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

Research output: Chapter in Book/Report/Conference proceedingChapter

## Abstract

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.

Original language English The Dual of L∞(X,L,λ), Finitely Additive Measures and Weak Convergence. Cham, Switzerland Springer Science and Business Media B.V. 77-86 10 9783030347321 9783030347314 https://doi.org/10.1007/978-3-030-34732-1_9 E-pub ahead of print - 3 Jan 2020

### Publication series

Name SpringerBriefs in Mathematics 2191-8198 2191-8201

## ASJC Scopus subject areas

• Mathematics(all)

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