We investigate the problem if every compact space K carrying a Radon measure of Maharam type k can be continuously mapped onto the Tikhonov cube [0,1]K (k being an uncountable cardinal). We show that for k > cf(?;) > o>2 this holds if and only if re is a precaliber of measure algebras. Assuming that there is a family of ui null sets in 2LJ% 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 k = lo . We also give alternative proofs of two related results due to Kunen and van Mill [18].
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