Curvature Measures and Random Sets,I

The paper deals with signed curvature measures as introduced by Federer for sets with positive reach. An integral representation and a local Steiner formula for these measures are given. The main result is the additive extension of the curvature measures to locally finite unions of compatible sets with positive reach. Within this comprehensive class of subsets of R^d a generalized Steiner polynomial (local version) and section theorems (principal kinematic formula, Crofton formula) for the curvature measures are derived.