Oriented distance point of view on random sets

Motivated by free boundary problems under uncertainties, we consider the oriented distance function as a way to define the expectation for a random compact or open set. In order to provide a law of large numbers and a central limit theorem for this notion of expectation, we also address the question of the convergence of the level sets of fn to the level sets of f when (fn) is a sequence of functions uniformly converging to f. We provide error estimates in term of Hausdorff convergence. We illustrate our results on a free boundary problem.