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, Volume 29, Issue 3, pp 419–441 | Cite as

Helly Spaces and Radon Measures on Complete Lines

  • Marianne Morillon
Article

Abstract

We work in the set theory without the Axiom of Choice ZF. Given a linearly ordered set X, the (closed) subset H(X,[0,1]) of the product topological space [0,1] X consisting of the isotonic mappings u:X →[0,1] is (Loeb-)compact. The compactness of \(H(\mathbb R,L)\) where L is the lexicographic order [0,1] ×{0,1} is not provable (in ZF). Radon measures on a complete linearly ordered set X are studied: they are of Radon–Stieltjes type; moreover, the “dual ball” of the Banach space C(X) is (Loeb-)compact in the weak* topology, and the Banach space C(X) satisfies the (effective) continuous Hahn–Banach property.

Keywords

Axiom of Choice Product topology Compactness Helly space Stieltjes–Radon measures Hahn–Banach property 

Mathematics Subject Classifications (2010)

Primary 03E25; Secondary 06A05 06F30 54B10 

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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.ERMIT, Département de Mathématiques et InformatiqueUniversité de La Réunion, Parc Technologique UniversitaireSainte-ClotildeFrance

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