  We define a class of random measures, spatially independent martingales,
which we view as a natural generalisation of the canonical random discrete set,
and which includes as special cases many variants of fractal percolation and
Poissonian cut-outs. We pair the random measures with deterministic families of
parametrised measures $\{\eta_t\}_t$, and show that under some natural
checkable conditions, a.s. the total measure of the intersections is H\"older
continuous as a function of $t$. This continuity phenomenon turns out to
underpin a large amount of geometric information about these measures, allowing
us to unify and substantially generalize a large number of existing results on
the geometry of random Cantor sets and measures, as well as obtaining many new
ones. Among other things, for large classes of random fractals we establish (a)
very strong versions of the Marstrand-Mattila projection and slicing results,
as well as dimension conservation, (b) slicing results with respect to
algebraic curves and self-similar sets, (c) smoothness of convolutions of
measures, including self-convolutions, and nonempty interior for sumsets, (d)
rapid Fourier decay. Among other applications, we obtain an answer to a
question of I. {\L}aba in connection to the restriction problem for fractal
measures.
