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

Let $X(\cdot )$ be a random field ${\mathbb{R}}^D\to {\mathbb{R}}^d$, $D\ge d$. We first studied the level set $X^{-1}(u)$, $u\in {\mathbb{R}}^d$. In particular we gave a weak condition for this level set to be rectifiable. Then, we established a Kac-Rice formula to compute the $D-d$ Hausdorff measure. Our results extend known results, particularly in the non-Gaussian case where we obtained a very general result. We conclude with several extensions and examples of application, including functions of Gaussian random field, zeroes of the likelihood, gravitational microlensing, shot-noise.