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.
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Morillon, M. Helly Spaces and Radon Measures on Complete Lines. Order 29, 419–441 (2012). https://doi.org/10.1007/s11083-011-9212-6
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DOI: https://doi.org/10.1007/s11083-011-9212-6