  We study the existence of non-separable compact spaces that support a measure
and are small from the topological point of view. In particular, we show that
under Martin's axiom there is a non-separable compact space supporting a
measure which has countable $\pi$-character and which cannot be mapped
continuously onto $[0,1]^{\omega_1}$. On the other hand, we prove that in the
random model there is no non-separable compact space having countable
$\pi$-character and supporting a measure.
