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Paracompact spaces and Radon spaces. (English) Zbl 0946.46012

From the text: “Recall that a cardinal \(\alpha\) is said to be of measure zero if, for any set \(A\) of this cardinal, every finite measure on the subsets of \(A\) which vanishes on the singletons is zero. The density of a topological space \(E\) is the smallest cardinal of the dense subsets of \(E\).
Let \(E\) be a regular topological space and let \({\mathcal H}\) be a family of closed sets of \(E\). Then a finite Borel measure \(\mu\) on \(E\) is said to be a Radon measure of type \(({\mathcal H})\) if \(\mu\) is a \(\tau\)-additive measure and it is innerly \({\mathcal H}\)-regular [P. Jiménez Guerra and B. Rodriguez-Salinas, “Medidas de Radon de tipo \(({\mathcal H})\) en espacios topológicos arbitrarios”, Mem. Real. Acad. Cienc. Exact. Fis. Natur. Madrid (1979); B. Rodriguez-Salinas, Rev. Mat. Hisp.-Am. 33, 257-274 (1973; Zbl 0278.28005)]. In particular, here we consider the case in which \({\mathcal H}\) is the family \({\mathcal F}\) of all the closed sets of \(E\). A regular topological space \(E\) is said to be Radon space of type \(({\mathcal H})\) if every finite Borel measure on \(E\) is a Radon measure of type \(({\mathcal H})\) [B. Rodriguez-Salinas and P. Jiménez Guerra, Rev. Acad. Ci. Madrid 69, 761-774 (1975; Zbl 0339.28004)].
We prove that if \(E\) is a subset of a Banach space whose density is of measure zero and such that \((E,weak)\) is a paracompact space, then \((E,weak)\) is a Radon space of type \(({\mathcal F})\) under very general conditions”.

MSC:

46A50 Compactness in topological linear spaces; angelic spaces, etc.
46G12 Measures and integration on abstract linear spaces
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