and level sets of N-dimensional Gaussian fields possessing "smooth" sample paths

(i.e., almost surely (a.s.) continuous, continuously differentiable, etc.). When some of these smoothness conditions are relaxed it is clear by analogy with the well-researched one-dimensional case that the topological properties of these sets that we have been studying are no longer appropriate concepts, as the sample paths become wildly erratic. However, in these situations it becomes interesting to somehow measure the "size" of the sample paths and level sets. The appropriate concept in this regard is that of Hausdorff dimension. A set E c RN is said to have Hausdorff dimension a if