit neighborhoods if for every w ∈ W there is v ∈ V such that ' ' f dθ ≤ g dθ + w whenever f ≤g+v X

X

for all functions f, g ∈ F(X,θ) (X, P), that is the subcone of all functions in F(X, P) that are integrable over X. Note that every R-bounded measure is continuous relative to some system of inductive limit neighborhoods, that is the system V = {vw | w ∈ W} (see Section II.4).

4. Measures on Locally Compact Spaces

283

Proposition 4.8. Let θ be continuous relative to the system V of inductive limit neighborhoods. (a) If for E ∈ R and every v ∈ V there is s ∈ v such that s(x) = ∞ for all x ∈ E, then θE = 0 ∈ L(P, Q). (b) Every f ∈ FV (X, P) is integrable over every Borel set F ∈ AR with respect to θ. (c) Every f ∈ FV0 (X, P) is strongly integrable over every Borel set F ∈ AR with respect to θ. Proof. Suppose that θ is continuous relative to V. Let E ∈ R as in Part (a). For w ∈ W choose the inductive limit neighborhood v ∈ V as in 4.7. Following our assumption then there is s ∈ v, such that s(x) = ∞ for all x ( ∈ E. Then for every a (∈ P we have χE ⊗a ≤ s and 0 ≤ χE ⊗a + s. Thus X χE ⊗a dθ ≤ w and( 0 ≤ X χE ⊗a dθ +w. As w ∈ W was arbitrarily chosen, this shows θE (a) = X χE ⊗a dθ = 0, as claimed in (a). For (b), let f ∈ FV (X, P). According to Deﬁnitions II.4.12 and II.4.13 we shall ﬁrst verify integrability with respect to θ over a set E ∈ R. For this, let w ∈ W and 0 < ε(≤ 1. There is an inductive limit neighborhood ( v ∈ V such that X g dθ ≤ X h dθ + (ε/6)w holds whenever g ≤ h + v for integrable functions g, h ∈ F(X, P). By our deﬁnition of the cone FV (X, P) as the closure of E(X, P) with respect to the symmetric relative topology generated by V there is g ∈ E(X, P) such that both f ∈ vsε (g), that is f ≤ γg + s

and

g ≤ γ f + s

for some 1 ≤ γ, γ ≤ 1 + ε and s, s ∈ v. The functions in v are supposed to be measurable, but can take the values 0 and ∞ ∈ V. To remedy this, we choose v ∈ V such that (ε/2)χE ⊗v ≤ v, and for any s ∈ v we set s˜ = χ(X\F ) ⊗s + χX ⊗v, where F ∈ AR is the set of all points in X where the function s takes the value ∞. Then the function s˜ is V-valued and measurable, hence contained in FR (X, V). Moreover, according to Lemma II.4.4 we have ' ' ε s˜ dθ = sup h dθ h ∈ SR (X, P), h ≤ s˜ ≤ w, 3 E E ( since h ≤ s˜ implies that χE ⊗h ≤ s + χE ⊗v ≤ 2v, hence E h dθ ≤ (ε/3)w. For Deﬁnition II.4.12 now we choose f(w,ε) = γg + s˜

and

s(w,ε) = (γ + 1)˜ s ≤ 3˜ s.

Then f(w,ε) ∈ FR (X, PV ) by Proposition 4.1(b) and s(w,ε) ∈ FR (X, V). Moreover, f (x) ≤ f(w,ε) (x) ≤ γf (x) + s(w,ε) (x)

284

III Measures on Locally Compact Spaces

holds for all x ∈ X \ F, hence '

≤ f(w,ε) a.e.E ≤ γf + s(w,ε) f a.e.E

s(w,ε) dθ ≤ εw,

and E

since F was seen to be a set of measure 0 in Part (a). This shows integrability of f over the set E ∈ R by II.4.12. For integrability over a set F ∈ AR we have to verify that the limit ' ' f dθ = lim f dθ F

E∈R

(E∩F )

exists in Q. According to Proposition I.5.41 (Q, W) is complete with respect to the symmetric relative topology of Q, and according to I.5.42, convergence in this topology implies order convergence as required in the deﬁnition of ( the integral. It suﬃces therefore to verify that (E∩F ) f dθ E∈R forms a Cauchy net( in Q. For this, let w ∈ W and 0 < ε ≤ 1. There is v ∈ V ( such that X j dθ ≤ X l dθ + w whenever j ≤ l + v for integrable functions j, l ∈ F(X, P), and in turn there is g ∈ E(X, P) such that ε f ≤ γg + v 3

and

ε g ≤ γ f + v 3

for some 1 ≤ γ, γ ≤ 1 + ε/3. Let E0 ∈ R be a set that contains the support of the function( g. Then (for E1 , E2 ∈ R such that both E0 ⊂ E1 and E0 ⊂ E2 we have E1 g dθ = E2 g dθ, hence '

'

ε f dθ ≤ γ g dθ + w = γ 3 E1 E1

and likewise

' E2

'

ε g dθ + w ≤ γγ 3 E2

f dθ ≤ γγ

' f dθ + εw, E2

' f dθ + εw. E1

( ( As 1 ≤ γγ ≤ 1 + ε, this shows E1 f dθ ∈ vsε E2 f dθ . The net ( (E∩F ) f dθ E∈R is therefore indeed a Cauchy net, hence the function f is integrable over F as claimed. For Part (c), all left to show is that, according to the deﬁnition of strong is a step integrability in II.5.18, for every f (∈ FV0 (X, P), ( and w ∈ W there ( such that f dθ ≤ h dθ + w and h dθ is wfunction h ∈ SR (X, P) E E E ( bounded relative to E f dθ for all E ∈ R. Let us ﬁrst consider the case that f ∈ E0 (X, P). Let K be the compact support of f and let v ∈ V such that |θ|(K, v) ≤ w. There is λ ≥ 0 such that 0 ≤ f (x) + λv for all x ∈ K. Because the function f is r-continuous, given 0 < ε ≤ 1/(1 + λ), there is a partition many disjoint sets E1 , . . . , En ∈ R such that of K into ﬁnitely f (x) ∈ vε f (y) , that is see Lemma I.4.1(b)

4. Measures on Locally Compact Spaces

285

f (x) ≤ (1 + ε)f (y) + ε(1 + λ)v ≤ (1 + ε)f (y) + v whenever x, y are contained in the same component Ei . We choose xi ∈ Ei , ai = (1 + ε)f (xi ) and h = ni=1 χEi ⊗ai ∈ SR (X, P). Then f ≤ h + χK ⊗v and h ≤ (1 + ε)2 f + 2χK ⊗v. ( ( ( ( 2 Thus indeed E f dθ ≤ E h dθ + w and E(h dθ ≤ (1 + ε) E f dθ + 2w, ( hence E h dθ + w is w-bounded relative to E f dθ, for all E ∈ R. Now case, that is f ∈ FV0 (X, P), ﬁrst choose v ∈ V such ( ( for the general that X f dθ ≤ X g dθ + w/3 whenever f ≤ g + v for f, g ∈ FR (X, P). For a given function f ∈ FV0 (X, P) and v ∈ V there is g ∈ E0 (X, P) such that both f ≤ γg + v and g ≤ γ f + v for some 1 ≤ γ, γ ≤ 2. For the function g ∈ E0 (X, P), however, we did verify the existence of a(step function h ∈ (SR (X, P) with the required properties, ( ( that is E g dθ ≤ E h dθ+w/3 and E h dθ is w-bounded relative to E g dθ for all E ∈ R. Set h = γh ∈ SR (X, P). Then ' ' ' f dθ ≤ γg dθ + w/3 ≤ h dθ + w E

E

(

E

for all E ∈ R. Moreover, E h dθ( is w-bounded relative to in turn is w-bounded relative to E f dθ. Our claim follows.

( E

g dθ which

In this way, a V-continuous L(P, Q)-valued Borel measure θ deﬁnes a continuous linear operator T from FV (X, P) into Q, that is ' f → f dθ : FV (X, P) → Q. F

Recall from Sections 6.12, 6.14 and 6.16 of Chapter II that in special cases certain additional properties transfer from the measure θ to this operator: If P is a locally convex topological vector space over K = R or K = C and if the operators θE ∈ L(P, Q) are linear over K, then T is also linear over K (see II.6.12). If P is a topological algebra and if Q is the standard lattice completion of another topological algebra, then Condition (A) from II.6.14 guarantees that T is an algebra homomorphism on some algebra of integrable functions. Condition (A*) yields compatibility with an involution. For the special case Q = K, that is for a functional-valued measure θ we observed in a remark after II.6.15 that (A) means that the functionals θE are multiplicative and either θE = 0 or θG = 0 holds for disjoint sets E, G ∈ R. The latter holds, because otherwise we could ﬁnd elements a, b ∈ P such that θE (a) = θG (b) = 1, contradicting the requirement θE (a)θG (b) = 0. Consequently, θ takes at most one non-zero value, that is some multiplicative K-linear functional in μ0 ∈ P ∗ . If θ = 0 is inner regular, this leads to the following

286

III Measures on Locally Compact Spaces

conclusion: First we observe that θ(E∩G) = μ0 whenever θE = θG = μ0 for E, G ∈ R. Let K0 be the intersection of all compact sets K ∈ R such that θK = μ0 . A compactness argument together with the above shows that K0 = ∅. Inner regularity then implies that θE = 0 whenever E ∈ R is disjoint to K0 . θE = μ0 on the other hand, implies that K0 ⊂ E. Indeed, every compact subset K of E is disjoint from K0 , hence θK = 0. Thus θE = θ(E∩K0 ) for every E ∈ R and therefore θK0 = μ0 . Now the assumption that K0 might contain more than one point is easily contradicted. We infer that K0 = {x0 } for some x0 ∈ X, and for E ∈ R we have θE = μ0 if x0 ∈ E, and θE = 0 else. θ is therefore a point-evaluation measure in this case. If P is a ∨-semilattice cone, then Condition (L) from II.6.16 implies that T is a ∨-semilattice homomorphism on some ∨-semilattice of integrable functions. Moreover, if Q = R and θ is inner regular, then an argument similar to the above shows that under Condition (L) θ is a point evaluation, its only non-zero value being a ∨-semilattice homomorphism in P ∗ . Conversely, we shall demonstrate in our main result of the following section that for an r-lower continuous inductive limit topology, every continuous linear operator from FV (X, P) into Q can be represented by a unique quasi regular measure. In Section 6 we shall show that in special cases like the above certain additional properties of the operator transfer to this measure.

5. Integral Representation For the following integral representation theorem we shall consider continuous linear operators T : FV (X, P) → Q, where (P, V) is a full and (Q, W) is a locally convex complete lattice cone such that the order continuous linear functionals support the separation property for Q. V is a system of r-lower continuous inductive limit neighborhoods for F(X, P). The locally convex cones (FV (X, P), V) and (FV0 (X, P), V) were introduced and investigated in Section 2. We took particular eﬀort to characterize their respective subcones of r-continuous functions. In Section 3 we investigated various situations where continuous linear operators deﬁned only on a subcone of FV (X, P) can be extended. These results apply in particular if Q = R or if (P, V) is the standard full extension of some locally convex vector subspace (P0 , V) and if the operator is deﬁned on a suitable cone of P0 -valued functions (see Theorems 3.2 and 3.3). Representation Theorem 5.1. Let (P, V) be a full locally convex cone and let (Q, W) be a locally convex complete lattice cone such that the order continuous linear functionals support the separation property for Q. Let X be a locally compact Hausdorﬀ space and let V be a basis for an r-lower continuous inductive limit topology on F(X, P). Then every continuous linear operator T : FV (X, P) → Q can be represented as an integral on X. More precisely: There exists a unique bounded quasi regular L(P, Q)-valued

5. Integral Representation

287

measure θ on the weak σ-ring R of all relatively compact Borel subsets of X such that θ is continuous relative to V, all functions in FV (X, P) are integrable, all functions in FV0 (X, P) are strongly integrable with respect to θ, and ' ' ϕ⊗a dθ ≤ T (ϕ⊗a) ≤ ϕ⊗a dθ + O θA (a) X

X

holds for all ϕ ∈ K(X) and a ∈ P, where A denotes the compact support of the function ϕ. Thus ' ' f dθ ≤ T (f ) and indeed g dθ = T (g) X

X

holds for all f ∈ FV (X, P) and all g ∈ FV0 (X, P), respectively. Proof. Under the assumptions of this theorem, let T : FV (X, P) → Q be a continuous linear operator, that is for every w ∈ W there is an r-lower continuous neighborhood uw ∈ V such that T (f ) ≤ T (g) + w

whenever

f ≤ g + uw

for f, g ∈ FV (X, P). We proceed to construct a bounded inner regular L(P, Q)-valued measure θ on R, the weak σ-ring of all relatively compact Borel subsets of X. We shall follow some of the main lines of the standard proof for the Riesz representation theorem (see for example [178]), though the presence of unbounded elements in P, and the non-availability of negatives will complicate matters considerably. We shall ﬁrst list a few notations and abbreviations: For a set E ∈ R and a function ϕ ∈ K(X) we write E ≺ ϕ if χE ≤ ϕ, and ϕ ≺ E if ϕ ≤ χE and supp(ϕ) ⊂ E. We are ready to begin with our step-by-step construction of the measure θ. In a ﬁrst step, we shall deﬁne the measure θ on all relatively compact open subsets of X. For a relatively compact open set O ∈ R and an element a ∈ P we set (i)

θO (a) = lim T (ϕ⊗a). ϕ≺O

This limit is taken over the net whose index set is the upward directed family of all functions ϕ ∈ K(X) such that ϕ ≺ O. For a positive element 0 ≤ a ∈ P this limit obviously exists, as the net T (ϕ⊗a) ϕ≺O is increasing. We shall use the criterion in Proposition I.5.23 to verify that this limit exists in general for an element a ∈ P. To prepare this, let us observe the following: (ii) If χO ⊗a ≤ χO ⊗b + uw for 0 ≤ a, b ∈ P, w ∈ W and an open set O ∈ R, then θO (a) ≤ θO (b) + w. In particular, if χO ⊗v ≤ uw for some v ∈ V, then θO (v) ≤ w. Indeed, let 0 ≤ a, b ∈ P, let O ∈ R be an open set and let χO ⊗a ≤ χO ⊗b+uw , that is χO ⊗a ≤ χO ⊗b + s for some s ∈ uw . Let ϕ ≺ O. Then

288

III Measures on Locally Compact Spaces

ϕ⊗a ≤ ϕ⊗b + ϕ⊗s ≤ ϕ⊗b + uw . By the continuity of the operator T, this implies T (ϕ⊗a) ≤ T (ϕ⊗b) + w. Taking the limit over all functions ϕ ≺ O in this last inequality yields (ii) see Lemma I.5.20(c) . Now let a ∈ P. Given any neighborhood w ∈ W there is v ∈ V such that χO ⊗v ≤ uw, and there is λ ≥ 0 such that 0 ≤ a + λv. As argued before, the nets T (ϕ⊗λv) ϕ≺O and T (ϕ⊗a) + T (ϕ⊗λv) ϕ≺O are convergent in Q, and θO (v) ≤ w by (ii). Thus lim T (ϕ⊗λv) = θO (λv) = λθO (v) ≤ λw.

ϕ≺O

Now Proposition I.5.23 yields the convergence of the net T (ϕ⊗a) ϕ≺O . Let us observe that θ∅ = 0 since ϕ = 0 is the only function in K(X) such that supp(ϕ) ⊂ ∅. Because T is linear on E(X, P), we have T ϕ⊗(αa) = αT ϕ⊗a and T ϕ⊗(a + b) = T ϕ⊗a + T ϕ⊗b for an open set O ∈ R, for all a, b ∈ P, α ≥ 0 and ϕ ≺ O. Following Lemmas I.5.19 and I.5.21 this implies θO (αa) = lim T ϕ⊗(αa) = α lim T (ϕ⊗a) = αθO (a) ϕ≺O

ϕ≺O

and θO (a + b) = lim T ϕ⊗(a + b) = lim T (ϕ⊗a) + lim T (ϕ⊗b) = θO (a) + θO (a). ϕ≺O

ϕ≺O

ϕ≺O

Moreover, if a ≤ b for a, b ∈ P, then ϕ⊗a ≤ ϕ⊗b, hence T (ϕ⊗a) ≤ T (ϕ⊗b), and therefore θO (a) ≤ θO (b). In this way θO deﬁnes a monotone linear operator from P to Q. Obviously, this operator is also continuous: Given w ∈ W we choose v ∈ V such that χO ⊗v ≤ uw . Then θO (v) ≤ w by (ii), and a ≤ b + v for elements a, b ∈ P therefore implies θO (a) ≤ θO (b + v) ≤ θO (b) + w. This yields (iii) θO ∈ L(P, Q) for all open sets O ∈ R. Next we observe (iv) If 0 ≤ a for a ∈ P, and if O ⊂ U for open sets O, U ∈ R, then 0 ≤ θO (a) ≤ θU (a).

5. Integral Representation

289

For this, let a, O, U be as in (iv). Then θO (a) = lim T (ϕ⊗a) = sup T (ϕ⊗a) ≤ sup T (ϕ⊗a) = θU (a). ϕ≺O

ϕ≺O

ϕ≺U

(v) If 0 ≤ are open sets such that a for a ∈ P, and if Oi ∈ R O = i∈N Oi ∈ R, then θO (a) ≤ ∞ i=1 θOi (a). Indeed, let ϕ ≺ O. Because the support of ϕ is a compact subset of O, it is covered by ﬁnitely many of the open sets O i .n Thus there are functions ϕ1 , . . . , ϕn ∈ K(X) such that ϕi ≺ Oi and i=1 ϕi (x) = 1 for all x ∈ supp(ϕ). Then ϕi ϕ ≺ Oi and ϕ = ni=1 ϕi ϕ. Using (iv) we infer T (ϕ⊗a) =

n n ∞ ! ! ! T (ϕi ϕ)⊗a ≤ θOi (a) ≤ θOi (a). i=1

i=1

i=1

Taking the supremum, that is the limit over all ϕ ≺ O, yields our claim. (vi) If the open sets Oi ∈ R are pairwise disjoint ∞ and if O = i∈N Oi ∈ R, ∞ then θO = i=1 θOi , that is θO (a) = i=1 θOi (a) for all a ∈ P. Let O, Oi ∈ R be as in (vi). First suppose that 0 ≤ a for a ∈ P. We ﬁx n ∈ N and choose functions ϕi ≺ Oi for i = 1, . . . , n. Then ϕ = ni=1 ϕi ≺ O, hence n ! T (ϕi ⊗a) = T (ϕ⊗a) ≤ θO (a). i=1

Taking the suprema overall such choices of the functions ϕi ≺ Oi yields n !

θOi (a) ≤ θO (a).

i=1

This holds for all n ∈ N, and as ∞ !

θOi (a) = sup n∈N

i=1

we infer that

∞

i=1 θOi (a)

n !

θOi (a) ,

i=1

≤ θO (a). Together with (v), this yields ∞ !

θOi (a) = θO (a).

i=1

Now for the general case, let a ∈ P and w ∈ W. There is v ∈ V such that χO ⊗v ≤ uw and λ ≥ 0 such that 0 ≤ a + λv. Together with the limit rules in Lemma I.5.19, the above shows

290

III Measures on Locally Compact Spaces

lim

n !

n→∞

n n ! ! θOi (a) + θO (λv) = lim θOi (a) + lim θOi (v) n→∞

i=1

≤ lim

i=1 n !

n→∞

=

∞ !

n→∞

θOi (a + λv)

i=1

i=1

θOi (a + λv)

i=1

= θO (a) + θO (λv)

and n !

θO (a) + θO (λv) = lim n→∞

i=1

n !

≤ lim

n→∞

= lim n→∞

i=1 n !

θOi (a + λv) n ! θOi (a) + lim θOi (v) n→∞

i=1

θOi (a) + θO (λv).

i=1

Because θO (λv) ≤ λw by (ii), the cancellation law in Proposition I.5.9 yields that lim

n→∞

n !

θOi (a) ≤ θO (a) + εw

and

θO (a) ≤ lim

n→∞

i=1

n !

θOi (a) + εw

i=1

for all ε > 0. As w ∈ W was arbitrarily chosen, we infer that ∞ !

θOi (a) = θO (a),

i=1

hence our claim follows. Now, in the second step of the construction of the measure θ we proceed to extend the deﬁnition of θ to all relatively compact subsets of X. For a relatively compact set E ⊂ X and an element a ∈ P we set (vii)

θE (a) = lim θO (a). O⊃E

This limit is taken over the net whose index set is the downward directed family of all open sets O ∈ R containing E. For a positive element 0 ≤ a ∈ P this limit obviously exists, since the net θO (a) O⊃E is decreasing in Q as we established in (iv). For the general case we shall again use the criterion in Proposition I.5.23 to verify convergence. First we observe the following:

5. Integral Representation

291

(viii) If χE ⊗a ≤ χE ⊗b + uw for 0 ≤ a, b ∈ P, w ∈ W and a set E ∈ R, then θE (a) ≤ θE (b) + w. In particular, if χE ⊗v ≤ uw for some v ∈ V, then θE (v) ≤ w. To verify this, let 0 ≤ a, b ∈ P, w ∈ W and E ∈ R, and suppose that χE ⊗a ≤ χE ⊗b + uw , that is χE ⊗a ≤ χE ⊗b + s for some r-lower continuous V-valued function s ∈ uw . Let O ∈ R be an open set such that E ⊂ O. There is v ∈ V such that χO ⊗v ≤ uw . Let ε > 0. By the r-lower continuity of the function s + χO ⊗b (see Lemma 1.1), for every x ∈ E there is an open neighborhood Ux ⊂ O of x such that a ≤ (1 + ε) s(y) + b + εv for all y ∈ Ux . If we set U = x∈E Ux , then U is an open subset of O, hence U ∈ R, and we have χU ⊗a ≤ (1 + ε)(s + χU ⊗b) + ε χU ⊗v ≤ (1 + ε)χU ⊗b + (1 + 2ε)uw . This shows θU (a) ≤ (1 + ε)θU (b) + (1 + 2ε)w ≤ (1 + 2ε) θU (b) + w by (ii). Because ε > 0 was arbitrarily chosen and because (Q, W) is endowed with its weak preorder, this yields θU (a) ≤ θU (b) + w. Taking the limit (inﬁmum) over all open sets E ⊂ U ⊂ O in this last inequality demonstrates (viii). Now in order to show convergence of the net in (vii) in the general case, let E ⊂ X be relatively compact, and let a ∈ P. Given any neighborhood w ∈ W there is v ∈ V such that χE ⊗v ≤ uw , and there is λ ≥ 0 such that 0 ≤ a + λv. As we remarked before, the net θO (λv) O⊃E is convergent, and lim O⊃E θO (λv) = θE (λv) ≤ λw by (viii). Likewise, the net θO (a) + θO (λv) O⊃E is decreasing, hence convergent in Q. Thus Proposition I.5.23 yields indeed the convergence of the net θO (a) O⊃E . The linearity of θE as an operator from P into Q follows immediately from the corresponding properties of the operators θO , for O ⊃ E, as stated in (iii), together with the limit rules in Lemma I.5.19. If a ≤ b for a, b ∈ P, then θO (a) ≤ θO (b) for all O ⊃ E, thus θE (a) ≤ θE (b). The operator θE is therefore monotone and also continuous: Given w ∈ W we choose v ∈ V such that χE ⊗v ≤ uw . Then θE (v) ≤ w by (viii), and a ≤ b + v for elements a, b ∈ P implies θE (a) ≤ θE (b + v) ≤ θE (b) + w. This yields (ix) θE ∈ L(P, Q) for all relatively compact sets E ∈ R. We have (x) If 0 ≤ a for a ∈ P, and if E ⊂ F for relatively compact sets E, F ⊂ X, then 0 ≤ θE (a) ≤ θF (a).

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III Measures on Locally Compact Spaces

Indeed, let a, E, F be as in (x). Then θE (a) = lim θO (a) = inf θO (a) ≤ inf θO (a) = θF (a). O⊃E

O⊃E

O⊃F

Next we shall verify that θ is inner regular on all open sets in R, that is (xi) θO (a) = lim θK (a) for every open set O ∈ R and a ∈ P. K⊂O

The limit in (x) is taken over the upward directed family of all compact sets K ⊂ O. To verify this, let O ∈ R be open. First, let us consider an element 0 ≤ a ∈ P. In this case the net θK (a) K⊂O is increasing, hence convergent in Q, and obviously limK⊂O θK (a) ≤ θO (a). For the reverse inequality, let ϕ ≺ O and K = supp(ϕ) ⊂ O. Then T (ϕ⊗a) ≤ θU (a) for every open set K ⊂ U ∈ R, hence T (ϕ⊗a) ≤ θK (a), and therefore T (ϕ⊗a) ≤ limK⊂O θK (a). Since this holds for all functions ϕ ≺ O, we conclude that θO (a) ≤ limK⊂O θK (a) holds as well. Now for the general case let a ∈ P. Given any w ∈ W there is v ∈ V such that χO ⊗v ≤ uw and λ ≥ 0 such that 0 ≤ a + λv. As we observed before, this yields lim θK (a) + θK (λv) = θO (a) + θO (λv) K⊂O

as well as lim θK (v) = θO (v).

K⊂O

Using the preceding and the limit rules in Lemma I.5.19 we infer that lim θK (a) + θO (λv) = lim θK (a) + lim θK (λv) K⊂O K⊂O ≤ lim θK (a) + θK (λv)

K⊂O

K⊂O

= θO (a) + θO (λv) and θO (a) + θO (λv) = lim θK (a) + θK (λv) K⊂O

≤ lim θK (a) + lim θK (λv) K⊂O

K⊂O

= lim θK (a) + θO (λv). K⊂O

As θK (v) ≤ λw by (xi), we conclude using the cancellation law in Proposition I.5.10 that lim θK (a) ≤ θO (a) + εw

K⊂O

and

θO (a) ≤ lim θK (a) + εw K⊂O

5. Integral Representation

293

for all ε > 0. As w ∈ W was arbitrary, this shows indeed θO (a) = lim θO (a). K⊂O

Moreover, for compact subsets of X we establish: (xii) θK (a) = lim T (ϕ⊗a) for every compact set K ∈ K and a ∈ P. ϕ#K

This limit is taken over the downward directed family of all functions ϕ ∈ K(X) such that K ≺ ϕ, that is χK ≤ ϕ. To verify this, let K ∈ K. First, let 0 ≤ a ∈ P. The net T (ϕ⊗a) ϕ#K is decreasing and bounded below, hence convergent in Q. For every open set O ∈ R containing K there is ϕ ∈ K(X) such that K ≺ ϕ ≺ O. This shows T (ϕ⊗a) ≤ θO (a), hence limϕ#K T (ϕ⊗a) ≤ θO (a) and therefore lim T (ϕ⊗a) ≤ θK (a)

ϕ#K

by (vii). For the converse inequality, let ϕ # K and γ > 1. There is an open set O ⊃ K such that O ≺ γϕ, hence T (ψ ⊗a) ≤ γT (ϕ⊗a) for every ψ ≺ O. Therefore θO (a) ≤ γT (ϕ⊗a) and θK (a) ≤ γT (ϕ⊗a) by (vii) and (x). As this holds for all ϕ # K we infer that θK (a) ≤ γ lim T (ϕ⊗a). ϕ#K

Finally, as γ > 1 was arbitrarily chosen, and as Q is a locally convex complete lattice cone, this yields together with the above that θK (a) = lim T (ϕ⊗a). ϕ#K

Now for the general case, let a ∈ P. Given any w ∈ W there is v ∈ V such that χK ⊗v ≤ uw and λ ≥ 0 such that 0 ≤ a + λv. As we observed before, the latter yields lim T (ϕ⊗a) + T (ϕ⊗λv) = θK (a) + θK (λv). ϕ#K

Using the preceding and the limit rules in Lemma I.5.19 we infer that lim T (ϕ⊗a) + θK (λv) ≤ θK (a) + θK (λv) ≤ lim T (ϕ⊗a)) + θK (λv).

ϕ#K

ϕ#K

As θK (v) ≤ λw by (viii) we conclude using the cancellation law in Proposition I.5.10 that lim T (ϕ⊗a) ≤ θK (a) + εw

ϕ#K

and

θK (a) ≤ lim T (ϕ⊗a) + εw ϕ#K

for all ε > 0. As w ∈ W was arbitrary, this yields θK (a) = limϕ#K T (ϕ⊗a). (xiii) If 0 ≤ a for a ∈ P, and if Ei ⊂ X such that E = i∈N Ei is relatively compact, then θE (a) ≤ ∞ i=1 θEi (a).

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III Measures on Locally Compact Spaces

To verify this, we shall make use of our assumption that the order continuous linear functionals support the separation property for Q. We shall use Proposition I.5.35: Let E, Ei be as in (xiii), and 0 ≤ a ∈ P. Let U ∈ R be a relatively compact open set containing E. For every i ∈ N let Ai = {θOi (a) | Oi ∈ R, Oi open, Ei ⊂ Oi ⊂ U } ⊂ Q. These sets Ai are directed downward, and because all of their elements are ∞ inf Ai is obviously convergent since its partial sums positive, the series i=1 form an increasing sequence. The assumptions of I.5.35 are therefore satisﬁed, and we have

∞ ∞ ∞ ! ! ! Ai = inf Ai = θEi (a) inf i=1

i=1

i=1

∞

∞

by I.5.35 and (vi). Now let i=1 θO i (a) ∈ i=1 Ai , that is Ei ⊂ Oi ⊂ U for open sets Oi ∈ R, and let O = i∈N Oi . Then E ⊂ O ⊂ U, hence O is open and relatively compact. Thus θE (a) ≤ θO (a) ≤

∞ !

θOi (a),

i=1

by (x) and by (v), and therefore θE (a) ≤ inf

∞ ! i=1

Ai

=

∞ !

θEi (a).

i=1

as claimed in (xiii).

(xiv) If Ei ⊂ X such that E = i∈N Ei is relatively compact, and if θE (a) ≥ ∞ (a) holds for all 0 ≤ a ∈ P, then θE = ∞ i=1 θEi i=1 θEi , that is θE (a) = ∞ i=1 θEi (a) for all a ∈ P. Indeed, the assumptions of (xiv) imply together with (xiii) that θE (a) = ∞ θ (a) holds for all 0 ≤ a ∈ P. Then a similar argument as in the E i i=1 second part of (vi) veriﬁes our claim. We proceed to deﬁne and investigate measurability with respect to θ for relatively compact subsets of X. (xv) A relatively compact subset E of X is said to be measurable if θF = θ(F ∩E) + θ(F \E) holds for all relatively compact sets F ⊂ X. In the light of (xii), for measurability of a set E we shall only have to check that θF (a) ≥ θ(F ∩E) (a) + θ(F \E) (a) holds for all relatively compact subsets F ⊂ X and all 0 ≤ a ∈ P. Moreover: (xvi) A relatively compact subset E of X is measurable if and only if θO (a) ≥ θ(O∩E) (a) + θ(O\E) (a) holds for all open sets O ∈ R and all 0 ≤ a ∈ P.

5. Integral Representation

295

The necessity of this condition is obvious. For its suﬃciency, let 0 ≤ a ∈ P and let E, F ⊂ X be relatively compact. Let O ∈ R be open such that O ⊃ F. Then θO (a) ≥ θ(O∩E) (a) + θ(O\E) (a) ≥ θ(F ∩E) (a) + θ(F \E) (a). Taking the inﬁmum on the left-hand side over all such sets O ⊃ F yields θF (a) ≥ θ(F ∩E) (a) + θ(F \E) (a) by (vii), hence our claim by the preceding remark. (xvii) Every relatively compact open set is measurable. Let E ∈ R be open. We shall verify the criterion in (xvi): For this, let O ∈ R be open and let 0 ≤ a ∈ P. Then E ∩ O is open. Let ϕ ≺ E ∩ O for ϕ ∈ K(X) and let K = supp(ϕ) ∈ K. The set O \ K ⊃ O \ E is also open, and for every ψ ≺ O \ K we have ψ + ϕ ≺ O, hence θO (a) ≥ T (ψ ⊗a) + T (ϕ⊗a) by (i). Taking the supremum over all such functions ψ ≺ O \ K on the right-hand side of the last inequality yields by (i) θO (a) ≥ θ(O\K) (a) + T (ϕ⊗a) ≥ θ(O\E) (a) + T (ϕ⊗a). Now taking the supremum over all functions ϕ ≺ E ∩ O yields θE (a) ≥ θ(E\O) (a) + θ(E∩O) (a), hence our claim. (xviii) The measurable relatively compact subsets of X form a weak σ% ring R. We shall verify properties (R1), (R2) and (R3) from II.1.1 for the collection % of all measurable relatively compact subsets of X. Obviously, ∅ ∈ R, % R % that is (R1). If E1 , E2 ∈ R, then for every relatively compact subset F of X we have θF = θ(F ∩E1 ) + θ(F \E1 )

and

θ(F \E1 ) = θ((F \E1 )∩E2 ) + θ((F \E1 )\E2 ) ,

by the measurability of E1 and E2 , respectively. Because (F \ E1 ) \ E2 = F \ (E1 ∪ E2 ), this yields θF = θ(F ∩E1 ) + θ((F \E1 )∩E2 ) + θ((F \(E1 ∪E2 )) = θ(F ∩E1 ) + θ((F ∩(E2 \E1 )) + θ((F \(E1 ∪E2 )) . First we observe that F ∩(E1 ∪E2 ) ∩E1 = F ∩E1 and F ∩(E1 ∪E2 ) \E1 = (F \ E1 ) ∩ E2 . We have

296

III Measures on Locally Compact Spaces

θ(F ∩(E1 ∪E2 )) = θ(F ∩E1 ) + θ((F \E1 )∩E2 ) , by the measurability of E1 , hence θF = θ(F ∩(E1 ∪E2 )) + θ((F \(E1 ∪E2 )) by the above. This of the set E1 ∪ E2 . Secondly, shows the measurability we observe that F \ (E2 \ E1 ) ∩ E1 = F ∩ E1 and F \ (E2 \ E1 ) \ E1 = F \ (E1 ∪ E2 ). This shows θ(F \(E2 \E1 )) = θ(F ∩E1 ) + θ(F \(E1 ∪E2 ) , by the measurability of E1 , hence θF = θ(F ∩(E2 \E1 )) + θ((F \(E2 \E2 )) again using the above. This shows the measurability of the set E2 \ E1 , hence (R2). All left to show is (R3). By induction the union of ﬁnitely many measurable subsets of X is again measurable. Let (Ei )i∈N be a sequence of % such that E = ∞ Ei is again relatively compact. Let disjoint sets in R i=1 % by the preceding. For 0 ≤ a ∈ P and any Gn = ni=1 Ei . Then Gn ∈ R relatively compact set F ⊂ X we have for all n ∈ N θF (a) = θF ∩Gn (a) + θ(F \Gn ) (a) ≥ θF ∩Gn (a) + θ(F \E) (a). Now Gn ∩ En = En and Gn \ En = Gn−1 . As En is measurable, this shows θF ∩Gn (a) = θF ∩En (a) + θF ∩Gn−1 (a). Induction then leads to θF ∩Gn (a) =

n !

θF ∩Ei (a),

i=1

hence θF (a) ≥

n !

θF ∩Ei (a) + θ(F \E) (a)

i=1

for all n ∈ N, and therefore θF (a) ≥

∞ !

θF ∩Ei (a) + θ(F \E) (a)

i=1

≥ θF ∩E (a) + θ(F \E) (a) % by (xvi). by (xiii). This demonstrates E ∈ R % (xix) R ⊂ R.

5. Integral Representation

297

% % Let AR % = {A ⊂ X | A∩E ∈ R for all E ∈ R} be the σ-algebra generated % by R. In order to demonstrate that AR % contains all Borel subsets of X, it suﬃces to sow that it contains all open sets. For this, let U ⊂ X be an % for all E ∈ R. % open set. We shall use (xvi) to demonstrate that U ∩ E ∈ R Let O ⊂ X be relatively compact and open, and let 0 ≤ a ∈ P. Then the % by (xvii). In turn, set O ∩ U is open and relatively compact, hence in R % since every weak σ-ring is closed for ﬁnite the set O ∩ U ∩ E in also in R, intersections. As O \ (O ∩ U ∩ E) = O \ (U ∩ E), the latter yields θO = θ(O∩U ∩E) + θ(O\(U ∩E)) . This show that the set U ∩ E is indeed measurable. Now, ﬁnally, let E be a relatively compact Borel subset of X, that is E ∈ R, and let O ⊂ X be % by (xvii) and E ∈ A % relatively compact such that E ⊂ O. Then O ∈ R R % % as claimed. by the above. Thus E = E ∩ O ∈ R. This yields R ⊂ R % are pairwise disjoint and if E = Ei ∈ R, % (xx) If the sets Ei ∈ R i∈N ∞ θ , that is θ (a) = θ (a) for all a ∈ P. then θE = ∞ E i=1 Ei i=1 Ei % . Then the measurability of E2 First let E1 and E2 be disjoint sets in R implies that θ(E1 ∪E2 ) = θ((E1 ∪E2 )∩E2 ) + θ((E1 ∪E2 )\E2 ) = θE1 + θE2 . % follows by induction. If E ∈ R % is the disjoint Finite additivity of θ on R % union of the sets Ei ∈ R for i ∈ N, then for every 0 ≤ a ∈ P and every n ∈ N we have n ! θEi (a), θE (a) ≥ θ(∪ni=1 Ei ) (a) = i=1

and therefore θE (a) ≥ sup

n !

n∈N i=1

θEi (a) =

∞ !

θEi (a).

i=1

Our claim follows from (xiv). Summarizing our observations from (xviii), (xix), (xx), (ix), (vii) and (xi) we realize that (xxi) θ is a quasi regular R-bounded L(P, Q)-valued measure on R. Furthermore, (xxii) θ is continuous relative to the given r-lower continuous system V of inductive limit neighborhoods; more precisely: for w ∈ W and integrable functions f, g ∈ F(X,θ) (X, P) such that f ≤ g + uw we ( ( have X f dθ ≤ X g dθ + w. In order to verify (xxii), given w ∈ W, recall that uw ∈ V is a corresponding r-lower continuous neighborhood such that

298

III Measures on Locally Compact Spaces

T (f ) ≤ T (g) + w

whenever

f ≤ g + uw

for f, g ∈ FV (X, P). In a ﬁrst step we shall generalize the statement of (viii) to (not necessarily positive) elements a, b ∈ P. For this, let a, b ∈ R, E ∈ R and suppose that χE ⊗a ≤ χE ⊗b + uw . For u ∈ W choose v ∈ V such that θE (v) ≤ u and λ ≥ 0 such that both 0 ≤ a + λv and 0 ≤ b + λv. Then χE ⊗(a + λv) ≤ χE ⊗(b + λv) + uw , hence θE (a + λv) ≤ θE (a + λv) + w by (viii). Using the cancellation law from Proposition I.5.10(a) for the element θE (λv) ≤ λu yields θE (a) ≤ θE (b) + w + εu for all ε > 0. Because this holds for all u ∈ W, we have indeed θE (a) ≤ θE (b) + w. Now in the second step of our argument for (xxii), let s be any of the Vvalued functions in uw . By the r-lower continuity of s the set Os = {x ∈ X | s(x) = +∞} is open in X. Moreover, for any subset E ∈ R of Os and any a ∈ P we have χE ⊗a ≤ εs as well as 0 ≤ χE ⊗a + εs, and therefore by the argument from our ﬁrst step θE (a) ≤ εw as well as 0 ≤ θE (a) + εw for all ε > 0. This shows ' ' f dθ ≤ εw as well as 0≤ f dθ + εw Os

Os

for every integrable function f( ∈ F(X, P) and all ε > 0. Next we shall verify that E s dθ ≤ w holds for every subset E ∈ R of X \ Os . First we recall from Proposition 4.1(b) that χE ⊗s ∈ FR (X, P), since the function s is r-lower continuous and χE ⊗s is P-valued. We shall calculate the integral of s over the set E II.4.9. according to Deﬁnition For ( this, let u ∈ W and as in II.4 let vu = r ∈ SR (X, V) | X r dθ ≤ u , and correspondingly ' ' (u) s dθ = sup h dθ h ∈ SR (X, P), h ≤ s + vu . E

E

Let h ∈ SR (X, P) such that h ≤ s + vu , that is h ≤ s + r for some r ∈ vu . We may express both step functions h and r as h=

n ! i=1

χEi ⊗ai

and

r=

n !

χEi ⊗vi

i=1

with ai ∈ P, vi ∈ V and disjoint sets Ei ∈ R. Let {Ki }ni=1 be a family of compact sets such that Ki ⊂ (Ei ∩ E). We have ai ≤ s(x) + vi for all x ∈ Ki . Let O ∈ R be an open set containing E and let v ∈ V such that χO ⊗v ≤ uw and let ε > 0. Recall from Lemma 1.1 that the functions s + χX ⊗vi for i = 1, . . . , n are also r-lower continuous. Thus there is a family {Oi }ni=1 of disjoint open sets Oi ∈ R such that Ki ⊂ Oi ⊂ O and

5. Integral Representation

299

ai ≤ (1 + ε)(s(x) + vi ) + εv holds for all x ∈ Oi . There exist functions ϕi ∈ K(X) such that Ki ≺ ϕi ≺ Oi for i = 1, . . . , n. Then ϕi ⊗ai ≤ (1 + ε)ϕi ⊗s + (1 + ε)ϕi ⊗vi + εϕi ⊗v, and we set n !

f=

ϕi ⊗ai

and

g = (1 + ε)

i=1

n !

ϕi ⊗vi .

i=1

n n Because i=1 ϕi ⊗s ≤ s ≤ uw and i=1 ϕi ⊗s ≤ χ0 ⊗v ≤ uw , we have f ≤ g + (1 + 2ε)uw , and as f, g ∈ E(X, P) ⊂ FV (X, P), the latter implies n !

T (ϕi ⊗ai ) = T (f ) ≤ T (g) + (1 + 2ε)w = (1 + ε)

i=1

n !

T (ϕi ⊗vi ) + (1 + 2ε)w.

i=1

Next, while keeping the sets Ki and Oi ﬁxed, for each i = 1, . . . , n we consider the downward directed net of all functions Ki ≺ ϕi ≺ Oi and recall from (xii) that θKi (ai ) = lim T (ϕ⊗ai )

and

ϕi #Ki

θKi (vi ) = lim T (ϕ⊗vi ). ϕi #Ki

Carrying out this limit process in the preceding inequality then leads to n !

θKi (ai ) ≤ (1 + ε)

n !

i=1

θKi (vi ) + (1 + 2ε)w.

i=1

The latter holds for all ε > 0, hence n !

θKi (ai ) ≤

i=1

n !

θKi (vi ) + w.

i=1

Moreover, as Ki ⊂ Ei , we have n !

θKi (vi ) ≤

i=1

hence

n !

'

n !

r dθ ≤ u,

θEi (vi ) =

i=1

X

θKi (ai ) ≤ u + w.

i=1

Now, for each i = 1, . . . , n we consider the upward directed net of all Ki ⊂ (Ei ∩ E) and recall from Proposition 4.4(c) that θ(Ei ∩E) (ai ) ≤ lim θKi (ai ) + O θ (Ei ∩E) (ai ) . Ki ⊂(Ei ∩E)

300

III Measures on Locally Compact Spaces

From the ﬁrst step of our argument we recall that χEi ⊗ai ≤ χEi ⊗vi + uw implies that θEi (ai ) ≤ θEi (vi ) + w ≤ u + w, hence O θ (Ei ∩E) (ai ) ≤ O θEi (ai ) ≤ ε(u + w) for all ε > 0. Taking the limit over all such nets of sets Ki ⊂ Ei leads to ' h dθ =

n !

E

θEi (ai ) ≤ (u + v) + ε(u + w) = (1 + ε)(u + w)

i=1

for all ε > 0, and therefore '

n !

h dθ = This demonstrates

E

i=1

'

' s dθ ≤

θEi (ai ) ≤ u + v.

(u)

s dθ ≤ u + w,

E

E

for all u ∈ W, hence

' s dθ ≤ w. E

Now in the third and ﬁnal step of this argument, let f, g ∈ F(X,θ) (X, P) and suppose that f ≤ g + uw , that is f ≤ g + s for some s ∈ uw . Let Os = {x ∈ X | s(x) = +∞}. For every E ∈ R both functions f and g are integrable over the sets E ∩ Os ∈ R and E \ Os ∈ R. We have ' ' f dθ ≤ εw ≤ g dθ + 2εw (E∩Os )

(E∩Os )

for all ε > 0 by the above, and ' ' f dθ ≤ (g + s) dθ (E\Os ) (E\Os ) ' ' g dθ + ≤ (E\Os )

Thus

' E

s dθ ≤

(E\Os )

' f dθ =

'

' f dθ +

(E∩Os )

' ≤

f dθ '

(E\Os )

g dθ + (E∩Os )

'

g dθ + (1 + 2ε)w (E\Os )

g dθ + (1 + 2ε)w

= E

g dθ + w. (E\Os )

5. Integral Representation

for all ε > 0, hence

301

'

' f dθ ≤ E

g dθ + w. E

This holds for all E ∈ R and yields ' ' f dθ ≤ g dθ + w X

X

by our deﬁnition of the integral over X in II.4.13. This completes our proof for (xxii). Note that Parts (b) and (c) of Proposition 4.8 now yield that all functions in FV (X, P) or in FV0 (X, P) are integrable or indeed strongly integrable over every Borel set F ⊂ X with respect to the measure θ. All left to show is that θ represents the given operator T as stated in our Theorem. (xxiii) Let ϕ⊗a ∈ E(X, P) for ϕ ∈ K(X) and a ∈ P. Then ' ' ϕ⊗a dθ ≤ T (ϕ⊗a) ≤ ϕ⊗a dθ + O θA (a) X

X

where A ∈ R denotes the support of the function ϕ. We may assume that 0 ≤ ϕ ≤ 1 an denote by A ⊂ X the compact support of ϕ. Let us ﬁrst consider the case that a ≥ 0. We ﬁx n ≥ 2 in N, and for i ≥ 1 deﬁne compact sets i n . Ai = x ∈ X | ϕ(x) ≥ n Then A = An0 ⊃ An1 ⊃ . . . ⊃ An+1 = ∅. For i ≥ 0 we deﬁne ψin ∈ K(X) by ⎧ 1 ⎪ if x ∈ Ani+1 ⎨n, n i ψi (x) = ϕ(x) − n , if x ∈ Ani \ Ani+1 ⎪ ⎩ 0, if x ∈ / Ani Then Ani+1 ≺ nψin ≺ Ani . All the functions ψin ⊗a are contained in E(X, P), hence 1 1 θAn (a) ≤ T (ψin ⊗a) ≤ θAni (a). n i+1 n The ﬁrst part of the last inequality follows from (xii), the second part from (vi) and the fact that nψin ≺ O for every open set O ∈ R containing Ani . Let x ∈ An0 such that nk ≤ ϕ(x) < k+1 n for some k = 0, . . . , (n+1), that is x ∈ Ank \Ank+1 . Then ψin (x) = n1 for all i = 0, . . . , (k −1), ψkn (x) = ϕ(x)− nk and ψin (x) = 0 for all i = k + 1, . . . , (n + 1). This shows n ! i=0

ψin (x) =

k k + ϕ(x) − = ϕ(x). n n

302

III Measures on Locally Compact Spaces

and

n !

ψin (x)

i=1

0 = ϕ(x) −

if ϕ(x) < if ϕ(x) ≥

1 n

1 n 1 n

We set ϕn = ni=1 ψi and conclude from the above that ϕn ⊗a ≤ ϕ⊗a and ϕn X ϕ as n tends to inﬁnity, in the sense of Section II.5.22. Indeed, given x ∈ X, v ∈ V and ε > 0, we have either ϕ(x) = 0, hence ϕ(x)a ∈ vε ϕn (x)a for all n ∈ N, or there is n0 ∈ N such that ϕ(x) ≤ (1 +ε)ϕn (x), hence ϕ(x)a ≤ (1 + ε)ϕn (x)a and therefore ϕ(x)a ∈ vε ϕn (x)a for all n ≥ n0 . We may therefore apply or Convergence Theorem II.5.25 with the measures θn = θ and the functions fn = ϕn ⊗a, f = ϕ⊗a, and f∗ = f∗∗ = 0. This yields ' ' ϕ⊗a dθ ≤ lim ϕn ⊗a dθ, n→∞

X

hence

'

X

' ϕ⊗a dθ = lim

n→∞

X

ϕn ⊗a dθ, X

as ϕn ⊗a ≤ ϕ⊗a for all n ∈ N. Next we choose the step functions 1! χAni ⊗a ∈ SR (X, P). n i=1 n

hn = As ψin ⊗a ≤ quently

1 n

χAni ⊗a for all i = 0, . . . , n, we have ϕn ⊗a ≤ hn , and conse-

'

'

n n 1! 1! θAni (a) = θAn (a) n i=1 n i=0 i+1 X " n # n ! ! ≤ T (ψin ⊗a) = T ψin ⊗a = T (ϕ⊗a).

ϕn ⊗a dθ ≤ X

hn dθ =

i=0

i=0

Thus the above demonstrates that ' ' ϕ⊗a dθ = lim ϕn ⊗a dθ ≤ T (ϕ⊗a). n→∞

X

X

For the reverse inequality, we observe that T (ϕ⊗a) =

n !

T (ψin ⊗a)

i=0

1! 1! 1 θAni (a) = θAn (a) + θA (a). n i=0 n i=1 i n n

≤

n

5. Integral Representation

303

As

1! χAi ⊗a ≤ ϕn ⊗a ≤ ϕ⊗a n i=1 n

hn = implies

'

1! θA (a) ≤ n i=1 i n

hn dθ = X

'

we realize that T (ϕ⊗a) ≤

ϕ⊗a dθ + X

' ϕ⊗a dθ, X

1 θA (a) n

holds for all n ∈ N. This shows ' T (ϕ⊗a) ≤ ϕ⊗a dθ + O θA (a) . X

Now suppose that the element a ∈ P is not necessarily positive. Given w ∈ W there is uw ∈ V such that T (f ) ≤ T (g) + w whenever f ≤ g + uw for f, g ∈ FV (X, P). In turn, there is v ∈ V such that χA ⊗v ≤ uw and λ ≥ 0 such that 0 ≤ a + λv. Then ' ϕ⊗(a + λv) dθ ≤ T ϕ⊗(a + λv) X ' ϕ⊗(a + λv) dθ + O θA (a + λv) ≤ X

and

'

'

ϕ⊗v dθ + O θA (v)

ϕ⊗v dθ ≤ T (ϕ⊗v) ≤ X

X

holds by the preceding argument for positive elements. Thus, ﬁrstly, ' ' ϕ⊗a dθ + λ ϕ⊗v dθ ≤ T (ϕ⊗a) + λT (ϕ⊗v) X X ' ≤ T (ϕ⊗a) + λ ϕ⊗v dθ + O θA (v) , X

hence

'

' ϕ⊗a dθ ≤ T (ϕ⊗a) + O ϕ⊗v dθ + O θA (v) ,

X

X

by the cancellation (law from Proposition I.5.10(a), and because ϕ⊗v ≤ χA ⊗v ≤ uw we have X ϕ⊗v dθ ≤ θA (v) ≤ w, hence ' ϕ⊗a dθ ≤ T (ϕ⊗a) + εw X

304

III Measures on Locally Compact Spaces

for all ε > 0. This holds for all w ∈ W and therefore yields ' ϕ⊗a dθ ≤ T (ϕ⊗a). X

Secondly, we have by the above ' ϕ⊗v dθ ≤ T (ϕ⊗a) + λT (ϕ⊗v) T (ϕ⊗a) + λ X ' ' ≤ ϕ⊗a dθ + λ ϕ⊗v dθ + O θA (a) + O θA (v) . X

X

Again using the cancellation law from Proposition I.5.10(a), this yields ' ' ϕ⊗a dθ + O ϕ⊗v dθ + O θA (a) + O θA (v) , T (ϕ⊗a) ≤ X

X

'

that is T (ϕ⊗a) ≤

ϕ⊗a dθ + O θA (a) + εw

X

for all ε > 0. This holds for all w ∈ W and therefore yields ' ϕ⊗a dθ + O θA (a) T (ϕ⊗a) ≤ X

as claimed. (xxiv) Let ϕ⊗a ∈ E0 (X, P) for ϕ ∈ K(X) and a ∈ P. Then ' ϕ⊗a dθ = T (ϕ⊗a). X

For this, let ϕ⊗a ∈ E0 (X, P) such that 0 ≤ ϕ ≤ 1 and let A ∈ R be the compact support of ϕ. Following Proposition 1.11(a) we have to distinguish two cases: In the ﬁrst case, let us assume that the element a ∈ P is bounded in F (X, P), V , hence θA (a) is bounded in P. Then χA ⊗a isbounded V in Q. This shows O θA (a) = 0 and therefore ' ϕ⊗a dθ = T (ϕ⊗a) X

by (xxiii). If on the other hand, the element a ∈ P is unbounded in P, then the set {x ∈ X | ϕ(x) > 0} is both open and closed in X, hence coincides with the support A of ϕ. Moreover, we have ϕ(x) ≥ ρ for all x ∈ A with some ρ > 0. Thus χA ≤ (1/ρ)ϕ. Given w ∈ W let v ∈ V such that χA ⊗v ≤ uw and λ ≥ 0 such that 0 ≤ a + λv. Then χA (a) ≤ χA ⊗(a + λv) ≤

1 ϕ⊗(a + λv), ρ

5. Integral Representation

hence θA (a) ≤

1 ρ

305

' ϕ⊗(a + λv) dθ ≤ X

(

1 ρ

' ϕ⊗a dθ + X

λ w. ρ

This ( shows θA (a) ∈ Bw X ϕ⊗a dθ for all w ∈ W, hence θA (a) ∈ B X ϕ⊗a dθ and therefore ' ' ϕ⊗a dθ + O θA (a) = ϕ⊗a dθ X

X

by Proposition I.5.14. Thus (xxiii) yields our claim in this case as well. Thus ﬁnally, both θ and T represent continuous linear operators from the locally convex cone FV (X, P), V into the locally convex complete lattice cone (Q, W). Both operators coincide ( on the subcone E0 (X, P), hence on its closure FV0 (X, P). Moreover, as X f dθ ≤ T (f ) holds for all functions f in the dense subcone E(X, P) of F(X, P), this inequality holds for all f ∈ F(X, P). Now all left to demonstrate is the uniqueness of the representing measure. This will follow with Proposition 4.5. Let ϑ be any representing measure for the operator T : FV (X, P) → Q satisfying the stated properties. Then ( (xxv) θE (a) = lim X ϕi ⊗a dϑ for every E ∈ R and a ∈ P, i∈I

where I consists of all ordered pairs (O, f ), where O ∈ O, the collection of all open supersets O ∈ R of E, and f : O → K(X) is a mapping such that f (O) ≺ O for all O ∈ O. This index set is ordered by (O1 , f1 ) ≤ (O2 , f2 ) if O1 ⊃ O2 and f1 (O) ≤ f2 (O) for all O ∈ O. We set ϕi = f (O) ∈ K(X) for i = (O, f ) ∈ I. Let Ai denote the support of the function ϕi ∈ K(X). Then Ai ⊂ O, hence O θAi (a) ≤ O θO (a) by I.5.11. Thus following Proposition 4.5(c) and the properties stated in Theorem 5.1 lim O θAi (a) ≤ lim O θO (a) ≤ O lim θO (a) = O θE (a) i∈I

O∈O

O∈O

by Proposition I.5.24. This yields ' lim T (ϕi ⊗a) ≤ lim i∈I

i∈I

'

ϕi ⊗a dϑ + O θAi (a)

X

ϕi ⊗a dϑ + lim O θAi (a) i∈I X i∈I ≤ θE (a) + O θE (a) = θE (a) ' = lim ϕi ⊗a dϑ

≤ lim

i∈I

X

≤ lim T (ϕi ⊗a) i∈I

306

III Measures on Locally Compact Spaces

in the order topology of Q. The value ϑE (a) ∈ Q, for all a ∈ P and E ∈ R, is therefore uniquely determined by the operator T. This completes our argument. Remarks 5.2. (a) Let us recollect and summarize the main steps in the construction of the quasi regular representing measure θ for the operator T : FV (X, P) → Q in Theorem 5.1. Using Proposition 4.5 we obtain: (i) For every open set O ∈ R and a ∈ P we have ' θO (a) = lim ϕ⊗a dθ = lim T (ϕ⊗a). ϕ≺O

X

ϕ≺O

(ii) For every compact set K ∈ R and a ∈ P we have ' θK (a) = lim ϕ⊗a dθ = lim T (ϕ⊗a). ϕ#K

X

ϕ#K

(iii) For every E ∈ R and every open set O ∈ R containing E there is a net (ϕi )i∈I in K(X) such that ϕi ≺ 0 for all i ∈ I, and ' θE (a) = lim ϕi ⊗a dθ = lim T (ϕi ⊗a) i∈I

X

i∈I

for all a ∈ P in the order topology of Q. (b) Part (a) yields the following implication: Let θ be the representing measure for the operator T : FV (X, P) → Q. Let O ∈ R be an open set. Part(iii) of 5.2(a) then implies that for every subset E ∈ R of O and every a ∈ P the element θE (a) ∈ Q is contained in the order closure in Q of the image under T of the subset A = {ϕ⊗a | ϕ ∈ K(X), ϕ ≺ O} of E(X, P). We observe that this set is relatively bounded in FV (X, P). Indeed,let ψ ∈ K(X) such that O ≺ ψ. Let v ∈ V and choose v ∈ V such that χF ⊗v ≤ v, where F ∈ R denotes the support of ψ. Let λ ≥ 0 such that 0 ≤ a + λv. Then for every function ϕ ∈ K(X) such that ϕ ≺ O we have ϕ ≤ ψ and 0 ≤ ϕ⊗(a + λv) ≤ ψ ⊗(a + λv) ≤ ψ ⊗a + λv, and therefore 0 ≤ ϕ⊗a + λv

and

ϕ⊗a ≤ ψ ⊗a + λv.

The set A is therefore bounded below and bounded above relative to the function ψ ⊗a, thus relatively bounded in FV (X, P) according to I.4.24.

6. Special Cases and Applications

307

Recall from a remark in I.4.24 that every continuous linear operator maps relatively bounded sets into relatively bounded sets.

6. Special Cases and Applications We shall discuss a range of special cases and applications of Theorem 5.1 in this ﬁnal section. Most of these settings had been earlier dealt with in Section 6 of Chapter II. Throughout the following, we shall assume that (P, V) is a quasi-full locally convex cone and that (Q, V) is a locally convex complete lattice cone whose order continuous linear functionals support the separation property. We shall apply Theorem 5.1 with the standard full extension (PV , V) in place of P and assume that the linear operator T : FV (X, P) → Q can be extended to FV (X, PV ). In Section 3 we investigated various situations where these extensions are guaranteed, in particular if Q = R or if (P, V) is a locally convex vector space. Throughout the following (Q0 , W0 ) will stand for a locally convex cone whose standard lattice completion in the sense of I.5.57 is (Q, W). We shall use the notations of Chapters I, II and the preceding sections of Chapter III. 6.1 The Case that Q Is the Standard Lattice Completion of Some of a Subcone Q0 . Suppose that (Q, W) is the standard lattice completion locally convex cone (Q0 , W0 ) (see I.5.57) and that T FV (X, P) ⊂ Q0 . Then T maps relatively bounded subsets of FV (X, P) into relatively bounded subsets of Q0 . Hence the bounded quasi regular L(P, Q)-valued measure θ that represents T as in 5.1 takes its values θE (a) for E ∈ R and a ∈ P, in the order closure in Q of relatively bounded subsets of Q0 see Remark 5.2(b) . According to I.5.57 these values are therefore elements of Q∗∗ 0 , the second dual of Q0 , and according to I.7.3 indeed contained in the dual of Q∗0 , if Q∗0 is endowed with the topology generated by the family Z of all relatively bounded subsets of Q0 . This cone was introduced as the (relative) strong second dual Q0∗∗sr of Q0 in Section I.7.3. Corollary 6.2. Suppose that (Q, W) is the standard lattice completion of some locally convex cone (Q0 , W0 ). Then, under the assumptions of Theorem 5.1,the representing measure for a linear operator T :FV (X,P)→Q0 , is L P, Q0∗∗sr -valued. We may obtain a further strengthening of this observation in special cases: 6.3 Compact and Weakly Compact Operators. Suppose that as in 6.1 (Q, W) is the standard lattice completion of a locally convex cone (Q0 , W0 ). A linear operator T : E(X, P) → Q is called compact (or weakly compact) if for every E ∈ R and every relatively bounded subset A of P the image under T of the set

308

III Measures on Locally Compact Spaces

n ! i=1

n ! ϕi ⊗ai ai ∈ A, ϕi ∈ K(X), ϕi ≤ χE i=1

is relatively compact in the symmetric relative topology or in the weak topol ∗ ogy σ(Q0 , Q0 ) of Q0 . (For the deﬁnition of the weak topology see I.4.6.) Recall from Lemma I.4.7 that the symmetric relative topology is ﬁner than σ(Q0 , Q∗0 ), and from I.5.57 that σ(Q0 , Q∗0 ) is ﬁner than the induced order topology on Q0 which is however still Hausdorﬀ. The latter two topologies coincide, if all elements of Q0 are bounded (see I.5.57). Moreover, σ(Q0 , Q∗0 ) coincides with its own relative topology (see I.4.6). Every subset M of Q0 which is relatively compact in the symmetric relative topology is also relatively weakly compact. Indeed, the closure M of M with respect to the w symmetric relative topology is contained in its closure M with respect to the weak topology. M is compact in the former, hence also in the latter topology, thus weakly closed since σ(Q0 , Q∗0 ) is Hausdorﬀ. We infer that w M = M , and our claim follows. Every compact operator is therefore also weakly compact. Remarks 6.4. (a) For any E ∈ R and every relatively bounded subset A of P the set

n n ! ! A= ϕi ⊗ai | ai ∈ A, ϕi ∈ K(X), ϕi ≤ χE i=1

i=1

from 6.2 is relatively bounded in FV (X, P). Indeed, let ψ ∈ K(X) such that E ≺ ψ. There is a0 ∈ P such that A is bounded above relative to a0 . Now let v ∈ V and choose v ∈ V such that χF ⊗v ≤ v, where F ∈ R is the support of the function ψ. Let λ, ρ ≥ 0 such that 0 ≤ a0 + λv, 0 ≤ a + λv and a ≤ ρa0 + λv for all a ∈ A. Now let ϕ1 , . . . , ϕn ∈ K(X) such that n i=1 ϕi ≤ χE and let a1 , . . . an ∈ A. Then 0≤

n ! i=1

ϕi ⊗(ai + λv) ≤

n ! i=1

ϕi ⊗ai + λχE ⊗v ≤

n !

ϕi ⊗ai + λv

i=1

and n ! i=1

ϕi ⊗ai ≤

n !

ϕi ⊗(ρa0 + λv) ≤ ψ ⊗(ρa0 + λv) ≤ ρψ ⊗a0 + λχF ⊗v ≤ ρψ ⊗a0 + λv.

i=1

The set A is therefore bounded below and bounded above relative to the function ψ ⊗a0 , thus relatively bounded in FV (X, P). (b) As a consequence of (a), every continuous linear operator T : FV (X, P) → Q0 that maps relatively bounded subsets of FV (X, P) into relatively (weakly) compact subsets of Q0 , that is every continuous linear operator which is (weakly) compact in the usual sense, is also (weakly) compact

6. Special Cases and Applications

309

in the sense of 6.3. The converse of this statement holds true in special cases: Suppose that X is compact, that is X ∈ R, and that FV (X, P) carries the topology of uniform convergence generated by a single neighborhood; more precisely: The inductive limit neighborhood system V consists of multiples of a single neighborhood v which in turn contains the single constant function x → v for a ﬁxed neighborhood v ∈ V, that is V = {εv | ε> 0}, where v = {χX ⊗v}. Then FV (X, P) = C r (X, P) see Remark 2.10(c) . In this situation, suppose that the operator T : FV (X, P) → Q0 is (weakly) compact in the sense of 6.2 and let B be a relatively bounded subset of FV (X, P), that is there is g ∈ FV (X, P) and λ ≥ 0 such that 0 ≤ f + λχX ⊗v and f ≤ g + λχX ⊗v for all f ∈ B. We may also assume that 0 ≤ g + λχX ⊗v. We proceed to construct a corresponding subset A of E(X, P) in the following way: Because FV (X, P) is the closure of the subcone E(X, P), for every f ∈ B and every 0 < ε ≤ 1 we can ﬁnd an element h(f,ε) ∈ E(X, P) such that h(f,ε) ∈ vsε (f ). According to Lemma I.4.1(b) and (c) this implies h(f,ε) ≤ (1 + ε)f + ε(1 + λ)χX ⊗v

and

f ≤ (1 + ε)h(f,ε) + ε(2 + λ)χX ⊗v.

Using an argument involving a partition of the unit, we may indeed assume that the functions h(f,ε) are all of the type n !

ϕi ⊗ai ,

where

ai ∈ P

and

i=1

n !

ϕi = 1,

i=1

and for each i = 1, . . . , n there is a point xi ∈ X such that ϕi (xi ) = 1. s Similarly, there is a function l = m k=1 ψk ⊗bk ∈ E(X, P) such that l ∈ v1 (g), which implies by I.4.1(b) and (c) l ≤ 2g + (1 + λ)χX ⊗v

and

g ≤ 2l + (2 + λ)χX ⊗v.

Now let A = {h(f,ε) | f ∈ B, ε > 0}. Following our construction, B is contained in the closure A of A, taken relative inductive limit topology. in FV (X, P) with respect to its symmetric Moreover, for any function h(f,ε) = ni=1 ϕi ⊗ai ∈ A, for any i = 1, . . . , n, we have h(xi ) = ai for some xi ∈ X, hence 0 ≤ f (xi ) + λv ≤ 2h(xi ) + (2 + λv) + λv = 2ai + 2(1 + λ)v, and 0 ≤ ai + (1 + λ)v. Furthermore, ai = h(xi ) ≤ 2f (xi ) + (1 + λ)v ≤ 2g(xi ) + (1 + 3λ)v ≤ 2l(xi ) + (5 + 5λ)v,

310

III Measures on Locally Compact Spaces

where l(xi ) = m k=1 ψk (xi )bk . There is ρ ≥ 0 such that 0 ≤ bk + ρv for k = 1, . . . , m. We set b = b1 + . . . + bm and realize that m !

ψk (xi )bk ≤

k=1

m !

ψk (xi )bk + 1 − ψk (xi ) (bk + ρv) ≤ b + mρv.

k=1

Thus ai ≤ 2b + (5 + 5λ + mρ)v for the elements ai ∈ P from above. We conclude that the set A ⊂ P of all values of the functions in A is bounded below and bounded above relative to the element b ∈ P. Therefore T (A) is contained in the relatively (weakly) compact image under T of the set

n n ! ! ϕi ⊗ai ai ∈ A, ϕi ∈ K(X), ϕi ≤ 1 . i=1

i=1

Because the operator T : FV (X, P) → Q0 is also continuous if we consider the respective symmetric relative topologies on FV (X, P) and Q0 (see Proposition I.4.5), we have T B ⊂ T A ⊂ T (A), where T (A) denotes the closure of T (A) in Q0 with respect to the symmetric relative topology. But this topology is ﬁner than the weak topology σ(Q0 , Q∗0 ) (see Lemma I.4.7), hence T (A) is contained in the weak closure of T (A) which was seen to be (weakly) compact. Thus T (B) is also relatively (weakly) compact in Q0 , and the operator T : FV (X, P) → Q0 is (weakly) compact in the usual sense, as claimed. For the next corollary, recall the introduction of various operator topologies from Example I.7.2(a). In particular, the strong operator topology on L(P, Q) is generated by the family Z of all ﬁnite subsets of P and the symmetric topology of Q see I.7.3(a)(ii) ; the weak operator topology by thefamily Z of allﬁnite subsets of P and the weak topology σ(Q, Q∗ ) of Q see I.7.3(a)(iii) . We shall use these deﬁnitions with the subcone Q0 of Q in place of Q. Corollary 6.5. Suppose that (Q, W) is the standard lattice completion of some locally convex cone (Q0 , W0 ) and that, under the assumptions of Theorem 5.1, the operator T : FV (X, P) → Q0 is compact (or weakly compact) on E(X, P). Then the following holds: (a) The representing measure θ for T is L(P, Q0 )-valued. (b) For every E ∈ R and a ∈ P the set {θG (a) | G ∈ R, G ⊂ E} is compact (or weakly compact) in Q0 .

6. Special Cases and Applications

311

(c) For every E ∈ R the set {θG | G ∈ R, G ⊂ E} of linear operators from P into Q0 is relatively compact in L(P, Q0 ), if endowed with the symmetric relative strong (or with the symmetric weak) operator topology. Proof. Suppose that the operator T : FV (X, P) → Q0 is compact (or weakly compact) in the sense of 6.3 and recall that compactness implies weak compactness. Let E ∈ R and let O ∈ R be any open set containing E. For Parts (a) and (b), let a ∈ P and let G ∈ R be a subset of E. According to 5.2(b) the element θG (a) ∈ Q is contained in the order closure in Q of the image under T of the subset A = {ϕ⊗a | ϕ ∈ K(X), ψ ≺ O} of E(X, P). This image is relatively compact (or relatively weakly compact) w in Q0 by 6.3. Let T (A) and T (A) denote its closure in Q0 with respect to the symmetric relative topology and with respect to the weak topology σ(Q0 , Q∗0 ), respectively. Let us ﬁrst consider the weakly compact case: w Then T (A) is weakly compact. Recall from I.5.57 that the weak topology ∗ σ(Q0 , Q0 ) is generally ﬁner than the induced order topology on Q0 which is w however still Hausdorﬀ. The subset T (A) of Q0 is therefore also compact in the coarser induced order topology, and indeed closed in Q as the order topology is Hausdorﬀ in this case. Similarly, in the compact case, because the symmetric relative topology on Q0 is also ﬁner than the induced order topology on Q0 , one argues that the set T (A) is closed in Q in the order topology. This demonstrates in particular that θG (a) ∈ T (A) ⊂ Q0 in the w compact case, or in θG (a) ∈ T (A) ⊂ Q0 in the weakly compact case. The set {θG | G ∈ R, G ⊂ E} is therefore compact (or weakly compact) and contained in Q0 as claimed. For Part (c), let O = {θG | G ∈ R, G ⊂ E} ⊂ L(P, Q0 ), and for every a ∈ P let O(a) = {θG (a) | G ∈ R, G ⊂ E}. According to Part (b), this set is relatively (weakly) compact in Q0 . Its closure O(a) in Q0 (taken either in the symmetric relative or in the weak topology)is therefore (weakly) compact. Following Tychonoﬀ’s theorem the set C = a∈P O(a) is compact in the product of the symmetric relative (or weak) topology. Now let (Si )i∈I be a net in O. Then the mapping i → Si (a) a∈P : I → C is a net in C which because of compactness permits a convergent subnet j → Sj (a) a∈P : J → C.

312

III Measures on Locally Compact Spaces

Let S(a) a∈P ∈ C be the limit of this net. We claim that the mapping a → S(a) : P → Q0 is a continuous linear operator. Indeed, linearity follows directly from the linearity of the operators Sj in the approximating net. For continuity, given w ∈ W0 there is v ∈ V such that |θ|(E, v) ≤ w. Then a ≤ b + v for a, b ∈ P0 implies that θG (a) ≤ θG (b) + w for all θG ∈ O. Because all the operators Sj are contained in O, this shows that S(a) ≤ S(b) + w holds as well. Thus S ∈ L(P, Q0 ), and the net (Sj )j∈J converges to S in the relative strong (or weak) operator topology of L(P, Q0 ). This demonstrates, as claimed, that the set O is relatively compact in this topology. For the next corollary recall the deﬁnition of a compact and a weakly compact measure from Chapter II.6.8. Corollary 6.6. Suppose that (Q, W) is the standard lattice completion of some locally convex cone (Q0 , W0 ) and that all elements of Q0 are bounded. Under the assumptions of Theorem 5.1, the linear operator T : FV (X, P) → Q0 is compact (or weakly compact) on E(X, P) if and only if its representing measure θ is L(P, Q0 )-valued and compact (or weakly compact). In this case, and if all elements of P are bounded as well, the measure θ is countably additive with respect to the strong operator topology. Proof. Let us ﬁrst assume that the operator T : FV (X, P) → Q0 is compact (or weakly compact), and let θ be its L(P, Q0 )-valued representing measure. Given E ∈ R and a relatively bounded subset A of P, let O ∈ R be an open set containing E. Then the image under T of the set

n n ! ! ϕi ⊗ai ai ∈ A, ϕi ∈ K(X), ϕi ≤ χO A= i=1

i=1

is relatively compact (or relatively weakly compact) in Q0 . Let us recall our earlier remarks: In the second, that is the relatively weakly compact w case, the weak closure T (A) of T (A) in Q0 is weakly compact. In the ﬁrst, that is the relatively compact case, the closure T (A) of T (A) in the w symmetric topology coincides with T (A) and is compact in both topologies. Now consider disjoint compact subsets K1 , . . . , Kn of E, and let O1 , . . . , On be disjoint open subsets of O such that Kj ⊂ Oj for j = 1, . . . , n. Let a1 , . . . , an ∈ A. For each j = 1, . . . , n let (ϕij )ij ∈Ij be a net in K(X) as in 5.2(a)(iii), with Kj in place of E and Oj in place of O, that is ϕij ≺ 0j for all ij ∈ Ij and j = 1, . . . , n, as well as θKj (a) = limij ∈Ij T (ϕij ⊗a) for all a ∈ P. Let I = I1 × . . . × In , endowed with the componentwise order, and for i = (i1 , . . . , in ) set fi =

n ! j=1

ϕij ⊗aj .

6. Special Cases and Applications

313

Then fi ∈ A for all i ∈ I and lim T (fi ) = i∈I

n !

lim T (ϕi ⊗aj ) =

j=1

i∈I

n ! j=1

lim T (ϕij ⊗aj ) =

ij ∈Ij

n !

θKj (aj )

j=1

by Proposition j = 1, I.5.22 and Lemma I.5.20(d), since for every . . . , n the net T (ϕi ⊗aj ) i∈I can be considered to be a subnet of T (ϕij ⊗aj ) ij ∈Ij . This n yields that the element j=1 θKj (aj ) is contained in the (compact or weakly w

compact) closure T (A) of T (A). Now let E1 , . . . , En ∈ R be disjoint subsets of E and let a1 , . . . , an ∈ A. Because all elements of Q0 are supposed to be bounded, Proposition 4.4 yields that θEj (aj ) = limKj ⊂Ej θKj (aj ) for every j = 1, . . . , n. Now an argument similar to the above yields lim

n !

K1 ⊂ E1 , ... j=1 Kn ⊂ En

θKj (aj ) =

n ! j=1

From this we infer that the element w

lim θKj (aj ) =

n !

Kj ⊂Ej

n

j=1 θEj (aj )

θEj (aj ).

j=1 w

is also contained in T (A) .

As a subset of T (A) the set

n ! θEi (ai ) ai ∈ A, Ei ∈ R disjoint subsets of E i=1

is therefore also relatively compact (or weakly compact), hence the measure θ is compact (or weakly compact) in the sense of II.6.8. If the elements of P are also bounded, then our version of Pettis’ theorem, that is Theorem II.3.11 applies: The representing L(P, Q0 )-valued measure θ is countably additive with respect to the strong operator topology of L(P, Q0 ) in this case. Now suppose that the measure θ is compact (or weakly compact), let E ∈ R and let A be a relatively bounded subset of P. According to Proposition II.6.9 then

n ' n ! ! ϕi ⊗ai ai ∈ A, 0 ≤ ψi measurable, ϕi ≤ χE i=1

X

i=1

is a relatively compact (or relatively weakly compact) subset of Q(0 . As all elements of Q0 are bounded, Theorem 5.1 yields that T (ϕ⊗a) = X ϕ⊗a dθ holds for all ϕ⊗a ∈ E(X, P). Thus the subset

n n ! ! T (ϕi ⊗ai ) ai ∈ A, ϕi ∈ K(X), ϕi ≤ χ E i=1

i=1

of the above is also relatively compact (or relatively weakly compact), hence our claim.

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III Measures on Locally Compact Spaces

Proposition II.6.11 states that in special circumstances every bounded L(P, Q0 )-valued measure is weakly compact. In combination with Corollary 6.6 this yields: Corollary 6.7. Suppose that (P, ) is a ﬁnite dimensional normed space and that (Q, W) is the standard lattice completion of a Banach space (Q0 , ). For a linear operator T : FV (X, P) → Q0 and its representing measure θ the following are equivalent: (a) T is weakly compact on E(X, P). (b) θ is weakly compact. (c) θ is L(P, Q0 )-valued. Each of these properties implies that the measure θ is countably additive with respect to the strong operator topology. Remark 6.8. If X carries the discrete topology, then all functions χE ⊗a for E ∈ R and a ∈ P are contained in E0 (X, P) ⊂ FV (X, P). Hence, for a continuous linear operator T : FV (X, P) → Q0 its representing measure θ is given by θE (a) = T (χE ⊗a) for all E ∈ R and a ∈ P. The measure θ is therefore L(P, Q0 )-valued for any choice of the operator T. This example shows that the implication (c) ⇒ (a) from Corollary 6.7 does not hold in more general circumstances. 6.9 Locally Convex Topological Vector Spaces. The case that both (P, V) and (Q0 , W0 ) are locally convex topological vector spaces over K = R or K = C is of particular interest. We shall assume that (Q, W) is the standard lattice completion of (Q0 , W0 ) and that the neighborhoods in V and in W0 are balanced and convex. The cones FV (X, P) and FV0 (X, P) then coincide, and Theorems 3.3 and 3.1 yield that a continuous linear operator T : FV (X, P) → Q0 can be extended into a continuous linear operator T : FV (X, PV ) → Q. We shall denote the strong second dual of a locally convex topological vector space (N , U) by Ns∗∗ , that is the dual of N ∗ if the latter is endowed with the topology that is generated by the family Z of all bounded subsets of N (see Section I.7.3). Recall from Example I.2.1(d) that there is a canonical correspondence between the dual cone N ∗ of a vector space N (considered as a locally convex cone) and its usual vector space dual, that is the space of all continuous K-linear functionals on N . ∗∗ may be The same holds for the second dual cone. Thus, in our context, Q0s considered to be either the strong second dual cone of (Q0 , W0 ) as a locally convex cone, hence a subcone of Q (recall the remarks in I.7.3), or the strong second vector space dual of (Q0 , W0 ). The statements of Theorem 5.1 (with PV in place of P) can therefore be reformulated using only the underlying vector spaces P and Q0 and their vector space duals. If the operator T : FV (X, P) → Q0 is indeed linear over K, then ∗∗ ) are also linear over Remark 5.2(a) yields that the operators θE ∈ L(P, Q0s K. In order to prove this claim, let E ∈ R. According to 5.2(a)(iii) there is

6. Special Cases and Applications

315

a net (ϕi )i∈I in K(X) such that θE (a) = lim T (ϕi ⊗a) for all a ∈ P in the i∈I

order topology of Q, that is θE a (μ) = lim T ϕi ⊗a (μ) i∈I

for all a ∈ P and μ ∈ Q∗0 . Thus for a ∈ P and α ∈ K, as T (ϕi ⊗a) ∈ Q0 , we have θE αa (μ) = lim μ T (ϕi ⊗(αa)) = α lim μ T (ϕi ⊗a) = αθE a (μ) i∈I

i∈I

∗∗ for all μ ∈ Q∗0 ; that is θE (αa) = αθE (a) as an element of Q0s ⊂ Q. In the light of this and the preceding observation the measure θ can be interpreted ∗∗ ), that is the space of all continuous K-linear as an element of LK (P, Q0s ∗∗ . operators from the vector space P into the vector space Q0s We shall formulate this important special case as a corollary. For our notions of various choices for operator topologies we refer to Section I.7.2(a). Recall that in case that both (P, V) and (Q0 , W0 ) are locally convex topological vector spaces and Q0 is reﬂexive, then every continuous linear operator T : FV (X, P) → Q0 is weakly compact see Corollary VI.4.3 in [55] .

Corollary 6.10. Let (P, V) and (Q0 , W0 ) be locally convex topological vector spaces over K = R or K = C. Let X be a locally compact Hausdorﬀ space and let V be a basis for a symmetric r-lower continuous inductive limit topology on F(X, P). Then every continuous K-linear operator T : FV (X, P) → Q0 can be represented as an integral on X. More precisely: ∗∗ )-valued measure θ on the weak There exists a unique bounded LK (P, Q0s σ-ring R of all relatively compact Borel subsets of X with the following properties: θ is countably additive and quasi regular with respect to the weak* ∗∗ ), continuous relative to V, all functions in operator topology of LK (P, Q0s FV (X, P) are strongly integrable with respect to θ, and ' f dθ = T (f ) for all f ∈ FV (X, P). X

The operator T is compact (or weakly compact) on E(X, P), if and only if the measure θ is LK (P, Q0 )-valued and compact (or weakly compact). In this case θ is countably additive with respect to the strong operator topology of LK (P, Q0 ). This corollary generalizes quite a few of the standard results that can be found in the literature. Theorem IV.7.2 in [55] (or Theorem 1 in Chapter VI.2 of [43]), for example, establishes an integral representation for a continuous linear operator from the space of continuous real-valued functions on a compact space into a Banach space. The representing integral takes its values in the second dual of this Banach space. The corresponding stronger results for weakly compact and compact operators can be found in Theorems IV.7.3

316

III Measures on Locally Compact Spaces

and IV.7.7 in [55] or in Theorems 5 and 18 in Section VI.2 of [43]. These are obviously special cases of our Corollary 6.4. Indeed, let P = K for K = R or K = C with the Euclidean topology. The values θE for E ∈ R of the representing measure θ from Corollary 6.4 then are K-linear operators from K ∗∗ ∗∗ , that is elements of Q0s or indeed elements of Q0 if the represented into Q0s operator T is weakly compact. The strong operator topology of LK (K, Q0 ) corresponds to the given (symmetric) topology of Q0 . We proceed to discuss a two special cases of Corollary 6.10: (i) The case that (Q0 , W0 )= FU (Y, K), U . Let (P, V) be a vector space over K and let (Q0 , W0 ) = FU (Y, K), U in its symmetric topology, where Y is a second locally compact space and U is a basis for a symmetric r-lower continuous inductive limit topology on F(Y, K). As before, we shall assume that (Q, W) is the standard lattice completion of (Q0 , W0 ). Recall that the neighborhoods for K are the strictly positive multiples of the unit ball B in K. We shall assume in addition that for every y ∈ Y there is a neighborhood uy ∈ U such that s(y) ≤ B for all s ∈ u. This condition implies that all point evaluations εy , that is the mappings g → g(y) : FU (Y, K) → K are continuous K-linear functionals on FU (Y, K), Hence elements of the vector space dual of FU (Y, K). Consequently, considering FU (Y, K), U as a locally convex cone, for every y ∈ Y and α ∈ K the real-valued linear functional αεy , that is the mapping g → e αg(y) : FU (Y, K) → R is an element of its dual cone FU (Y, K)∗ . Every element l of the second ∗∗ of Q0 can be projected onto a K-valued function (vector space) dual Q0s ϕl on Y in a canonical way: We set ϕl (y) = l(εy ) for all y ∈ Y. If applied to this situation, Corollary 6.10 yields an integral representation for a continuous linear operator T : FV (X, P) → FU (Y, K). Let θ be its representing measure. Following the preceding remark, the ∗∗ elements of LK (P, Q0s ), that is the values θE of θ, can be identiﬁed with K-linear operators from P into F(Y, K). The integral of a P-valued step function h = ni=1 χEi ⊗ai ∈ SR (X, P) then is given by ' h dθ = X

n !

θEi (ai ) ∈ F(Y, K),

i=1

since the evaluations θEi (ai ) of the measure θ are functions in F(Y, K). For every y ∈ Y and α ∈ K the functional αεy is an order continuous lattice homomorphism from Q into R. Thus the P ∗ -valued composition measure

6. Special Cases and Applications

317

E → αεy ◦ θE : R → P ∗ is well deﬁned (see II.3.9). According to Theorem II.5.35 we have ' ' f dθ (αεy ) = f d(αεy ◦ θ) X

X

for every f ∈ FU (Y, K). Consequently, for every y ∈ Y the measure ϑy on R, that is E → (εy ◦ θ)

E → (εy ◦ θ) − i(iεy ◦ θ)

or

in the real or the complex case, respectively, is LK (P, K)-valued, that is its values are in the vector space dual of P. If we interpret the integral which ∗∗ as a K-valued function on Y, this yields is an element of Q0s ' ' f dθ (y) = f dϑy T f (y) = X

X

for all f ∈ FV (X, P) and all y ∈ Y, where ' ' ' f dϑy = f d(εy ◦ θ) = f dθ (εy ) X

in the real, and '

X

X

'

' f d(εy ◦ θ) − i f d(iεy ◦ θ) X X ' ' f dθ (y) − i f dθ (iεy ) =

f dϑy = X

X

X

in the complex case. Finally, let us consider the mapping y → ϑy from the locally compact space Y into the set of LK (P, K)-valued measures. For this let us assume in addition that the representation measure θ from above is indeed LK P, CU (Y, K) -valued. This is guaranteed, for example, if the range of the operator T is contained in CU (Y, K) and if T is weakly compact. Then the mapping y → ϑy is seen to be continuous with respect to the given topology of Y and the topology of setwise convergence for LK (P, K)valued measures (see II.5.13). Indeed, let (yi )i∈I be a net in Y converging towards y ∈ Y. Then for every E ∈ R and a ∈ P we have lim ϑyi E (a) = lim θE (a (yi ) = θE a (y) i∈I

i∈I

by the continuity of the function θE (a) ∈ CU (Y, K). This yields our claim. (ii) The case that P = K and (Q0 , W0 ) = FU (Y, K), U . This special case of the preceding one is of particular interest. Under the same conditions

318

III Measures on Locally Compact Spaces

on FU (Y, K), U we obtain an integral representation for a continuous linear operator T : FV (X, K) → FU (Y, K). The values θE of the representing measure θ, that is the elements of ∗∗ ) may now be identiﬁed with functions ψ ∈ F(Y, K), acting as LK (P, Q0s α → αψ : K → F(Y, K). The integral of a K-valued step function h = ni=1 αi χEi ∈ SR (X, K) then is given by ' n ! h dθ = αi θEi , X

i=1

where the evaluations θEi of the measure θ are functions in F(Y, K). For Y = X and U = V, for example, the identity operator on FV (X, K) is represented by the measure E → χE : R → F(X, K). For comprehensiveness we shall formulate this special case as a further corollary. Corollary 6.11. Let K = R or K = C. Let X and Y be locally compact Hausdorﬀ spaces and let V and U be bases for symmetric r-lower continuous inductive limit topologies on F(X, K) and F(Y, K), respectively. Suppose that for every y ∈ Y there is uy ∈ U such that s(y) ≤ B for all s ∈ u. Then every continuous K-linear operator T : FV (X, K) → FU (Y, K) can be represented as an integral on X. More precisely: There exists a unique bounded quasi regular F(Y, K)-valued measure θ on the weak σ-ring R of all relatively compact Borel subsets of X with the following properties: θ is countably additive and quasi regular with respect to pointwise convergence on X for the functions in F(Y, K), continuous relative to V, all functions in FV (X, P) are strongly integrable with respect to θ, and ' f dθ = T (f ) for all f ∈ FV (X, K). X

The operator T is compact (or weakly compact) on E(X, P), if and only if the measure θ is FU (Y, K)-valued and compact (or weakly compact). In this case θ is countably additive with respect to the symmetric topology of FU (Y, K). 6.12 Algebra Homomorphisms. Now suppose that the locally convex vector spaces (P, V) and (Q0 , W0 ) are indeed topological algebras over K = R or K = C and that the continuous linear operator T : FV (X, P) → Q0 is multiplicative on E(X, P). As before we assume that (Q, W) is the standard lattice completion of (Q0 , W0 ). We shall verify that the representing measure

6. Special Cases and Applications

319

θ satisﬁes Condition (A) from II.6.14 in this case. First let us recall some facts and techniques that were introduced in II.6.14: A topological algebra A is an algebra and a locally convex topological vector space such that for a ﬁxed element a ∈ A (or b ∈ A) the linear operator c → ac (or c → cb) from A into A is continuous (see for example 8.1 in [137]). Because for a linear operator continuity implies weak continuity, A is also a topological algebra in its weak topology. Moreover, on Q0 ⊂ Q order convergence and week convergence coincide (see I.5.57). An extension of the multiplication from Q0 to Q was introduced in II.6.14: For l, m ∈ Q we denote by l • m the set of all elements q ∈ Q for which we can ﬁnd nets (li )i∈I and (mj )j∈J in Q0 ⊂ Q such that limi∈I li = l, limj∈J mj = m and lim lim li mj = lim lim li mj = lim lim li mj = lim lim li mj = q. i∈I j∈J

i∈I j∈J

j∈J i∈I

j∈J i∈I

The set l • m may be empty or contain more than one element of Q. For l, m ∈ Q0 we have l • m = {lm}. Similarly, if the algebra Q0 has an involution a → a∗ , then for every l ∈ Q we denote by l the set of all elements q ∈ Q for which we can ﬁnd a net (li )i∈I in Q0 ⊂ Q such that limi∈I li = l and lim li∗ = q. i∈I

Obviously, if P is an algebra, so is E(X, P), endowed with the canonical, that is pointwise multiplication for P-valued functions. Now suppose that the continuous linear operator T : FV (X, P) → Q0 is multiplicative on E(X, P) and let θ be its representing measure from Corollary 6.10. We shall establish that θ satisﬁes (A) θE (ab) ∈ θE (a) • θE (b) and disjoint sets E, G ∈ R.

0 ∈ θE (a) • θG (b) for all a, b ∈ P and

For this, let E ∈ R and a, b ∈ P. Let U ∈ R be an open set containing E and choose the net (ϕi )i∈I in K(X) as in Proposition 4.5(c) with U in place of O, that is ϕi ≺ U for all i ∈ I and θE (a) = lim T ϕi ⊗a , θE (b) = lim T ϕi ⊗b and θE (ab) = lim T ϕi ⊗(ab) . i∈I

i∈I

i∈I

Recall from 4.5(c) that the index set I consists of all ordered pairs (O, f ), where O ∈ O, the family of all open sets O ∈ R such that E ⊂ O and O ⊂ U, and f : O → K(X) is a mapping such that f (O) ≺ O for all O ∈ O. I is ordered by (O1 , f1 ) ≤ (O2 , f2 ) if O1 ⊃ O2 and f1 (O) ≤ f2 (O) for all O ∈ O. We have ϕi = f (O) ∈ K(X) for i = (O, f ) ∈ I. Now let w ∈ W and v ∈ V such that θU (v) ≤ w. There is λ ≥ 0 such that 0 ≤ ab + λv. For a ﬁrst step in our argument we ﬁx a set O0 ∈ O and a compact subset K0 of E. Let ϕ0 ∈ K(X) such that K0 ≺ ϕ0 ≺ O0 . Let i0 = (O0 , f0 ) ∈ I, where f0 : O → K(X) is a mapping such that K0 ≺ f0 (O) ≺ O for all O ∈ O. Thus χK0 ≤ ϕi ϕ0 ≤ χO0 for all i ≥ i0 .

320

III Measures on Locally Compact Spaces

This yields χK0 ⊗(ab + λv) ≤ ϕi ϕ0 ⊗(ab + λv), hence θK0 (ab) + λθK0 (v) ≤ T ϕi ϕ0 ⊗(ab + λv) ≤ T (ϕi ⊗a) T (ϕ0 ⊗b) + λθ00 (v). The latter demonstrates

θK0 (ab) + λθK0 (v) ≤ lim T (ϕi ⊗a) T (ϕ0 ⊗b) + λθ00 (v) i∈I

and, similarly, lim T (ϕi ⊗a) T (ϕ0 ⊗b) + λθK0 (v) ≤ θO0 (ab) + λθO0 (v). i∈I

Moreover, since K0 ≤ ϕj ≺ O0 holds for all i0 ≤ j ∈ I, the above yields indeed θK0 (ab) + λθK0 (v) ≤ lim lim T (ϕi ⊗a) T (ϕj ⊗b) + λθ00 (v) j∈I i∈I

and

lim lim T (ϕi ⊗a) T (ϕj ⊗b) + λθK0 (v) ≤ θO0 (ab) + λθO0 (v). j∈I i∈I

Finally, we take the supremum and the inﬁmum over all compact sets K0 ⊂ E and open sets E ⊂ O0 , respectively, in the preceding inequalities. We have inf θO0 (v) = θE (v) ≤ sup θK0 (v) + O θU (v) O0 ⊃E

K0 ⊂E

by Proposition 4.4. Using this, the cancellation law from Proposition I.5.10(a), the fact that θE (v) ≤ θU (v) ≤ w and that the neighborhood w ∈ W was arbitrarily chosen, we obtain θE (ab) = lim lim T (ϕi ⊗a) T (ϕj ⊗b) = lim lim T (ϕi ⊗a) T (ϕj ⊗b) . j∈I i∈I

j∈I i∈I

Similarly, one veriﬁes θE (ab) = lim lim T (ϕi ⊗a) T (ϕj ⊗b) = lim lim T (ϕi ⊗a) T (ϕj ⊗b) . i∈I j∈I

i∈I j∈I

The ﬁrst part of (A) follows. For the second part, let E, G ∈ R be disjoint sets and let a, b ∈ P. Let O ∈ R be an open set containing both E and G, and choose the nets (ϕi )i∈I and (ψj )ij∈J in K(X) as in Proposition 4.5(c) for the sets E and G, respectively, that is ϕi , ψj ≺ O for all i ∈ I and j ∈ J and

6. Special Cases and Applications

θE (a) = lim T (ϕi ⊗a) i∈I

321

and

θG (b) = lim T (ψj ⊗b). j∈J

Given w ∈ W we choose v ∈ V such that θO (v) ≤ w and λ ≥ 0 such that 0 ≤ a + λv. For any choice of open sets U, V ⊂ O such that E ⊂ U and G ⊂ V there are i0 ∈ I and j0 ∈ J such that ϕi ≺ U and ψj ≺ V, hence ϕi ψj ≺ U ∩ V for all i ≥ i0 and j ≥ j0 . Thus 0 ≤ T ϕi ψj ⊗(ab + λv) ≤ T (ϕi ⊗a) T (ψj ⊗a) + λθ(U ∩V ) (v) T (ϕi ⊗a) T (ψj ⊗a) ≤ T ϕi ψj ⊗(ab + λv) ≤ θ(U ∩V ) (ab + λv). This shows 0 ≤ lim lim T (ϕi ⊗a) T (ψj ⊗a) + λθ(U ∩V ) (v) and

j∈J i∈I

lim lim T (ϕi ⊗a) T (ψj ⊗a) ≤ θ(U ∩V ) (ab + λv).

and

j∈J i∈I

Next we observe that θ(E∪G) (c) + θ(U ∩V ) (c) ≤ θ(U ∪V ) (c) + θ(U ∩V ) (c) = θU (c) + θV (c) holds for all elements c ≥ 0 in the standard full extension PV of P. Taking the inﬁmum over all open sets U and V containing the given sets E and G, respectively, yields θ(E∪G) (c) + inf θ(U ∩V ) (c) = θE (c) + θG (c) = θ(E∪G) (c), U ⊃E, V ⊃G

hence

0 ≤ inf θ(U ∩V ) (c) ≤ O θ(E∪G) (c) ≤ O θO (c) U ⊃E, V ⊃G

by Proposition I.5.10(a). As P is a vector space, its elements are bounded, and we have O θO (ab + v) = O θO (v) ≤ w. Using this in the above inequalities together with the fact that the neighborhood w ∈ W was arbitrarily chosen then yields lim lim T (ϕi ⊗a) T (ψj ⊗a) = lim lim T (ϕi ⊗a) T (ψj ⊗a) = 0. j∈J i∈I

j∈J i∈I

Similarly, one veriﬁes lim lim T (ϕi ⊗a) T (ψj ⊗a) = lim lim T (ϕi ⊗a) T (ψj ⊗a) = 0, i∈J j∈I

hence the second part of (A).

i∈J j∈I

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III Measures on Locally Compact Spaces

If (P, V) is a topological algebra with an involution a → a∗ , then canon ∗ ically, for a function f ∈ F(X, P) we denote the mapping x → f (x) by f ∗ . If (Q0 , W) has also an involution and if the continuous linear operator T : E(X,P) → Q0 is compatible with the respective involutions, that is if ∗ T (f ∗ ) = T (f ) holds for all f ∈ E(X, P), then the representing measure θ satisﬁes (A*) θE (a∗ ) ∈ θE (a) for all E ∈ R and a ∈ P. (See Section II.6.14.) This claim is readily veriﬁed: For E ∈ R and a ∈ P let the net (ϕi )i∈I in K(X) be as in Proposition 4.5(c), that is θE (a) = lim T ϕi ⊗a i∈I

and

∗ θE (a∗ ) = lim T (ϕi ⊗a∗ ) = lim T (ϕi ⊗a) . i∈I

i∈I

∗ According to our previously introduced notation, this means θE (a ) ∈ θE (a) . We summarize:

Corollary 6.13. Let (P, V) and (Q0 , W0 ) be topological algebras over K = R or K = C. Let X be a locally compact Hausdorﬀ space and let V be a basis for a symmetric r-lower continuous inductive limit topology on F(X, P). Then every continuous K-linear operator T : FV (X, P) → Q0 that is multiplicative on E(X, P) can be represented as an integral on X. More pre∗∗ )-valued measure θ on the cisely: There exists a unique bounded LK (P, Q0s weak σ-ring R of all relatively compact Borel subsets of X with the following properties: θ is countably additive and quasi regular with respect to the ∗∗ ). We have weak* operator topology of LK (P, Q0s θE (ab) ∈ θE (a) • θE (b)

and

0 ∈ θE (a) • θG (b)

for all a, b ∈ P and disjoint sets E, G ∈ R. If T with respec is compatible tive involutions in P and Q0 , then θE (a∗ ) ∈ θE (a) holds in addition. The measure θ is continuous relative to V, all functions in FV (X, P) are strongly integrable with respect to θ, and ' f dθ = T (f ) for all f ∈ FV (X, P). X

The operator T is compact (or weakly compact) on E(X, P), if and only if the measure θ is LK (P, Q0 )-valued and compact (or weakly compact). In this case θ is countably additive with respect to the strong operator topology of LK (P, Q0 ).

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We proceed to discuss three special cases of Corollary 6.13: (i) The case that Q0 = K. If Q0 = K, that is if the values θE of the representation measure θ are K-linear functionals on the algebra P, then Condition (A) means that all functionals θE are multiplicative and that for disjoint sets E, G ∈ R we have either θE = 0 or θG = 0. In the conclusion of Section 4 we inferred that θ is therefore indeed some point evaluation measure in this case. Inner regularity of θ is guaranteed by the fact that all values in Q0 are bounded (see Proposition 4.4(a). (ii) The case that (Q0 , W0 ) = FU (Y, K), U . This situation was consid- ered earlier as a special case of Corollary 6.10. The conditions on FU (Y, K), U from 6.10(i) guarantee that the point evaluations εy are elements of the vector space dual of FU (Y, K). We obtain additional information if (P, V) is topological algebra and if the operator T : FV (X, P) → FU (Y, K) is multiplicative on E(X, P). Let θ be its representing measure from Corollary 6.13. Recall that the values θE of θ may be identiﬁed with Klinear operators from P into F(Y, K). The extended multiplication • in Q then corresponds to the pointwise multiplication in F(Y, K). Thus for every y ∈ Y the mapping f → T f (y) : FV (X, P) → K on is a continuous linear functional on FV (X, P) which is multiplicative E(X, P). The representation measure δy see the remark in 6.10(i) for this functional therefore satisﬁes Conditions (A), and following case (i) is some point evaluation measure on X. More precisely: There exist mappings y → xy and y → μy from Y into X and into the set of continuous multiplicative linear functionals on P, respectively, such that ' f dϑy = μy f (xy ) T f (y) = X r CV (X, P)

for all f ∈ and all y ∈ Y. Now let us assume in addition that the range of the operator T is contained in CU (Y, K) and consider the mapping y → (xy , μy ) : Y → X × P ∗ . Let (yi )i∈I be a net in Y converging towards y ∈ Y. Then lim μyi f (xyi ) = lim T f (yi ) = T f (y) = μy f (xy ) i∈I

i∈I

for all f ∈ FV (X, P). Thus the mapping y → (xy , μy ) is continuous with respect to the given topology of Y and the weak topology induced on X ×P ∗ ∗ by the functions f ∈ FV (X,P) acting on X × P in the canonical way, that is (x, μ) → μ f (x) ∈ K. See the deﬁnition of the weak topology in 8.9

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in [198] or 1.4.8 in [59]. Two special cases are of particular interest. For these let us assume in addition that for every y ∈ Y there is f ∈ FV (X, P) such that T f (y) = 0. (i) If P = K, then the identity is the only no-zero multiplicative linear functional on P, that is μy = 1 for all y ∈ Y. Then the mapping y → xy : Y → X is continuous with respect to the given topology of Y and the weak topology on X generated by the functions in FV (X, P). (ii) If X = {x} is a singleton set, then FV (X, P) = P and the mapping y → xy = x is constant. Given a ∈ P there is f ∈ FV (X, P) such that f (x) = a, hence the above shows that the mapping y → μy : Y → P ∗ is continuous with respect to the given topology of Y and the topology w(P ∗ , P) of P ∗ . We formulate this observation as a further corollary: Corollary 6.14. Let (P, V) be a topological algebra over K = R or K = C. Let X and Y be locally compact Hausdorﬀ spaces and let V and U be bases for symmetric r-lower continuous inductive limit topologies on F(X, P) and F(Y, K), respectively. Suppose that for every y ∈ Y there is uy ∈ U such that s(y) ≤ B for all s ∈ u. Then for every continuous K-linear operator T : FV (X, P) → F(Y, K) that is multiplicative on E(X, P) there exist mappings y → xy and y → μy from Y into X and into the set of continuous multiplicative linear functionals on P, respectively, such that T f (y) = μy f (xy ) r for all f ∈ CV (X, P) and all y ∈ Y. If the range of the operator T is contained in CU (Y, K), then the mapping y → (xy , μy ) is continuous with respect to the given topology of Y and the weak topology induced on X × P ∗ by the functions f ∈ FV (X, P).

(iii) B ∗ -algebras. Let P = C and let (Q0 , ) be a commutative B ∗ algebra, that is a commutative complex Banach algebra with identity and an involution a → a∗ satisfying a∗ a = a∗ a see IX.3.1 in [56] . Let X be the spectrum of Q0 , that is the σ(Q∗0 , Q0 )-compact subset of all multi∗ plicative linear functionals in Q0 . According to the Gelfand-Naˇ ımark theorem see Theorem IX.3.7 in [56] or Theorem III.16.1 in [137] then Q0 is isometrically *-isomorphic to C(X, C), that is to say there is a continuous multiplicative linear operator T : C(X, C) → Q0 such that T (f ∗ ) = T (f )∗ for all f ∈ C(X, C). As P = C, the complex linear operators from P into ∗∗ ∗∗ ∗∗ are indeed the elements a of Q0s , acting as α → αa : C → Q0s . Thus Q0s ∗∗ -valued measure according to Corollary 6.13, T can be represented by a Q0s θ with the stated properties. In addition, we have Condition (A*) from 6.12, for all E ∈ R and α ∈ K. As (αθE ) = αθE that is αθE ∈ αθE ∗ (see II.6.14), this yields θE ∈ θE . On the subset X of Q0 of all multiplicative linear functionals, for each E ∈ R the element θE is therefore a real-valued function taking only the values 0 or 1, that is the characteristic function of some subset Φ(E) of X. Moreover, for disjoint sets E, G ∈ R the subsets Φ(E) and Φ(G) are disjoint.

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325

∗∗ Summarizing, there is an Q0s -valued measure θ on the spectrum X of Q0 whose values θE yield characteristic functions of subsets of X, disjoint for disjoint sets E, G ∈ R, and such that the integrals of the functions in C(X, C) with respect to θ are the elements of Q0 .

6.15 Lattice Homomorphisms. Now suppose that the quasi-full locally convex cone (P, V) is indeed a locally convex ∨-semilattice cone (see Section 5.1 of Chapter I and the additional remarks concerning quasi-full cones in Section II.6.16), that is the order in P is antisymmetric, for any two elements a, b ∈ P their supremum a ∨ b exists in P and (∨1) (a + c) ∨ (b + c) = a ∨ b + c holds for all a, b, c ∈ P. (∨2 ) a ≤ v for a ∈ P and v ∈ V implies that a ∨ 0 ≤ v. Topological vector lattices and locally convex complete lattice cones in the sense of I.5 are of course locally convex ∨-semilattice cones in this sense. Further examples include R and R+ Examples I.1.4(a) and (b) and cones of non-empty convex subsets of a topological vector space with the set-inclusion as order Example I.1.4(c) . The supremum f ∨ g ∈ F(X, P) of two functions f, g ∈ F(X, P) is canonically deﬁned as the mapping x → f (x) ∨ g(x). A brief review of Deﬁnition 2.4 conﬁrms that the subcone FV (X, P) of F(X, P) is indeed σ (X, P) the subclosed for suprema. As in Section II.6.16 we denote by SR cone of all functions f ∈ FR (X, P) for which there exists a sequence (hn )n∈N of step functions that is bounded below and bounded above relative to f and such that hn( −→ f. According to Corollary II.5.26, this implies that ( lim X hn dθ = X f dθ holds for every R-bounded L(P, Q)-valued mean→∞

σ (X, P) and FV (X, P) by sure θ. We shall denote the intersection of SR σ SV (X, P). Lemma II.5.27 and Corollary II.5.28 yield that E(X, P) is conσ (X, P). tained in SV Now suppose that the continuous linear operator T : FV (X, P) → Q is a σ (X, P) in the sense of I.5.30, that is ∨-semilattice homomorphism on SV

T (f ∨ g) = T (f ) ∨ T (g) σ holds for all f, g ∈ SV (X, P), and that T can be extended to FV (X, PV ). Let θ be the representing measure for T from Theorem 5.1. We shall establish that θ satisﬁes the following condition:

(L) θE (a) ∨ θE (b) = θE (a ∨ b) for all E ∈ R and a, b ∈P, and θE (a) ∨ θG (b) ≤ θE (a) + θG (b) ≤ θE (a) ∨ θG (b) + O θ(E∪G) (a ∨ b) for all a, b ≥ 0 in P and disjoint sets E, G ∈ R. (Recall the corresponding Condition (L) from II.6.16, which is equivalent to the above in case that all elements of P are bounded, but slightly stronger in its second part for the general case.)

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In order to verify (L), let E ∈ R and a, b ∈ P. Let O ∈ R be an open set containing E and choose the net (ϕi )i∈I in K(X) as in Proposition 4.5(c), that is ϕi ≺ O for all i ∈ I and θE (a) = lim T ϕi ⊗a , i∈I

θE (b) = lim T ϕi ⊗b i∈I

and θE (a ∨ b) = lim T ϕi ⊗(a ∨ b) . i∈I

We have ϕi ⊗(a∨b) = (ϕi ⊗a)∨(ϕi ⊗b), hence T ϕi ⊗(a∨b) = T (ϕi ⊗a)∨T (ϕi ⊗b), σ (X, P). Now the continuity since T is a ∨-semilattice homomorphism on SV of the lattice operation in Q see Proposition I.5.25(a) yields lim T (ϕi ⊗a) ∨ T (ϕi ⊗b) = lim T (ϕi ⊗a) ∨ lim T (ϕi ⊗b) , i∈I

i∈I

i∈I

hence θE (a) ∨ θE (b) = θE (a ∨ b) as claimed in the ﬁrst part of (L). For the second part of (L), let E, G ∈ R be disjoint sets an let 0 ≤ a, b ∈ P. Because θE (a) + θG (b) = θE (a) ∨ θG (b) + θE (a) ∧ θG (b) by Proposition I.5.3, and because θE (a), θG (b) ≥ 0, we have θE (a) ∨ θG (b) ≤ θE (a) + θG (b). For the right-hand side of the inequality in (L) let O ∈ R be an open set containing both E ∪ G, and choose the nets (ϕi )i∈I and (ψj )ij∈J in K(X) as in Proposition 4.5(c) for the sets E an G, respectively, that is ϕi , ψj ≺ O for all i ∈ I and j ∈ J and θE (a) = lim T (ϕi ⊗a) i∈I

and

θG (b) = lim T (ψj ⊗b). j∈J

Before we proceed, let us observe that in a semilattice cone P we always have αa + βb ≤ αa ∨ βb + (α ∧ β)(a ∨ b) for any choice of positive elements a, b ∈ P and α, β ≥ 0. Indeed, if β ≤ α, then αa ≤ αa ∨ βb and βb ≤ (α ∧ β)(a ∨ b), hence our claim. Using this, we infer that ϕi ⊗a + ψj ⊗b ≤ (ϕi ⊗a) ∨ (ψj ⊗b) + (ϕi ∧ ψj )⊗(a ∨ b), hence

T (ϕi ⊗a) + T (ψj ⊗b) = T (ϕi ⊗a) ∨ T (ψj ⊗b) + T (ϕi ∧ ψj )⊗(a ∨ b) .

For any choice of open sets U, V ⊂ O such that E ⊂ U and G ⊂ V there are i0 ∈ I and j0 ∈ J such that ϕi ≺ U and ψj ≺ V, hence ϕi ∧ ψj ≺ U ∩ V and T (ϕi ∧ ψj )⊗(a ∨ b) ≤ θ(U ∩V ) (a ∨ b) for all i ≥ i0 and j ≥ j0 . Now passing to the limits over i ∈ I and j ∈ J in the preceding equation and using Proposition I.5.25(a) leads to

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327

θE (a) + θG (b) ≤ θE (a) ∨ θG (b) + θ(U ∩V ) (a ∨ b) for any choice of such open sets U and V. Thus θE (a) + θG (b) ≤ θE (a) ∨ θG (b) + inf θ(U ∩V ) (a ∨ b). U ⊃E, V ⊃G

We abbreviate c = a ∨ b ≥ 0 and observe that θ(E∪G) (c) + θ(U ∩V ) (c) ≤ θ(U ∪V ) (c) + θ(U ∩V ) (c) = θU (c) + θV (c). Taking the inﬁmum over all open sets U and V containing the given sets E and G, respectively, yields θ(E∪G) (c) + inf θ(U ∩V ) (c) ≤ θE (c) + θG (c) = θ(E∪G) (c), U ⊃E, V ⊃G

inf θ(U ∩V ) (c) ≤ O θ(E∪G) (c)

hence

U ⊃E, V ⊃G

by Proposition I.5.10(a). Together with the above, this yields θE (a) + θG (b) ≤ θE (a) ∨ θG (b) + O θ(E∪G) (a ∨ b) as claimed. We summarize: Corollary 6.16. Let (P, V) be a quasi-full semilattice cone and suppose that the order continuous linear functionals support the separation property for Q. Let X be a locally compact Hausdorﬀ space and let V be a basis for a symmetric r-lower continuous inductive limit topology on F(X, P). Let T : σ (X, P) → Q be a continuous linear ∨-semilattice homomorphism that can SV be extended to a continuous linear operator on FV (X, PV ). Then there exists a unique bounded quasi regular L(PV , Q)-valued measure θ on the weak σ-ring R of all relatively compact Borel subsets of X with the following properties: θ is continuous relative to V and satisﬁes Condition (L), all functions in FV (X, P) are integrable, all functions in FV0 (X, P) are strongly integrable with respect to θ, and ' ' ϕ⊗a dθ ≤ T (ϕ⊗a) ≤ ϕ⊗a dθ + O θA (a) X

X

holds for all ϕ ∈ K(X) and a ∈ P, where A denotes the compact support of the function ϕ. Thus ' ' f dθ ≤ T (f ) and indeed g dθ = T (g) X

X

holds for all f ∈ FV (X, P) and all g ∈ FV0 (X, P), respectively.

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Signiﬁcant simpliﬁcations occur if (P, V) is indeed a topological vector lattice over R . We shall formulate these as an additional corollary. Corollary 6.17. Let (P, V) be a topological vector lattice and suppose that the order continuous linear functionals support the separation property for Q. Let X be a locally compact Hausdorﬀ space and let V be a basis for a symmetric r-lower continuous inductive limit topology on F(X, P). Let T : FV (X, P) → Q be a continuous linear operator that is a ∨-semilattice homoσ (X, P). Then there exists a unique bounded L(P, Q)-valued morphism on SV measure θ on the weak σ-ring R of all relatively compact Borel subsets of X with the following properties: θ is countably additive and quasi regular. We have θE (a ∨ b) = θE (a) ∨ θE (b)

and

θE (a) ∨ θG (b) = θE (a) + θE (b)

for all 0 ≤ a, b ∈ P and disjoint sets E, G ∈ R. The measure θ is continuous relative to V, all functions in FV (X, P) are strongly integrable with respect to θ, and ' f dθ = T (f ) for all f ∈ FV (X, P). X

We shall discuss two special cases of Corollary 6.17: (i) The case that (P, V) is a topological vector lattice and that Q = R. If Q = R, that is if the values θE of the measure θ are elements of P ∗ , then Condition (L) means that (i) all functionals θE are ∨-semilattice homomorphisms and (ii) for disjoint sets E, G ∈ R we have either θE = 0 or θG = 0. In the conclusion of Section 4 we inferred that θ is therefore indeed some point evaluation measure in this case. Inner regularity of θ is guaranteed by the fact that all values of θ are bounded (see Proposition 4.4(c). lattice and that (Q, W) = that (P, V) is a topological vector (ii) The case FU (Y, R), U . Let (Q, W) = FU (Y, R), U , where Y is a second locally compact space and U is a basis for a symmetric r-lower continuous inductive limit topology on F(Y, R). Then FU (Y, R), U is a locally convex complete lattice cone. We shall assume in addition that for every y ∈ Y there is a neighborhood uy ∈ U such that s(y) ≤ 1 for all s ∈ u. This condition implies that all point evaluations εy , that is the mappings g → g(y) : FU (Y, R) → R are continuous linear functionals on FU (Y, R). Let T : FV (X, P)→FU (Y, R) be a continuous linear operator that is a ∨-semilattice homomorphism on σ (X, P) and let θ be its representing measure from Corollary 6.17. The SV values θE of θ are linear operators from P into F(Y, R). Then for every y ∈ Y the mapping

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329

f→ T f (y) : FV (X, P) → R is a continuous linear functional on FV (X, P) which is a ∨-semilattice hoσ (X, P). Following case (i), this functional is represented momorphism on SV by some point evaluation measure on X. Consequently, there exist mappings y → xy and y → μy from Y into X and into the set of continuous linear ∨-semilattice homomorphisms on P, respectively, such that T f (y) = μy f (xy ) r for all f ∈ CV (X, P) and all y ∈ Y. Now let us assume in addition that the range of the operator T is contained in CU (Y, R). the mapping

y → (xy , μy ) : Y → X × P ∗ is continuous with respect to the given topology of Y and the weak topology which is generated on X × P ∗ by the functions f ∈ FV (X, P). Two special cases are of particular interest. (i) If P = R, then a continuous linear ∨-semilattice homomorphism on P is the multiplication with some non-negative real number, that is μy = ϕ(y) for some real-valued function ϕ : Y → [0, +∞), and we have T f (y) = ϕ(y) f (xy ) r for all f ∈ CV (X, P) and y ∈ Y. Moreover, the mapping y → ϕ(y)εxy : r to the given topology of Y and Y → CV (X, P)∗ is continuous with respect the topology σ FV (X, P)∗ , FV (X, P) on FV (X, P)∗ . (ii) If X = {x} is a singleton set, then FV (X, P) = P and the mapping y → xy = x is constant. Given a ∈ P there is f ∈ FV (X, P) such that f (x) = a, hence the above shows that the mapping y → μy : Y → P ∗ is continuous with respect to the given topology of Y and the topology w(P ∗ , P) of P ∗ . We formulate this observation as a further corollary:

Corollary 6.18. Let (P, V) be a topological vector lattice. Let X and Y be locally compact Hausdorﬀ spaces and let V and U be bases for symmetric rlower continuous inductive limit topologies on F(X, P) and F(Y, R), respectively. Suppose that for every y ∈ Y there is uy ∈ U such that s(y) ≤ 1 for all s ∈ u. Let T : FV (X, P) → F(Y, R) be a continuous linear operator σ (X, P) and can be extended that is a ∨-semilattice homomorphism on SV to FV (X, PV ). Then there exist mappings y → xy and y → μy from Y into X and into the set of continuous ∨-semilattice homomorphisms on P, respectively, such that T f (y) = μy f (xy ) r (X, P) and all y ∈ Y. If the range of the operator T is for all f ∈ CV contained in CU (Y, R), then the mapping y → (xy , μy ) is continuous with respect to the given topology of Y and the weak topology induced on X × P ∗ by the functions f ∈ FV (X, P).

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III Measures on Locally Compact Spaces

6.19 The Case P = R. An elementary function ϕ⊗(+∞) with ϕ ∈ K(X) is the characteristic function χO ⊗(+∞) for some relatively compact open subset O of X in this case. This function is relatively continuous if and only if the set O ∈ R is both open and compact see Proposition 1.11(b) . For real-valued functions, on the other hand, continuity with respect to the symmetric relative topology of R coincides with the usual (Euclidean) notion. The cone E(X, R) therefore consists of all sums of continuous real-valued functions and characteristic functions χO ⊗(+∞) for some open set O ∈ R. The subcone E0 (X, R) of E(X, R) consists of all sums of continuous realvalued functions and characteristic functions χO ⊗(+∞) for some open and compact set O ∈ R. The values θE for E ∈ R of a representing measure in the sense of 5.1 are continuous linear operators from R into some locally convex complete lattice cone Q in this case. Such an operator θE maps real numbers into invertible (and therefore bounded) elements of Q; more precisely: We have θE (α) = αl with some positive invertible element 0 ≤ l ∈ Q for all α ∈ R. The image of +∞ ∈ R on the other hand is some zero component q∞ ∈ Q, (that is q∞ ≥ 0 and αq∞ = q∞ for all α > 0, see I.5.8) such that θE (α) ≤ q∞ for all α ∈ R. If (Q, W) = FU (Y, R), U , where Y is a second locally compact space and U is a basis for a symmetric r-lower continuous inductive limit topology on F(Y, R), then its invertible elements are real-valued, whereas its zero components are functions on Y that take only the values 0 and +∞. 6.20 The Case P = R+ . Here we choose P = R+ , endowed with the singleton neighborhood system V = {0} (see Example 1.4(b) in Chapter I). The topolsymmetric relative topology on P = R+ coincides with the Euclidean ogy on (0, +∞) and renders both 0 and +∞ as isolated points see Exam ple I.4.37(b) . The cone E(X, R) therefore consists of all sums of continuous non-negative real-valued functions and characteristic functions χO ⊗(+∞) for some open set O ∈ R. The subcone E0 (X, R) of E(X, R) consists of all such functions that take each of the values 0 and +∞ on both open and compact subsets of X, respectively. An operator θE , corresponding to an L(R+ , Q)valued measure θ, maps positive reals α → αl with some positive (not necessarily invertible) element 0 ≤ l ∈ Q The image of +∞ ∈ R is again some zero component q∞ ∈ Q, such that θE (α) ≤ q∞ for all α > 0. 6.21 The Case Q = R. We choose Q = R with the canonical order and the neighborhoods V = {ε ∈ R | ε > 0} and consider the representation of a continuous linear functional on the cone FV (X, P) by a P ∗ -valued measure (see Section II.6.1). We may use any locally convex cone (P, V) in this case, since a linear functional on the cone FV (X, P) can be extended into a linear functional on FV (X, PV ) where (PV , V) denotes the standard full extension of P. We may than use Theorem 5.1 for this full cone PV and obtain:

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331

Corollary 6.22. Let (P, V) be a locally convex cone. Let X be a locally compact Hausdorﬀ space and let V be a basis for an r-lower continuous inductive limit topology on F(X, P). Then every continuous linear functional μ ∈ FV (X, P)∗ can be represented as an integral on X. More precisely: There exists a bounded quasi regular P ∗-valued measure θ on the weak σ-ring R of all relatively compact Borel subsets of X such that θ is continuous relative to V. All functions in FV (X, P) are integrable with respect to θ, and ' f dθ ≤ μ(f ) for all f ∈ FV (X, P). X

All functions in FV0 (X, P) are strongly integrable with respect to θ, and ' f dθ = μ(f ) for all f ∈ FV0 (X, P). X

This corollary recovers and generalizes the result from Theorem 4.2 in [171]. 6.23 Sequence Cones. In order to obtain sequence cones we choose X = N with the discrete topology see also Example 2.11(f) . Let (P, V) be a quasi-full locally convex cone. The functions in E(N, P) = E0 (N, P) then are ﬁnite sequences in P. Depending on our choice for the inductive limit neighborhood system V we obtain a variety of sequence cones for FV (N, P), for example lp -type cones as elaborated in 2.11(f). An L(P, Q)-valued measure θ on R, that is the collection of ﬁnite subsets of N, corresponds to a sequence (θi )i∈N of operators in L(P, Q). Every such measure is bounded, as for E = {x1 , . . . , xn } ∈ R and each neighborhood w ∈ W there is v ∈ V such that θxi (s) ≤ (1/n)w whenever s ≤ v for s ∈ P and all i = 1, . . . , n. Hence |θ|(E, v) ≤ w. According to Proposition 3.4 every continuous linear operator T : FV (N, P) → Q can be extended to FV (N, PV ) in this case. Thus, as stated in Theorem 5.1 there exists a unique sequence (θi )i∈N of linear operators in L(P, Q) such that ∞ ! T (ai )i∈N = θi (ai ) i=1

holds for every element (sequence) (ai )i∈N ∈ FV (N, P). 6.24 The Convergence Theorems. For a continuous linear operator T : FV (X, P) → Q and its L(P, Q)-valued representing measure θ, the convergence theorems of Chapter II.5 if applied to the measure θ may be reinterpreted in terms of the operator T. Theorem II.5.25, for example, yields the following: Corollary 6.25. Let (P, V) be a full locally convex cone and let (Q, W) be a locally convex complete lattice cone such that the order continuous linear

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III Measures on Locally Compact Spaces

functionals support the separation property for Q. Let X be a locally compact Hausdorﬀ space and let V be a basis for an r-lower continuous inductive limit topology on F(X, P). Let T : FV (X, P) → Q be a continuous linear operator than can be extended to FV (X, PV ). Let fn , f, f∗∗ , f∗ , f ∗∗ , f ∗ ∈ E0 (X, PV ) such that f∗∗ ≤ fn + f∗ and fn + f ∗∗ ≤ f ∗ for all n ∈ N, and that fn −→ f. Then T (f ) ≤ lim T (fn ) + O (T (f∗ ) n→∞

and lim T (fn ) ≤ T (f ) + O (T (f ∗ )) .

n→∞

If (Q, W) is the standard lattice completion of a locally convex cone (Q0 , W0 ), and the range of T is contained in Q0 , then the above convergence statements refer to weak convergence in Q0 . If P and Q0 are indeed locally convex vector spaces and if the operator T is weakly compact, then lim T (fn ) = T (f )

n→∞

holds with respect to the symmetric topology of Q0 . The last part of this corollary follows from Theorem II.5.36. It assumptions are satisﬁed if P and Q0 are locally convex vector spaces and if the operator T is weakly compact. 6.26 The Case that Q Is the Standard Lattice Completion of Some , M) is the simpliﬁed standard Operator Cone. Suppose that Q = H(N lattice completion of some locally convex cone H(N , M) of linear operators from a cone N into a locally convex cone (M, W) as introduced in Section I.7.1, endowed with a locally convex cone topology generated by a family Z of subsets of N . We suppose that the union of the sets Z ∈ Z is all of N . Let (P, V) be a quasi-full locally convex cone and let T : FV (X, P) → H(N , M) be a continuous linear operator that permits an , M). Let θ be the bounded quasi reguextension T : FV (X, PV ) → H(N lar L PV , H(N , M) -valued measure θ that represents the operator T as in , M) are R-valTheorem 5.1. Recall from I.7.1 that the elements ϕ of H(N ∗ $ for H(N , M) ued functions on N × M , and the neighborhood system W is generated by the functions ϕ(Z,w) for Z ∈ Z and w ∈ W such that ϕ(Z,w) (z, μ) = 1 if (z, μ) ∈ Z × w◦ and ϕ(Z,w) (z, μ) = +∞, else. According to Remark 5.2(b), for ﬁxed E ∈ R and a ∈ P the value θE (a) of the , M) of the representing measure θ is contained in the order closure in H(N image T (A) of some relatively bounded subset A of FV (X, P). We shall proceed to verify that the elements of this order closure, that is in particular the element θE (a), may again be identiﬁed with linear operators from N

6. Special Cases and Applications

333

into M∗∗ sr , the (relative) strong second dual of M. For this, suppose that the set A is bounded below and bounded above relative to the function f0 ∈ FV (X, P). Then, obviously, the set T (A) ⊂ H(N , M) is bounded below and bounded above relative to the operator L0 = T (f0 ) in H(N , M). , M) be in the order closure of T (A), and let (ϕi )i∈I be a net Let ϕ ∈ H(N in T (A) converging to ϕ with respect to the order topology. This implies pointwise convergence on N × M∗ and yields the following: (i) For every z ∈ N the mapping μ → ϕi (z, μ) : M∗ → R is linear for every i ∈ I, hence the mapping μ → ϕ(z, μ) : M∗ → R is linear as well, hence may be interpreted as an element of M∗∗ , the second dual of M. (ii) More precisely, for a ﬁxed element z ∈ N the set T A (z) = L(z) | L ∈ T (A) is a relatively bounded subset of M. Indeed, by the above T (A) is bounded below and boundedabove relative to the operator L0 = T (f0 ) ∈ H(N , M). This implies that T A (z) is bounded below and bounded above relative to the element L0 (z) ∈ M. To demonstrate this, let w ∈ W. There is Z ∈ Z such that z ∈ Z, and according to I.4.24(iv) there are λ, ρ ≥ 0 such that 0 ≤ L + λV(Z,w) and L ≤ ρL0 + λV(Z,w) for all L ∈ L. As z ∈ Z, this implies in particular that 0 ≤ L(z) + w and L(z) ≤ ρL0 (z) + w for all L ∈ T (A), hence our claim. Because the mapping μ → ϕ(z, μ) : M∗ → R is the pointwise limit of the mappings μ → ϕi (z, μ) : M∗ → R, according to I.7.3, the former may therefore be identiﬁed with an element of M∗∗ sr , the (relative) strong second dual of M. (iii) For every μ ∈ M∗ , the mapping z → ϕ(z, μ) : N → R is clearly linear as it is the pointwise limit of linear mappings. Thus we realize that for every E ∈ R and a ∈ P the value θE (a) of the representing measure θ is indeed a linear operator from N into M∗∗ sr . (iv) Now let us suppose in addition that all the operators in H(N , M) are bounded on Z, that is for every L ∈ H(N , M) and Z ∈ Z the set L(Z) is bounded in M. Let Y be the family of all neighborhoods w◦ ⊂ M∗ for ∗ w ∈ W, and let us endow the cone M∗∗ sr of linear functionals on M with ∗∗ the neighborhood system W generated by this family. In this way, (M, W) ∗∗ is a subcone of (M∗∗ sr , W ). We shall proceed to establish that, under these circumstances, the operator θE (a) ∈ L(N , M∗∗ sr ) is also bounded on Z. In order to demonstrate this claim, let Z ∈ Z and w◦ ∈ Y. There are λ, ρ ≥ 0 such that 0 ≤ L + λV(Z,w) and L ≤ ρL0 + λV(Z,w) for all L ∈ T (A), and there is σ ≥ 0 such that L0 (z) ≤ σw for all z ∈ Z. This implies in particular that 0 ≤ L(z) + λw and L(z) ≤ L0 (z) + λw ≤ (λ + σ)w holds for all z ∈ Z and all L ∈ T (A). Because the operator θE (a) is the pointwise limit (as functions on N × M∗ ) of operators in T (A), the same relations hold true for the operator θE (a) ∈ L(N , M∗∗ sr ), that is our claim. $ W) $ be the standard lattice completion of (M, W), that is a (v) Let (M, cone of R-valued functions as elaborated in I.5.57. For a ﬁxed element z ∈ N $ that maps an let us denote by T z the operator from FV (X, PV ) into M element f ∈ FV (X, PV ) to the function

334

III Measures on Locally Compact Spaces

μ → T f (z, μ) : M∗ → R. $ (Recall that T (f ) is an element This function is indeed an element of M. of H(N , M), that is an R-valued function on N × M∗ .) The operator $ is clearly linear and continuous with respect to the T z : FV (X, PV ) → M $ respectively. $ for FV (X, P ) and M, given neighborhood systems V and W V Indeed, let Z ∈ Z such that z ∈ Z and w ∈ W. By the continuity of the operator , M) T : FV (X, PV ) → H(N there is v ∈ V such that T (f ) ≤ T (g) + V(Z,w) whenever f ≤ g + v for f, g ∈ FV (X, PV ). This implies in particular that T f (z, μ) ≤ T g (z, μ) + 1, that is

T z f (μ) ≤ T g (μ) + 1

for all μ ∈ w◦ . Likewise, for a ﬁxed element z ∈ N and the representing $ measure θz in the following measure θ we may deﬁne an L(PV , M)-valued z the operator from PV into way: For every ﬁxed E ∈ R let us denote by θE $ M that maps an element a ∈ PV to the function μ → θE a (z, μ) : M∗ → R. As before, it is easy to verify that this operator is linear and continuous with respect to the given neighborhood systems of PV and M. Thus the set function z $ : R → L(PV , M) E → θE $ is an L(PV , M)-valued measure on R in the sense of Section II.3. Countable additivity and boundedness follow immediately from the corresponding , M) -valued measure θ. properties of the L PV , H(N It can now be shown using Theorem II.5.35 that the measure θz represents the operator T z in the sense of Theorem 5.1. We omit the details. (vi) If in addition to the assumptions of (iv), for every z ∈ N the re$ to the subcone P, that is striction of the operator T z : FV (X, PV ) → M z the operator T : FV (X, P) → M is compact (or weakly compact), then a similar argument as in Corollary 6.5 shows that the values θE (a) of the representing measure θ, for E ∈ R and a ∈ P, are indeed linear operators in L(N , M) that are bounded on Z. Moreover, for every E ∈ R the set z | G ∈ R, G ⊂ E} of linear operators from P into M is seen to be {θG relatively compact in L(P, M) if endowed with the symmetric strong (or with the symmetric weak) operator topology.

6. Special Cases and Applications

335

(vii) If the values θE (a) of the representing measure θ are linear operators in L(N , M) that are bounded on Z (see (vi) or the reﬂexive case that M∗∗ s = M), and if all elements of P are bounded, then our version of Pettis’ theorem, that is Theorem II.3.11 applies: For every ﬁxed element z ∈ N the L(P, M)-valued measure θz is countably additive with respect to the strong operator topology of L(P, M) in this case. We shall formulate the special case that the cones P, N and M are indeed locally convex vector spaces over K = R or K = C as another corollary of Theorem 5.1. We shall assume that all the operators involved are linear over K in this case and say that a linear operator L : N → M is bounded if it maps bounded subsets of N into bounded subsets of M. Note that this notion of boundedness does not guarantee that the operator L is continuous. We shall use the family of all bounded subsets of N for Z. This family Z generates the uniform operator topology see I.7.2(i) on HK (N , M), the space of all bounded K-linear operators from N into M. Theorems 3.3 and 3.1 yield that a continuous linear op , M) can be extended into a continuous linear erator T : FV (X, P) → H(N , M) in this case. A similar argument as operator T : FV (X, PV ) → H(N in 6.9 demonstrates that the values θE (a) for E ∈ R and a ∈ P of the representing measure theta are indeed bounded K-linear operators from N into M∗∗ s in this case. Corollary 6.27. Let (P, V), (N , U) and (M, W) be locally convex topological vector spaces over K = R or K = C. Let HK (N , M) be the space of all bounded K-linear operators from N into M, endowed with the uniform operator topology. Let X be a locally compact Hausdorﬀ space and let V be a basis for an r-lower continuous inductive limit topology on F(X, P). Then every continuous K-linear operator T : FV (X, P) → HK (N , M) can be represented as an integral on X. More precisely: There exists a unique bounded measure θ on the weak σ-ring R of all relatively compact Borel subsets of X whose values θE for E ∈ R are continuous K-linear operators from P into HK (N , M∗∗ s ), the space of all bounded K-linear operators from N into the strong second dual M∗∗ s of M, with the following properties: For every z ∈ N the L(P, M)-valued measure θz is countably additive and quasi regular with respect to the weak* operator topology of LK (P, M∗∗ s ). θ is continuous relative to V, all functions in FV (X, P) are strongly integrable with respect to θ, and ' f dθ = T (f ) for all f ∈ FV (X, P). X

If for every z ∈ N the operator T z is compact (or weakly compact), then the values of θ are indeed continuous linear operators from P into HK (N , M), and in this case for every z ∈ N the measure θz is countably additive with respect to the strong operator topology of LK (P, M). Moreover, for every z | G ∈ R, G ⊂ E} of linear operators from P into E ∈ R the set {θG

336

III Measures on Locally Compact Spaces

M is relatively compact in L(P, M) endowed with the strong (or the weak) operator topology. The following special case of Corollary 6.27 is of particular interest: The case that (P, V) is a topological algebra and that N = M. Suppose that (P, V) is a topological algebra over K and that N = M, that is a locally convex topological vector space over K and endowed with the neighborhood system W. The vector space HK (N ) = LK (N ) of all bounded Klinear operators on N then forms a topological (non-commutative) algebra, where the canonical multiplication is the composition of the concerned oplinear operator that erators. Let T : FV (X, P) → HK (N) be a continuous is multiplicative on E(X, P). Its LK P, HK (N ) -valued representation measure θ then satisﬁes condition (A) from 6.12. In order to understand this condition, let us investigate how the operator multiplication extends to the K (N ) of HK (N ) in the sense of 6.12. First standard lattice completion H let us recall that an operator L ∈ HK (N ) is represented as an element of K (N ), that is a real-valued function on N × N ∗ as (z, μ) → e μ(L(z)) . H Now let L, M ∈ HK (N ). (z, μ) ∈ N × N ∗ we have L ◦ M (z, μ) = e μ L(M (z)) = L M (z), μ . % ∈ LK (N , N ∗∗ ). Its adjoint operator L % ∗ maps N ∗∗∗ into N ∗ . For Now let L s s % % ∗ μ (z) = μ L(z) every μ ∈ Ns∗∗∗ we have L for all z ∈ N . Similarly, the second adjoint L∗∗ maps N ∗∗ into Ns∗∗∗∗ . This yields for the representation K (N ) % as an element of H of L % μ) = e L % ∗ μ (z) L(z, for all (z, μ) ∈ N × N ∗ . If both L, M ∈ LK (N , Ns∗∗ ), then the composition operator L∗∗ ◦ M is in LK (N , Ns∗∗∗∗ ) and can be represented as an element K (N ) as of the standard lattice completion H ∗∗ L ◦ M (z, μ) = e μ (L∗∗ ◦ M )(z) . for (z, μ) ∈ N × N ∗ . The latter operation is well deﬁned, since N ∗ ⊂ Ns∗∗∗ % M &∈ and the elements of Ns∗∗∗∗ are linear functionals on Ns∗∗∗ . Now let L, ∗∗ LK (N , Ns ) ⊂ HK (N ) and let (Li )i∈I and (Mj )j∈J be nets in HK (N ) such % and limj∈J Mj = M & with respect to order convergence that limi∈I Li = L in HK (N ), that is pointwise convergence on N × N ∗ . Then for all (z, μ) ∈ N × N ∗ this implies that % Mj (z), μ lim Li Mj (z), μ = L i∈I

6. Special Cases and Applications

337

for every j ∈ J , and % Mj (z ), μ = lim e L % ∗ (μ) = M % ∗ (μ) % ∗ μ Mj (z ) = lim Mj z, L & z, L lim L j∈J j∈J j∈J &)(z ) = (L % ∗∗ ◦ M &)(z, μ). &(z ) = e μ (L % ∗∗ ◦ M % ∗ μ (M = e L

Thus

% ∗∗ ◦ M &)(z, μ). lim lim Li ◦ Mj (z, μ) = (L

j∈J i∈I

Similarly, one veriﬁes % ∗∗ ◦ M &)(z, μ). lim lim Li ◦ Mj (z, μ) = (L i∈I j∈J

K (N ) is therefore The extension of the multiplication to elements L, M ∈ H given by L • M = L∗∗ ◦ M and Condition (A) from 6.12 for the representing measure θ then reads as follows: (A’) θE (ab) = θE (a)∗∗ ◦ θE (b) and θE (a)∗∗ ◦ θG (b) = 0 for all a, b ∈ P and disjoint sets E, G ∈ R. The case that P = K and that N = M. This case is of particular interest as it will lead to the Spectral representation theorem. If P = K, then linear operators from P into HK (N ) are indeed elements L of HK (N ) acting as in addition that the representation α → αL : K → HK (N ). We shall assume measure θ is indeed LK K, HK (N ) -, that is HK (N )-valued. According to Corollary 6.27, this circumstance is guaranteed if the operator T is weakly compact or if the locally convex vector space N is reﬂexive. Condition (A) then is further simpliﬁed and reads: 2 = θE and θE ◦ θG = 0 for disjoint sets E, G ∈ R. (A”) θE

We shall formulate this as an additional corollary: Corollary 6.28. Let (N , U) be a locally convex topological vector space over K = R or K = C and let HK (N ) be the space of all bounded K-linear operators on N , endowed with the uniform operator topology. Let X be a locally compact Hausdorﬀ space and let V be a basis for an r-lower continuous inductive limit topology on F(X, P). Let T : FV (X, K) → HK (N ) be a continuous multiplicative K-linear operator, and suppose that either T is weakly compact or that N is reﬂexive. Then there exists a unique bounded HK (N )-valued measure θ on the weak σ-ring R of all relatively compact Borel subsets with the following properties: For every z ∈ N the N -valued measure θz is countably additive and quasi regular with respect to the given topology of N . We have

338

III Measures on Locally Compact Spaces 2 θE = θE

and

θE ◦ θG = 0

for disjoint sets E, G ∈ R. The measure θ is continuous relative to V, all functions in FV (X, P) are strongly integrable with respect to θ, and ' f dθ = T (f ) for all f ∈ FV (X, P). X

If for every z ∈ N the operator T z is compact (or weakly compact), then z for every E ∈ R the subset {θG | G ∈ R, G ⊂ E} of N relatively compact (or relatively weakly compact). This corollary leads directly to the spectral representation theorem, our ﬁnal application. 6.29 The Spectral Theorem. We choose a complex Hilbert space H, $, % for N in Corollary 6.28. Let L be a normal continuous linear operator on H, that is L ◦ L∗ = L∗ ◦ L, and let X ⊂ C be its compact spectrum. For the neighborhood system V we choose the collection of singleton sets v, each containing a constant function x → α for some α > 0. Thus FV (X, P) = C(X, C), endowed with the topology of uniform convergence. The closed subalgebra Λ of HK (H) = LK (H) generated by the identity operator I and the two elements L and L∗ is a commutative B ∗ -algebra with involution (see IX.3.1 in [56]), and its spectrum coincides with the spectrum of L (see Corollaries IX.3.10 and IX.3.11 in [56]). According to the Gelfand- Naˇımark theorem see Theorem IX.3.7 in [56] or Theorem III.16.1 in [137] then Λ is isometrically *-isomorphic to C(X, C), that is to say there is a continuous multiplicative linear operator T : C(X, C) → LK (H) that maps the constant function x → 1 into I and the identity function x → x into the operator L. Furthermore, T (f ∗ ) = T (f )∗ holds for all f ∈ C(X, C). According to Corollary 6.27, T can be represented by an LK (H)-valued measure θ with the stated properties. In addition, we have Condition (A*) from 6.12, ∗ , this that is αθE = (αθE )∗ for all E ∈ R and α ∈ K. As (αθE )∗ = αθE ∗ yields θE = θE . Summarizing, Conditions (A) and (A*) demonstrate that the operators θE are indeed projections and that θE and θG are orthogonal whenever E, G ∈ R are disjoint sets. This is of course the classical Spectral representation theorem for normal operators on a Hilbert space see Theo rem X.2.1 in [56] or Theorem II.44.1 in [82] .

7. Notes and Remarks The topology τ of a locally convex topological vector space N is called the inductive limit of the topologies τi of a directed (with respect to inclusion) family {Ni }i∈I of subspaces of N , if N = i∈I Ni , and if τ is the ﬁnest locally convex topology on N whose trace topology on each of the subspaces

7. Notes and Remarks

339

Ni is coarser than τi (see II.6 in [185]). The inductive limit is called strict, if these trace topologies coincide with the τi . Now let N be the space of all continuous real-valued functions with compact support on a locally compact Hausdorﬀ space X, and for any compact subset E of X let NE be the subspace consisting of all functions in N whose support is contained in E. If each subspace NE is endowed with the supremum norm for the functions on E, then the corresponding inductive limit topology on N can be described by the following system of 0-neighborhoods: Let v be a convex subset of nonnegative lower semicontinuous R-valued functions on X such that for every compact subset E of X there is ε > 0 and s ∈ v such that ε χE ≤ s, where χE denotes the characteristic function of the set E. The corresponding 0 -neighborhood vN in N then is given by vN = f ∈ N |f | ≤ s for some s ∈ v . It is now straightforward to verify that the family of all these sets vN establishes a 0-neighborhood system for the inductive limit topology of N . This is of course our approach to inductive limit topologies on cones of functions from Section 2.1. The neighborhood system V of P = R consists of all positive constants in this case, and the system V of all convex sets v of V -valued functions from above deﬁnes an inductive limit topology on N in the sense of 2.1. The functions in N of course vanish at inﬁnity. The concept of weighted spaces of continuous real-valued functions on a locally compact set is due to Nachbin [136] and Prolla [155]. In brief, it works as follows: A family W of non-negative real-valued upper semicontinuous functions on a locally compact Hausdorﬀ space X is called a family of weights if for all w1 , w2 ∈ W there are w3 ∈ W and ρ > 0 such that w1 ≤ ρ w3

and

w2 ≤ ρ w3 .

The associated the subspace of C(X) CW (X) = {f ∈ C(X) | wf vanishes at inﬁnity for all w ∈ W}, together with the locally convex topology generated by the seminorms pw (f ) = sup |wf (x)| x ∈ X for w ∈ W and f ∈ CW , is called a weighted space of functions. The neighborhood system V of P = R consists of all positive constants in this case. The V-valued functions sw (x) = 1/w(x), for w ∈ W, are lower semicontinuous, hence bounded below by a positive constant on every relatively compact subset of X. Thus, as explained in Example 2.11(e), the inductive limit neighborhoods vw = {sw }, for all w ∈ W, model this situation in the settings of Section 2.1 and lead to a representation of a weighted space of functions as a cone of real-valued functions endowed with a lower semicontinuous inductive limit topology.

340

III Measures on Locally Compact Spaces

In this way, the concept of inductive limit topologies from Section 2.1 combines the corresponding classical notion with the notion of weighted spaces of functions. The classical Riesz representation theorem [164] is of course the prototype of integral representation theorems for linear operators on function spaces. It states that every positive linear functional on the space N of all continuous real-valued functions with compact support on a locally compact Hausdorﬀ space X can be represented by the integral with respect to some regular positive Borel measure on X (Theorem 13.23 in [178]). Every such positive positive linear functional is of course continuous with respect to the inductive limit topology from above, and an immediate generalization of this theorem states that every real-valued linear functional on N , which is continuous with respect to the inductive limit topology, can be represented by a regular Borel measure on X. There are a number of generalizations of this result, most of them are concerned with operators on spaces of continuous real-valued functions and representations by vector-valued measures. One of the betterknown versions is the representation theorem by Bartle-Dunford-Schwartz ([7], Theorem iV.5 in [43]). It states that a weakly compact continuous linear operator from a space of continuous functions on a compact space (endowed with the supremum norm) into some Banach space L can be represented by the integral with respect to a regular L-valued Borel measure (see [26]). In case that the operator T is not weakly compact, the representing measure has to be allowed to take values in the second dual of L. These results are recovered as special cases of Theorem 5.1 in Section 6 (see Corollary 6.5) of this chapter. For further studies, it might be interesting and possibly rewarding, to investigate potential adaptations of Choquet’s theorem to cone- or vectorvalued functions. In this case, one looks for an integral representation of a linear operator which is deﬁned only on a given subspace of continuous functions. There are usually many measures doing this, but some can be singled out for having a particularly small and identiﬁable support in X. Choquet’s theorem (see [4] or [148]) states that, given a subspace L of the space N of continuous real-valued functions on a compact Hausdorﬀ space X and a continuous linear functional μ : L → (R, there is a real-valued regular Borel measure θ on X such that μ(f ) = f dθ holds for all f ∈ L, and such that θ is supported (in a delicate way) by some subset of X, called the Choquet boundary of L. Though classical by now, this is still considered to be a deep result, and the arguments involved are quite subtle, in particular if the compact space X carries a non-metric topology. The order structure of the involved function and measure spaces is used extensively, since the sought after representing measures are maximal in some sense. This book uses order structures as its main means to treat integration theory and might therefore lead to a line of approach for the generalization of Choquet’s theorem to the vector- or cone-valued case.

List of Symbols

Standard Symbols N = {1, 2, 3, . . .} Z = {. . . − 1, 0, 1, 2, . . .} R R = R ∪ {+∞} R+ = {α ∈ R | α ≥ 0} C Γ = {γ ∈ C | |γ| = 1} K B B∗

The natural numbers The integer numbers The real numbers The extended real numbers The non-negative extended real numbers The complex numbers The unit circle of C Stands for either R or C Unit ball of a normed space Dual unit ball of a normed space

Special Symbols P, Q, N , M V, W, U (P, V), (Q, W), . . . V = V ∪ {0, ∞} F(X, P) FV (X, P), V b

V) (P, (PV , V) σ(P, P ∗ ) O(P)

Cones, I.1 Abstract neighborhood systems for cones, I.1 Locally convex cones, I.1 augmented neighborhood system, I.1.4, II.2.2 Cone of all P-valued functions on X, I.1.4 Locally convex cone of P-valued functions on X, I.1.4 Standard completion of a locally convex cone (P, V), I.5.57 Standard full extension of a quasi-full locally convex cone (P, V), I.6.2 Weak topology on P, I.4.6 Order topology on P, I.5.43

341

342

List of Symbols

o(P, P ∗ ) L(N , M) L(P, Q) V(Z,W) H(N , M), V ∗∗ ∗∗ , Psl , P ∗∗ Pw∗∗ , Ps∗∗ , Psr

Weak order topology on P, I.5.49 Cone of linear operators form N to M, I.7 Cone of continuous linear operators form P to Q, II.3 Neighborhood system for L(N , M), I.7 Locally convex cone of linear operators from N to M, I.7 Second duals of a locally convex cone, I.7.3

Integral-Related Special Symbols R AR χE SR (X, P) ER (X, P) FR (X, P) |θ|(E, v) ( f dθ F ( F

f dθ

F(F,θ) (X, P) F(F,Θ) (X, P) F(|F |,Θ) (X, P) (X, P), V(F, Θ) F(F,Θ) Rs θn , F, f var (θa , X) Z(A, E), I(A, E)

Weak σ-ring of subsets, II.1.1 σ-algebra of measurable subsets, II.1.1 Characteristic function of a subset E ⊂ X, II.1.1 Cone of all P-valued step functions supported by R, II.1.1 Subcone generated by all P-valued elementary functions, II.6.16 A cone of measurable functions, II.2.3 The modulus of the measure θ, II.3.2 Integral of a function f ∈ FR (X, P) over a set F ∈ AR , II.4.9 Integral of a function f ∈ F(X, P) over a set F ∈ AR , II.4.13 Functions in F(X, P) that are integrable over F ∈ AR , II.4.13 Functions integrable with respect to a family of measures, II.5.3 Functions integrable with respect to a family of measures, II.5.3 Locally convex cone of integrable functions, II.5.5 Residual component of a function, II.5.16 Variation of a real-valued measure, II.5.34 II.6.8

Order Relations ≤ v

Standard order relation (reﬂexive, transitive and compatible with algebraic operations) (global) weak preorder, I.3 local weak preorder (referring to a neighborhood v ∈ V), I.3

List of Symbols

343

v ≤ a.e.F

(global) preorder, I.8 local preorder (referring to a neighborhood v ∈ V), I.8 Almost everywhere order relation for functions, II.4.11 Order relation for measures, II.5.11 Order relation for measures, II.5.17

≤P ≺

F F

Operations on Elements B(a), (a)B, B s (a) Bv (a), (a)Bv , Bvs (a) O(a) O(a b)

Boundedness components of a, I.4.9 Local boundedness components of a (referring to a neighborhood v ∈ V ), I.4 Zero component of a, I.5.8 Zero component of a relative to b, I.5.16

Operations on Sets ↓A = {x ∈ E | x ≤ a for some

a ∈ A}

↑A = {x ∈ E | x ≥ a for some

a ∈ A}

conv(A) Ex(A) A, A◦ , ∂A = A \ A A(l) A(u) v(A),

(A)v

decreasing hull of a set A ⊂ P, I.1.4 increasing hull of a set A ⊂ P,

Convex hull of a set A ⊂ P, I.5.7 Set of extreme points of a convex set A ⊂ P, I.5.33 topological closure, interior and boundary of a set A Closure of A with respect to the lower relative topology, I.4.24 Closure of A with respect to the upper relative topology, I.4.24 upper and lower relative neighborhoods of a set A ⊂ P, I.4.28

Convergence limi∈I ai , limi∈I ai , fn f, fn → f v F v F

f, fn a.e.F f fn −→ a.e.F

fn f, v F

Order convergence for a net (ai )i∈I , I.5.18

Upper, lower and symmetric pointwise convergence for functions on a set F, II.1.7 fn a.e.F f, Upper, lower and symmetric almost everywhere pointwise convergence for functions on a set F, II.5.22

344

θn θ, θn −→ θ

List of Symbols

θn θ,

Upper, lower and symmetric convergence of sequences of measures, II.5.13

Symbols Related to Continuous Cone-valued Functions supp(f ) = {x ∈ X | f (x) = 0}

The support of a function, III.1

supp*(f ) = {x ∈ X | f (x) = 0} The core support of a function, III.1 Cone of r-continuous P-valued functions, III.1 C r (X, P) Elementary function, III.1.10 ϕ⊗ a K Family of all compact subsets of X, III.2 Family of all both open and compact subsets of K0 X, III.2 K(X) Continuous positive real-valued functions with support in K, III.2 Continuous positive real-valued functions with K0 (X) core support in K0 , III.2 Cones generated by elementary functions, III.2.3 E(X, P), E0 (X, P) FV (X, P), FV0 (X, P) Closures of E(X, P) and E0 (X, P), III.2.4 r r (X, P), CV (X, P) The r-continuous functions in FV (X, P), and CV 0 FV0 (X, P), III.2.5 E ≺ ϕ, ϕ≺E for E ∈ R and ϕ ∈ K(X), III.4.5

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Index

abstract neighborhood system, 10 algebra homomorphism, 220, 285, 318 of operators, 237 topological, 220, 285, 318 almost everywhere, 149 convergence of sequences of functions, 182 with respect to a family of measures, 159 Baire, 246 balanced, 11 Bartle, 141, 247, 340 basis, 11 Bessel, 238 Bochner, 218, 231, 246 Borel, 193, 246 bounded above, 11 above relative to, 32 below, 10 below relative to, 32 relative to, 36 sup-bounded below, 84 v-bounded (above), 26 v-bounded above relative to, 40 bounded above subset of functions, 190 boundedness at inﬁnity, 264 boundedness component local, 40 local lower, 40 local symmetric, 40 local upper, 40 lower, 32 symmetric, 36 uniform, 42 upper, 32

Bourbaki, 1 B ∗ -algebra, 324 cancellation law, 10 Carath´eodory, 1, 246 Cauchy net, 90 Choquet, 6, 340 boundary, 340 theorem, 340 Clarkson, 246 compact linear operator, 307 component, 41 quasi-, 41 residual, 176 zero, 62 zero relative to, 66 cone, 10 dual, 18 full locally convex, 10 locally convex, 11 locally convex complete lattice, 59 locally convex lattice, 57 locally convex semilattice, 325 locally convex -semilattice, 58 locally convex ∨-semilattice, 57 locally convex -semilattice, 59 locally convex ∧-semilattice, 57 ordered, 10 quasi-full locally convex, 107 second dual, 114 semilattice, 224, 286 connected, 39 locally, 41 pathwise, 40 convergence compact, 265 of sequences in F(X, P), 182 353

354 convergence (contd.) of sequences in F(X, P) almost everywhere, 182 of sequences of measures, 174 order, 69 pointwise, 265 uniform, 265 weak order, 93 convergence theorems, 183, 186, 189, 204 convex quasiuniform structure, 13 Daniell, 1 Day, 117 decreasing, 11 decreasing hull, 14 Dedekind, 96 Diestel, 1, 246 Dieudonn´e, 135 dual cone, 18 Dunford, 1, 120, 141, 218, 246, 340 embedding, 20 equicontinuous, 17 extended real number system R, 14 Extension Theorem, 21 extreme points, 106 family of weights, 266, 339 Fatou’s lemma, 182 Fuchssteiner, 116 function aﬃne, 22 bounded below sequence, 129 concave, 22 convex, 21 elementary, 191, 256 measurable, 120 r-lower, r-upper, r-continuous, 250 relatively continuous (r-continuous), 250 relativley bounded above subset, 190 supported by a set, 119 vanishing at inﬁnity, 258 with compact support, 257 functional continuous linear, 18 extended superlinear, 21 linear, 18 sublinear, 21 supporting the separation property, 80 Gelfand, 246 Gelfand-Naˇımark theorem, 324, 338 Gierz, 116 Gowurin, 246 Graves, 1

Index Hahn, 1, 194, 246 Hahn-Banach, 21, 96 Halmos, 1, 246 Harnak, 246 Hilbert, 237 space, 237, 338 Hofmann, 116 increasing, 11 inductive limit topology, 257 integrable Bochner, 218, 231 Dunford$Schwartz, 218 Dunford&Schwartz, 231 extended, 245 strongly, 178 subcone-based, 161 with respect to a family of measures, 159 integral for a function f ∈ FR (X, P), 147 for a step function, 143 over a set E ∈ R, 150 over a set F ∈ AR, 150 integral representation, 286 isomorphism, 21 Jordan, 246 Kantoroviˇc, 96 Keimel, 2, 116 Korovkin, 2, 116 lattice homomorphism, 78, 224, 325 Lawson, 116 Lebesgue, 1, 246 locally convex topological vector space, 16, 216, 251, 314 ordered, 14, 267 Lusky, 116 measurability scalar, 125 weak, 125 measure algebra-valued, 242 bounded semivariation, 134 compact, 214 composition with a linear operator, 137 continuous relative to a system of neighborhoods, 282 equibounded family, 159 extension, 136 family of, 159 ﬁnite p-semivariation, 135

Index inner regular, 274 lattice-valued, 242 L(P, Q)-valued, 131 modulus, 132 multiples, 171 operator-valued, 131 order relation, 171 outer regular, 274 p-domination, 135 quasi regular, 275 R-bounded, 134 restriction to a subset, 173 semivariation, 132 spectral, 237 strong additivity, 138 strongly additive, 194 sums, 171 total variation, 135, 137 uniformly strongly additive family, 194 weakly compact, 141, 214 weakly sequentially compact set, 197 Meyer-Nieberg, 117 Mislove, 116 Morse, 246 Nachbin, 3, 5, 116, 249, 339 neighborhood abstract, 10 augmented system, 127 connected, 41 inductive limit, 128, 144 inﬁnity, 127 lower, 10 lower relative, 26 modular, 16 R-compatible inductive limit, 128 symmetric, 10 symmetric relative, 26 upper, 10 upper relative, 26 net almost order convergent, 88 bounded below, 69 inferior limit, 69 superior limit, 69 Nikod´ ym, 6, 194 operator compact, 307 continuous linear, 17 equicontinuous family, 17 linear, 17 monotone, 17 order continuous linear, 77

355 weakly compact, 307 operator algebra, 237 order convex, 11 Peano, 246 Pettis, 138, 194, 246 theorem, 141 polar, 18 preorder, 116 local, 117 weak, 23 weak global, 23 weak local, 25 Prolla, 6, 135, 249, 339 Radon, 1, 246 Range Theorem, 23 Representation Theorem, 286 residual component, 176 Riemann, 246 Riesz, 6, 96 representation theorem, 287, 340 Rosenthal, 1, 246 Saks, 1, 194, 246 Sandwich Theorem, 21 Sch¨ afer, 117 Schwartz, 1, 120, 141, 218, 340 Scott, 116 second dual, 114 lower strong, 115 relative strong, 115 strong, 115 weak, 115 semilattice homomorphism, 77, 286 semivariation, 246 Separation Theorem, 22 sequence cone, 17, 331 series convergent, 74 of subsets, 75 set of measure zero, 149 σ-algebra, 120 σ-ring, 120 weak, 120 spectral measure, 237 Spectral representation theorem, 338 spectrum, 338 standard full extension, 109 standard lattice completion, 98 simpliﬁed, 104

356 step function integral for, 143 P-valued, 120 standard representation, 120 strongly additive measure, 194 subcone locally convex complete lattice, 84 locally convex lattice, 84 subset balanced, 11 bounded, 43 bounded above, 42 bounded above relative to, 43 bounded below, 42 closed convex, 44 decreasing, 11 increasing, 11 measurable, 120 of measure zero, 149 order convex, 11 relatively bounded, 43 relatively bounded above, 43 solid, 56 sup-bounded below, 84 tightly covered by bounded elements, 80 topological algebra, 220 topological vector lattice, 56, 328 topology inductive limit, 257 lower, 10 lower relative, 28 lower relative v-topology, 31 modular symmetric, 15

Index order, 91 quasiuniform, 10 r-(lower,upper) continuous inductive limit, 257 strong operator, 113 symmetric, 10 symmetric relative, 28 symmetric relative v-topology, 31 uniform operator, 113 upper, 10 upper relative, 28 upper relative v-topology, 31 weak, 31 weak operator, 113 weak order, 93 weak* operator, 113 total variation, 210, 246 Uhl, 1, 246 uniformly strongly additive family of measures, 194 Vitali, 194 weak compactness relative, 141 weakly compact linear operator, 307 weakly sequentially compact set of measures, 197 weighted space of functions, 339 weights, 266 zero component, 62 zero component relative to, 66

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