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Abstract

We investigate the problem if every compact space $K$ 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].