The Dual of L∞(X,L,λ), finitely additive measures and weak convergence : a primer / John Toland

By:

Toland, John [VerfasserIn]

Contributor(s):

Springer Nature Switzerland AG [Verlag]

Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: SpringerBriefs in mathematics | Springer eBook CollectionPublisher: Cham : Springer Nature Switzerland, [2020]Publisher: Cham : Imprint: Springer, [2020]Copyright date: © 2020Description: 1 Online-Ressource (X, 99 p. 1 illus.)ISBN:

9783030347321

Subject(s):

Additional physical formats: 9783030347314 | 9783030347338 | Erscheint auch als: 9783030347314 Druck-Ausgabe | Erscheint auch als: 9783030347338 Druck-AusgabeDOI: DOI: 10.1007/978-3-030-34732-1Online resources:

Summary: 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.Summary: 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.PPN: PPN: 1689104430Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SMA | ZDB-2-SXMS

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