  The translative intersection formula of integral geometry yields an
expression for the mean Euler characteristic of a stationary random closed set
intersected with a fixed observation window. We formulate this result in the
setting of sets with positive reach and using flag measures which yield
curvature measures as marginals. As an application, we consider excursion sets
of stationary random fields with $C^{1,1}$ realizations, in particular,
stationary Gaussian fields, and obtain results which extend those known from
the literature.
