  We obtain a Principal Kinematic Formula and a Crofton Formula for surface
area measures of convex bodies, both involving linear operators on the vector
space of signed measures on the unit sphere $S^{d-1}$. These formulas are
related to a localization of Hadwiger's Integral Geometric Theorem. The
operators, mentioned above, will be shown to be compositions of spherical
Fourier transforms originating in the work of Koldobsky. As an application of
our Crofton Formula, we will find an extension of Koldobsky's orthogonality
relation for such transforms from the case of even spherical functions to
centered functions.
