Finite Borel measures on spaces of cardinality less than 𝔠

Gardner, R. J.; Gruenhage, G.

doi:10.1090/S0002-9939-1981-0601743-5

Finite Borel measures on spaces of cardinality less than $\mathfrak {c}$
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by R. J. Gardner and G. Gruenhage PDF

Proc. Amer. Math. Soc. 81 (1981), 624-628 Request permission

Abstract:

Let $\kappa < c$ be uncountable. We prove, among other results, that every $\alpha$-realcompact space of cardinality $\kappa$ is Borel measure-compact if and only if there is a set of reals of cardinality $\kappa$ whose Lebesgue measure is not zero.

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