  Hadwiger integrals employ the intrinsic volumes as measures for integration
of real-valued functions. We provide a formula for the expected values of
Hadwiger integrals of Gaussian-related random fields. The expected Hadwiger
integrals of random fields are both theoretically interesting and potentially
useful in applications such as sensor networks, image processing, and cell
dynamics. Furthermore, combining the expected integrals with a functional
version of Hadwiger's theorem, we obtain expected values of more general
valuations on Gaussian-related random fields.

Probability