We examine the concept of essential intersection of a random set in the framework of robust optimization programs and ergodic theory. Using a recent extension of Birkhoff's Ergodic Theorem developed by the present authors, it is shown that essential intersection can be represented as the countable intersection of random sets involving an asymptotically mean stationary transformation. This is applied to the approximation of a robust optimization program by a sequence of simpler programs with only a finite number of constraints. We also discuss some formulations of robust optimization programs that have appeared in the literature and we make them more precise, especially from the probabilistic point of view. We show that the essential intersection appears naturally in the correct formulation.