ABSTRACT.   Probability function maximization and quantile function minimization problems are approximated starting from weak convergence of discrete measures with increasing dimension. It is assumed that solutions of both problems depend on a random parameter, i.e., solutions are sought as decision rules from the class of bounded measurable functions L  . Both problems are approximated by sequences of finite dimensional extremum problems with discrete measures and with increasing dimensions. Convergence conditions for optimal values and solutions of both problems are presented.