Measures and fibers

We study measures on compact spaces by analyzing the properties of fibers of continuous mappings into 2ω2ω. We show that if a compact zerodimensional space K   carries a measure of uncountable Maharam type, then such a mapping has a non-scattered fiber and, if we assume additionally a weak version of Martin's Axiom, such a mapping has a fiber carrying a measure of uncountable Maharam type. Also, we prove that every compact zerodimensional space which supports a strictly positive measure and which can be mapped into 2ω2ω by a finite-to-one function is separable.