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In measure theory, a familiar representation theorem due to F. Riesz identifies the dual space Lp(X,L,λ)* with Lq(X,L,λ), where 1/p+1/q=1, as long as 1 ≤ p<∞. However, L∞(X,L,λ)* cannot be similarly described, and is instead represented as a class of finitely additive measures.

This book provides a reasonably elementary account of the representation theory of L∞(X,L,λ)*, examining pathologies and paradoxes, and uncovering some surprising consequences. For instance, a necessary and sufficient condition for a bounded sequence in L∞(X,L,λ) to be weakly convergent, applicable in the one-point compactification of X, is given.

With a clear summary of prerequisites, and illustrated by examples including L∞(Rn) and the sequence space l∞, this book makes possibly unfamiliar material, some of which may be new, accessible to students and researchers in the mathematical sciences.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
This chapter explains a strategy for characterising weakly convergent sequences in \(L_\infty \) from what is essentially a measure-theoretic viewpoint. The underlying setting is the Banach space \(L_\infty \) of (equivalence classes of) functions which on an arbitrary set are measurable with respect to a sigma-algebra, and essentially bounded with respect to a complete, sigma-finite measure. The discussion emphasises the differences between finitely additive and countably additive measures on sigma-algebras and engages with various issues that arise naturally as the theory is developed.
John Toland

Chapter 2. Notation and Preliminaries

Abstract
This chapter is a brief survey of background material and terminology collected from various sources and organised in a consistent notation that will be used in the chapters that follow. Although most of the material is entirely standard and does not need to be digested until necessary, some specialised items are explained in detail for future reference.
John Toland

Chapter 3. and Its Dual

Abstract
This chapter is a statement of the Yosida–Hewitt representation of the dual of \(L_\infty \) as a space of finitely additive measures and some immediate consequences which reflect the delicacy of the theorem and the differences between countably additive and finitely additive measures.
John Toland

Chapter 4. Finitely Additive Measures

Abstract
This chapter introduces basic notation and definitions, for example, of partial ordering, lattice operations, absolute continuity and singularity, for finitely additive measures. The important notion of pure finite additivity of measures follows, and it is shown that every finitely additive measure is uniquely the sum of a countably additive and a purely finitely additive measure. This sets the scene for the material that follows.
John Toland

Chapter 5. : 0–1 Finitely Additive Measures

Abstract
This chapter is devoted to the set \(\mathfrak G\) of finitely additive measures \(\omega \) which take only the values 0 or 1 and explains the sense in which every essentially bounded function is constant \(\omega \)-almost everywhere. The existence of elements of \(\mathfrak G\) with prescribed properties is established using Zorn’s lemma and a relation between elements of \(\mathfrak G\) and maximal filters. This observation and its consequences dominate subsequent developments.
John Toland

Chapter 6. Integration and Finitely Additive Measures

Abstract
This chapter deals with the integration of essentially bounded measurable functions with respect to finitely additive measures like those that featured in the Yosida–Hewitt representation theorem in Chap. 3. The chapter includes a study of integration of an essentially bounded function u with respect to elements of \(\mathfrak G\), which leads to the multivalued essential range of u, and ends with an example of finitely additive measures that are not countably additive, but which coincide with Lebesgue measure when integrating continuous functions.
John Toland

Chapter 7. Topology on

Abstract
This chapter introduces a compact Hausdorff topology \(\tau \) on \(\mathfrak G\) and from theory already developed derives the existence of an isometric isomorphism between the Banach algebra \(\text {L}_{\infty }\) and the space of real-valued continuous functions C(\(\mathfrak G\), \(\tau \)). This leads to, among other things, a duality between weak convergence in \(\text {L}_{\infty }\) and weak convergence in C(\(\mathfrak G\), \(\tau \)).
John Toland

Chapter 8. Weak Convergence in

Abstract
This chapter opens by observing the relation between Dirac measures acting on C(\(\mathfrak G\),\(\tau \)) and elements of \(\mathfrak G\) acting on \(L_\infty \). This leads to a pointwise characterisation of weakly convergent sequences in \(L_\infty \) that settles some quite subtle questions about the weak convergence of specific sequences that are pointwise convergent.
John Toland

Chapter 9. When X is a Topological Space

Abstract
This chapter deals with refinement of the theory to Borel measure spaces when the underlying space X is locally compact and Hausdorff. Prototypical examples are open sets in Euclidian space with Lebesgue measure and the natural numbers with the discrete topology and counting measure. The key observation now is that \(\mathfrak G\) is the disjoint union of compacts sets \(\mathfrak G\)(x), where x is an element of the one-point compactification of X and elements of \(\mathfrak G\)(x) are zero outside every open neighbourhood of x. This localisation result leads to a characterization of weakly convergent sequences in terms of the pointwise behaviour of related sequences of functions in neighbourhoods of x. Here, “pointwise” has its usual measure-theoretic meaning, whereas “localisation” refers to the behaviour of Borel measures and functions restricted to open neighbourhoods of points in a topological space. Similarly, the essential range is localised to reflect the fine structure of u at points x which is related to the isometric isomorphism in Chap. 7.
John Toland

Chapter 10. Reconciling Representations

Abstract
This chapter deals with the fact that in a general setting \(\text {L}_{\infty }\) and C(\(\mathfrak G\), \(\tau \)) are isometrically isomorphic even though the dual of \(\text {L}_{\infty }\) is represented by finitely additive measures, while the dual of C(\(\mathfrak G\), \(\tau \)) is represented by countably additive measures. Then, in the special case when X is a locally compact Hausdorff topological space, bounded linear functionals which on \(\text {L}_{\infty }\) are represented by finitely additive measures are represented by countably additive measures when restricted to continuous functions. The relation between these two measures is explored in detail.
John Toland

Backmatter

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