Helly spaces and Radon measures on complete lines
The author works in \textbf{ZF} (without \textbf{AC} (= axiom of choice)). Apart from proving some results in \textbf{ZF}, a few results are noted to be not provable in \textbf{ZF}. A line is a linearly ordered set (with order topology). \(I = [0, 1]\), \(\mathbb R\) = real numbers. For an ordinal \(\alpha\), \(L_\alpha = \{0, 1\}^\alpha\) with lexicographic order. A Loeb-function on a space \(X\) is a choice function on the collection of nonempty closed sets of \(X\). A compact \(X\) with Loeb-function on it is Loeb-compact. The results of the first part of the paper concern lines. For lines \(X\) and \(Z\), the generalised Helly space is the (closed) subspace \(H(X, Z)\) of the product space \(ZX\) consisting of isotonic mappings from \(X\) to \(Z\). \(H(X, L_\omega)\) is proved to be Loeb-compact. It turns out that the Helly space \(H(X, I)\) is Loeb-compact (in \textbf{ZF}). (It easily follows, when using a weaker form of \textbf{AC}). Using the result (proved here) that, for an ordinal \(\alpha\) and a line \(X\) satisfying a certain condition, which \(\mathbb R\) is shown to satisfy when \(\alpha = \omega\), if \(H(X, L_{\alpha+1})\) is compact (Loeb-compact), then \(\{0, 1\}^X\) is compact (Loeb-compact), compactness of \(H(\mathbb R, L_{\omega+1})\) is shown to be not provable in \textbf{ZF}. Some of the results of the second part are proved for a compact Hausdorff space. They apply to a line which is complete (as a poset), as it holds (in \(\mathbf{ZF}\)) that a line is compact (Loeb-compact) iff it is complete. For a compact Hausdorff space \(X\) satisfying a certain condition, the `dual ball' of the Banach space \(C(X)\) over \(\mathbb R\) is compact (Loeb-compact) in the weak\({^\ast}\) topology. It is shown that if \(X\) equals the one-point compactification of a discrete space whose underlying set is equipotent with the power set of \(\mathbb R\), then there exists a model of \textbf{ZF} where the compactness of the dual ball of \(C(X)\) is not provable. (Though the result holds for every \(X\) using a weaker form of \textbf{AC}.) It is proved that for every complete line \(X\), the normed space \(C(X)\) satisfies the (effective) continuous Hahn-Banach property. An existing proof of the Stone-Weierstrass theorem for a compact Hausdorff space has been modified so as to work in \(\mathbf{ZF}\).
scientific article; zbMATH DE number 6102619
scientific article