Nonseparable Radon measures and small compact spaces
Volume 153 / 1997
Fundamenta Mathematicae 153 (1997), 25-40
DOI: 10.4064/fm-153-1-25-40
Abstract
We investigate the problem if every compact space carrying a Radon measure of Maharam type \kappa can be continuously mapped onto the Tikhonov cube [0, 1]^\kappa (\kappa being an uncountable cardinal). We show that for \kappa ≥ cf(\kappa) ≥ \kappa this holds if and only if \kappa is a precaliber of measure algebras. Assuming that there is a family of ω_1 null sets in 2^{ω1} such that every perfect set meets one of them, we construct a compact space showing that the answer to the above problem is "no" for \kappa = ω. We also give alternative proofs of two related results due to Kunen and van Mill [18].