  Fractal percolation exhibits a dramatic topological phase transition,
changing abruptly from a dust-like set to a system spanning cluster. The
transition points are unknown and difficult to estimate. In many classical
percolation models the percolation thresholds have been approximated well using
additive geometric functionals, known as intrinsic volumes. Motivated by the
question whether a similar approach is possible for fractal models, we
introduce corresponding geometric functionals for the fractal percolation
process $F$. They arise as limits of expected functionals of finite
approximations of $F$. We establish the existence of these limit functionals
and obtain explicit formulas for them as well as for their finite
approximations.
