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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.

##### MSC:
 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.)
##### Keywords:
Radon measure; Maharam type; $$m$$-precaliber; measure algebra
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