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

A Primer

Paperback Engels 2020 9783030347314
Verwachte levertijd ongeveer 9 werkdagen

Samenvatting

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.

Specificaties

ISBN13:9783030347314
Taal:Engels
Bindwijze:paperback
Uitgever:Springer International Publishing

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Inhoudsopgave

1 Introduction.- 2 Notation and Preliminaries.- 3&nbsp;L<sub>∞</sub>&nbsp;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&nbsp;L<sub>∞</sub>(X,L,λ).- 9&nbsp;L<sub>∞</sub>* when X is a Topological Space.- 10 Reconciling Representations.- References.- Index.

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