TY - CHAP

T1 - $$L:\infty ^*$$ When X is a Topological Space

AU - Toland, John

N1 - Publisher Copyright:
© 2020, The Author(s), under exclusive license to Springer Nature Switzerland AG.
Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2020/1/3

Y1 - 2020/1/3

N2 - This chapter deals with refinement of the theory to Borel measure spaces when the underlying space X is locally compact and Hausdorff. Prototypical examples are open sets in Euclidian space with Lebesgue measure and the natural numbers with the discrete topology and counting measure. The key observation now is that is the disjoint union of compacts sets(x), where x is an element of the one-point compactification of X and elements of(x) are zero outside every open neighbourhood of x. This localisation result leads to a characterization of weakly convergent sequences in terms of the pointwise behaviour of related sequences of functions in neighbourhoods of x. Here, “pointwise” has its usual measure-theoretic meaning, whereas “localisation” refers to the behaviour of Borel measures and functions restricted to open neighbourhoods of points in a topological space. Similarly, the essential range is localised to reflect the fine structure of u at points x which is related to the isometric isomorphism in Chap. 7.

AB - This chapter deals with refinement of the theory to Borel measure spaces when the underlying space X is locally compact and Hausdorff. Prototypical examples are open sets in Euclidian space with Lebesgue measure and the natural numbers with the discrete topology and counting measure. The key observation now is that is the disjoint union of compacts sets(x), where x is an element of the one-point compactification of X and elements of(x) are zero outside every open neighbourhood of x. This localisation result leads to a characterization of weakly convergent sequences in terms of the pointwise behaviour of related sequences of functions in neighbourhoods of x. Here, “pointwise” has its usual measure-theoretic meaning, whereas “localisation” refers to the behaviour of Borel measures and functions restricted to open neighbourhoods of points in a topological space. Similarly, the essential range is localised to reflect the fine structure of u at points x which is related to the isometric isomorphism in Chap. 7.

UR - http://www.scopus.com/inward/record.url?scp=85101183315&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-34732-1_9

DO - 10.1007/978-3-030-34732-1_9

M3 - Chapter or section

AN - SCOPUS:85101183315

SN - 9783030347314

T3 - SpringerBriefs in Mathematics

SP - 77

EP - 86

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

PB - Springer Science and Business Media B.V.

CY - Cham, Switzerland

ER -