The purpose of this paper is to develop an integration theory for positive functions with respect to a probabilistic measure in the meaning of Serstnev, using submeasures with probabilistic structures and the topological ring associated to them. The probabilistic measures are introduced for modelling those situations in which we have only probabilistic information about the measure of a set [7]. The point of view adopted to define the integral and the set of integrable functions belongs to Sion, Bartle, Dunford and Schwartz. It is shown that the classical theory of integration with respect to a positive measure is naturally included in our general case.