On a general Kac-Rice formula for the measure of a level set.

Let X be a random field from RD to Rd, where D≥d. We study the level set X−1(u), where u∈Rd. Specifically, we provide a weak condition for this level set to be rectifiable. Next, we establish the Kac-Rice formula to compute the (D−d)-dimensional Hausdorff measure. Our results extend previous work, particularly in the non-Gaussian case where we obtain a very general result. We conclude with several extensions and examples of applications, including functions of Gaussian random fields, zeros of the likelihood functions, gravitational microlensing, and shot-noise.