Stochastic linear optimization under partial uncertainty and incomplete information using the notion of probability multimeasure

We consider a scalar stochastic linear optimization problem subject to linear constraints. We introduce the notion of deterministic equivalent formulation when the underlying probability space is equipped with a probability multimeasure. The initial problem is then transformed into a set-valued optimization problem with linear constraints. We also provide a method for estimating the expected value with respect to a probability multimeasure and prove extensions of the classical strong law of large numbers, the Glivenko-Cantelli theorem, and the central limit theorem to this setting. The notion of sampling with respect to a probability multimeasure and the definition of cumulative distribution multifunction are also discussed. Finally, we show some properties of the deterministic equivalent problem.