  We consider the action of Mandelbrot multiplicative cascades on probability
measures supported on a symbolic space. For general probability measures, we
obtain almost a sharp criterion of non-degeneracy of the limiting measure; it
relies on the lower and upper Hausdorff dimensions of the measure and the
entropy of the random weights. We also obtain sharp bounds for the lower
Hausdorff and upper packing dimensions of the limiting measure. When the
original measure is a Gibbs measure associated with a potential of certain
modulus of continuity (weaker than H\"older), all our results are sharp. This
improves results previously obtained by Kahane and Peyri\`ere, Ben Nasr, and
Fan. We exploit our results to derive dimension estimates and absolute
continuity for some random fractal measures.
