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On compact spaces carrying random [Radon] measures of large Maharam type. (English) Zbl 1078.28009
Let \(K\) be a compact space. Denote by \(C(K)\) the Banach space of all continuous functions on \(K\) and by \(P(K)\) the space of all probability Radon measures on \(K\). We endow \(P(K)\) with weak\({}^*\) topology inherited from \(C(K)^*\). Given an infinite cardinal number \(\kappa\) and a compact set \(K\), consider the following four statements: (i) there is a continuous surjection from \(K\) onto \([0,1]^\kappa\); (ii) \(l^1(\kappa)\) can be isomorphically embedded into \(C(K)\); (iii) \(K\) carries a homogeneous Radon measure of type \(\kappa\); and (iv) there is a continuous surjection from \(P(K)\) onto \([0,1]^\kappa\). An almost complete picture of their mutual relationships has been already drawn by efforts of several mathematicians including the author. The purpose of the paper is to present a survey of this subject and give new proofs or improvements of some of them. The new proofs are achieved by taking an approach based on a measure theoretic lemma due to Fremlin and Haydon, and improvements are made by using the notion of an \(m\)-precaliber of measure algebras introduced by Haydon.

28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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