Mean Euler characteristic of stationary random closed sets

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