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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 a result of Feng and Hu 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 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.

Exact dimensionality and projections of random self‐similar measures and sets