The Dual of L∞(X,L,λ), Finitely Additive Measures and Weak Convergence : A Primer. 1st ed. 2020

種類:

電子ブック

責任表示:

by John Toland

出版情報:

Cham : Springer International Publishing : Imprint: Springer, 2020

著者名:

• Toland, John.
• SpringerLink (Online service)

シリーズ名:

SpringerBriefs in Mathematics ;

ISBN:

9783030347321 [303034732X]    

注記:

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,λ) -- 9 L∞* when X is a Topological Space -- 10 Reconciling Representations -- References -- Index.
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.

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