Approximation of Random Functions by Random Polynomials in the Framework of Choquet's Theory of Integration

Given a submodular capacity space, we prove the uniform convergence in capacity and also the uniform convergence in the Choquet-mean of order $p\ge 1$ with a quantitative estimate, of the multivariate Bernstein polynomials associated to a random function. Applications to quantitative estimates concerning the uniform convergence in capacity in the univariate case are given.