  We study measures on compact spaces by analyzing the properties of fibers of
continuous mappings into 2^omega. 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 2^omega by a
finite-to-one function is separable.
