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In stochastic convex programming numerous examples are to be found where the cost functional to be minimized is of the form of a mean value functional Ef(x)= f (x, ) dP () where ×  n and  is an uncertain quantity-element of a probability space. The problem of minimizing Ef is a deterministic problem related to the stochastic convex program. To be able to apply the methods of convex optimization and the theorems of convex analysis, it is important to know the properties of Ef, both topological and algebraic. The aim of this paper is to determine the main properties and characteristics of the mean value functional Ef resulting from these corresponding to the functions f (., ). By the conjugacy operation, the mean value functional is closely related to the continuous infimal convolution of which we shall also give some properties. Finally the different results obtained are applied to stochastic optimization problems.

About properties of the mean value functional and of the continuous infimal convolution in stochastic convex analysis