On the structure of stationary flat processes. III

Summary For arbitrary k and d with 1 k < d, sufficient conditions in terms of the second order moment measure are found for a stationary random measure in the space of k-flats in R^dto be a.s. invariant. Some of these conditions are further shown to be almost sharp, in the sense of being nearly fulfilled for a certain class of stationary random measures which fail to be invariant. The latter results are based on estimates of the distributions under the homogeneous probability measure of certain rotational invariants for pairs of linear subspaces.