  We study the geometric properties of random multiplicative cascade measures
defined on self-similar sets. We show that such measures and their projections
and sections are almost surely exact-dimensional, generalizing Feng and Hu's
result \cite{FeHu09} for self-similar measures. This, together with a compact
group extension argument, enables us to generalize Hochman and Shmerkin's
theorems on projections of deterministic self-similar measures \cite{HoSh12} to
these random measures without requiring any separation conditions on the
underlying sets. We give applications to self-similar sets and fractal
percolation, including new results on projections, $C^1$-images and distance
sets.
