Helly Spaces and Radon Measures on Complete Lines

摘要

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(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.