The Dual of L∞(X,L,λ), Finitely Additive Measures and Weak Convergence
Springer International Publishing (Verlag)
978-3-030-34731-4 (ISBN)
In measure theory, a familiar representation theorem due to F. Riesz identifies the dual space Lp(X,L, )* with Lq(X,L, ), where 1/p+1/q=1, as long as 1 p< . However, L (X,L, )* 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, )*, examining pathologies and paradoxes, and uncovering some surprising consequences. For instance, a necessary and sufficient condition for a bounded sequence in L (X,L, ) 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.
John Toland FRS is a mathematical analyst who worked in nonlinear partial differential equations and served as Director of the Isaac Newton Institute for Mathematical Sciences in Cambridge (2011-2016). He was awarded the London Mathematical Society Berwick Prize (2000) and the Royal Society Sylvester Medal (2012).
1 Introduction.- 2 Notation and Preliminaries.- 3 L and its Dual.- 4 Finitely Additive Measures.- 5 G: 0-1 Finitely Additive Measures.- 6 Integration and Finitely Additive Measures.- 7 Topology on G.- 8 Weak Convergence in L (X,L, ).- 9 L * when X is a Topological Space.- 10 Reconciling Representations.- References.- Index.
| Erscheinungsdatum | 17.02.2020 |
|---|---|
| Reihe/Serie | SpringerBriefs in Mathematics |
| Zusatzinfo | X, 99 p. 1 illus. |
| Verlagsort | Cham |
| Sprache | englisch |
| Maße | 155 x 235 mm |
| Gewicht | 185 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| Mathematik / Informatik ► Mathematik ► Angewandte Mathematik | |
| Schlagworte | Essential range • Extreme points • Finitely additive measures • Riesz Representation • weak convergence • Yosida-Hewitt |
| ISBN-10 | 3-030-34731-1 / 3030347311 |
| ISBN-13 | 978-3-030-34731-4 / 9783030347314 |
| Zustand | Neuware |
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