Nonseparable Radon measures and small compact spaces

We investigate the problem if every compact space K carrying a Radon measure of Maharam type κ can be continuously mapped onto the Tikhonov cube ${\displaystyle {[0,1]}^{\kappa }}$ (κ being an uncountable cardinal). We show that for κ ≥ cf(κ) ≥ κ this holds if and only if κ is a precaliber of measure algebras. Assuming that there is a family of ${\displaystyle {\omega }_1}$ null sets in ${\displaystyle 2^{\omega 1}}$ 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 κ = ω. We also give alternative proofs of two related results due to Kunen and van Mill [18].