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Radon measures on Banach spaces with their weak topologies. (English) Zbl 0848.46028

The authors present some improvements to results on the existence or non-existence of Borel measures that are not Radon measures on Banach spaces equipped with their weak topologies. Especially, they give some general criteria on a compact Hausdorff space \(K\) that ensure that \((C (K), \text{weak})\) admits a Borel measure that is not a Radon measure.
The main theorem concerning this is as follows: Let \(K\) be a compact Hausdorff space with a non-empty family \({\mathcal D}\) of non-empty proper clopen subsets with the two following properties.
(a) The union of any increasing sequence of members of \({\mathcal D}\) is properly contained in a member of \({\mathcal D}\).
(b) If \(S_1, S_2, \dots\), and \(T_1, T_2, \dots\) are two increasing sequences of clopen sets, all contained in a fixed set of \({\mathcal D}\), with \(S_n \cap T_n= \emptyset\), for \(n\geq 1\), then there are disjoint clopen sets \(S_0\) and \(T_0\) with \(S_n \subset S_0\) and \(T_n \subset T_0\), for \(n\geq 1\).
Then \((C (K), \text{weak})\) admits a Borel measure that is not a Radon measure.
In consequence, this theorem also yields some known results such as Talagrand’s theorem and a theorem due to de Maria and Rodriguez-Salinas.

MSC:

46G12 Measures and integration on abstract linear spaces
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
28C99 Set functions and measures on spaces with additional structure
46B99 Normed linear spaces and Banach spaces; Banach lattices
28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
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