The Dual of L∞(X,L,λ), Finitely Additive Measures and Weak Convergence

A Primer

  • John Toland

Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Table of contents

  1. Front Matter
    Pages i-x
  2. John Toland
    Pages 1-5
  3. John Toland
    Pages 7-26
  4. John Toland
    Pages 27-29
  5. John Toland
    Pages 31-39
  6. John Toland
    Pages 57-66
  7. John Toland
    Pages 87-93
  8. Back Matter
    Pages 95-99

About this book


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.


Riesz Representation Finitely additive measures Weak convergence Yosida-Hewitt Essential range Extreme points

Authors and affiliations

  • John Toland
    • 1
  1. 1.Department of Mathematical SciencesUniversity of BathBathUK

Bibliographic information

  • DOI
  • Copyright Information The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-030-34731-4
  • Online ISBN 978-3-030-34732-1
  • Series Print ISSN 2191-8198
  • Series Online ISSN 2191-8201
  • Buy this book on publisher's site